research paper of gravity

Einstein's Theory of Gravitation

Our modern understanding of gravity comes from Albert Einstein’s theory of general relativity, which stands as one of the best-tested theories in science. General relativity predicted many phenomena years before they were observed, including black holes , gravitational waves , gravitational lensing , the expansion of the universe , and the different rates clocks run in a gravitational field. Today, researchers continue to test the theory’s predictions for a better understanding of how gravity works.

Center for Astrophysics | Harvard & Smithsonian astrophysicists research the predictions of general relativity in many ways:

Capturing the first image of a supermassive black hole using the Event Horizon Telescope (EHT). This image of the black hole at the center of the nearby galaxy M87 reveals how gravitation affects the matter in orbit and the light that material emits, providing a novel test of general relativity in a regime where gravity is very strong. CfA Plays Central Role In Capturing Landmark Black Hole Image

Using gravitational lensing to search for the earliest galaxies in the universe. While they’re too faint to be seen directly, closer-by galaxies and clusters sometimes magnify their light, allowing us to learn about the ancestors of the Milky Way and other modern galaxies. Discovering Distant Radio Galaxies via Gravitational Lensing

Reconstructing the location of most of the mass in the universe using gravitational lensing. Next-generation observatories like the Large Synoptic Survey Telescope (LSST) will provide a census of millions of galaxies from their gravitational distortions. Mapping Dark Matter

Performing follow-up observations of gravitational wave events, to confirm the nature of the source. Collisions between neutron stars produce a lot of light in the form of short duration gamma ray bursts in addition to gravitational waves. Astronomers observed such a collision in 2017 using Dark Energy Camera on the Blanco Telescope in Chile, providing complementary data to the observation from LIGO. Astronomers See Light Show Associated With Gravitational Waves

Studying gravitational wave sources that aren’t visible to LIGO, but will be to future gravitational observatories. Using visible light telescopes, astronomers have observed that white dwarf binaries are relatively common in the galaxy, and some of them are in sufficiently tight orbits to be emitters of gravitational waves. One pair in particular orbits every 12.75 minutes, which will make it the strongest source for the future Laser Interferometer Space Antenna (LISA). Space-Warping White Dwarfs Produce Gravitational Waves

Determining whether black holes are actually what GR predicts. While GR is very clear that black holes exist, alternative theories propose different objects that behave in different ways. The challenge is that black holes appear very small in our telescopes, so it’s hard to observe their behavior. However, researchers have ruled out a number of alternative explanations, based on many observations of black holes. Do Stars Fall Quietly into Black Holes, or Crash into Something Utterly Unknown?

Testing general relativity’s prediction about the shape of a black hole. The Event Horizon Telescope is designed to take a picture of the “shadow” of the Milky Way’s supermassive black hole, which is the dark region through which no light passes. The shape of this shadow is predicted by GR, so the EHT will provide the first precision measurement of a fundamental property of a black hole. Event Horizon Telescope Reveals Magnetic Fields at Milky Way's Central Black Hole

A Century of Relativity

Albert Einstein published his full theory of general relativity in 1915, followed by a flurry of research papers by Einstein and others exploring the predictions of the theory. In general relativity (GR), concentrations of mass and energy curve the structure of spacetime, affecting the motion of anything passing near — including light. The theory explained the anomalous orbit of Mercury, but the first major triumph came in 1919 when Arthur Eddington and his colleagues measured the influence of the Sun’s gravity on light from stars during a total solar eclipse.

Physicists made many exotic predictions using general relativity. The bending of light around the Sun is small, but researchers realized the effect would be much larger for galaxies, to the point where gravity would form images of more distant objects — the phenomenon now called gravitational lensing. GR also predicted the existence of black holes: objects with gravity so intense that nothing getting too close can escape again, not even light.

General relativity showed that gravitation has a speed, which is the same as the speed of light. Catastrophic events like collisions between black holes or neutron stars produce gravitational waves. Researchers finally detected these waves in 2015 using the Laser Interferometer Gravitational Observatory (LIGO), a sensitive laboratory that took decades to develop.

For many aspects of astronomy — the motion of planets around stars, the structure of galaxies, etc. — researchers don’t need to use general relativity. However, in places where gravity is strong, and to describe the structure of the universe itself, GR is necessary. For that reason, researchers continue to use GR and probe its limits.

Black holes are extremely common in the universe. Stellar-mass black holes, the remnants of massive stars that exploded, are sometimes the source of powerful X-ray emissions when they are in binary systems with stars. In addition, nearly every galaxy harbors a supermassive black hole at its center, some of which produce powerful jets of matter visible from across the universe. GR is essential to understanding how these objects become so bright, as well as studying how black holes form and grow. The Event Horizon Telescope (EHT) is a world-spanning array of observatories that captured the first image of a supermassive black hole, providing a new arena for testing GR’s predictions.

Gravitational waves are a new branch of astronomy, providing a complementary way to study astrophysical systems to the standard light-based observations. Researchers use GR to provide “templates” of many possible gravitational wave signals, which is how they identify the source and its properties. Gravitational wave astronomy combines with light-based astronomy to characterize some of the most extreme events in the cosmos: collisions of black holes and neutron stars .

Astronomers use gravitational lensing to locate some of the earliest galaxies in the universe, which are too faint to be seen without the magnification provided by gravity. In addition, the distortion created by lensing allows researchers to study dark matter , and map the structure of the universe on the largest scales.

Not long after Einstein published GR, researchers realized the theory predicts that the universe changes in time. Observations in the 1920s found that prediction was true: the universe is expanding, with galaxies moving away from each other. Using GR, cosmologists found the cosmos had a beginning, and was once hotter and denser than it is today. GR provides the mathematical framework for describing the structure and evolution of the universe from its beginnings 13.8 billion years ago, and into the future.

This artist’s illustration depicts two merging neutron stars and the gravitational waves they emit. As the LIGO and Virgo gravitational wave observatories have confirmed, collisions of black holes and neutron stars emit enough gravitational waves to be seen billions of light-years away.

• Why do we need an extremely large telescope like the Giant Magellan Telescope?
• What happens to space time when cosmic objects collide?
• Black Holes
• Gravitational Lensing
• Gravitational Waves
• Very Long Baseline Interferometry
• High Energy Astrophysics
• Optical and Infrared Astronomy
• Theoretical Astrophysics

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Quantum physics and Einstein’s theory of general relativity are the two solid pillars that underlie much of modern physics. Understanding how these two well-established theories are related remains a central open question in theoretical physics.  Over the last several decades, efforts in this direction have led to a broad range of new physical ideas and mathematical tools.  In recent years, string theory and quantum field theory have converged in the context of holography, which connects quantum gravity in certain space-times with corresponding (conformal) field theories on a lower-dimensional space-time. These developments and connections have deepened our understanding not only of quantum gravity, cosmology, and particle physics, but also of intermediate scale physics, such as condensed matter systems, the quark-gluon plasma, and disordered systems.  String theory has also led to new insights to problems in many areas of mathematics.

The interface of quantum physics and gravity is currently leading to exciting new areas of progress, and is expected to remain vibrant in the coming decade.  Researchers in the Center for Theoretical Physics (CTP) have been at the forefront of many of the developments in these directions.  CTP faculty members work on string theory foundations, the range of solutions of the theory, general relativity and quantum cosmology, problems relating quantum physics to black holes, and the application of holographic methods to strongly coupled field theories.  The group in the CTP has close connections to condensed matter physicists, astrophysicists, and mathematicians both at MIT and elsewhere.

In recent years a set of new developments has begun to draw unexpected connections between a number of problems relating aspects of gravity, black holes, quantum information, and condensed matter systems. It is becoming clear that quantum entanglement, quantum error correction, and computational complexity play a fundamental role in the emergence of spacetime geometry through holographic duality.  Moreover these tools have led to substantial progress on the famous black hole information problem, giving new avenues for searching for a resolution of the tension between the physics of black holes and quantum mechanics.  CTP faculty members Netta Engelhardt and Daniel Harlow have been at the vanguard of these developments, which also tie into the research activity of several other CTP faculty members, including Aram Harrow , whose primary research focus is on quantum information, and Hong Liu , whose research connects black holes and quantum many-body dynamics.

Holographic dualities give both a new perspective into quantum gravitational phenomena as encoded in quantum field theory, and a way to explore aspects of strongly coupled field theories using the gravitational dual. CTP faculty have played a pioneering role in several applications of holographic duality. Hong Liu and Krishna Rajagopal are at the forefront of efforts that use holography to find new insights into the physics of the quark-gluon plasma. Liu was among the first to point out possible connections between black hole physics and the strange metal phase of high temperature superconductors, and in recent years has been combining insights from effective field theories, holography, and condensed matter physics to address various issues concerning far-from-equilibrium systems including superfluid turbulence, entanglement growth, quantum chaos, thermalization, and a complete formulation of fluctuating hydrodynamics. Gravitational effective field theories play a key role in the interpretation of gravitational wave observations. Mikhail Ivanov works at the intersection of these fields with the aim of testing strong field gravity at a new precision frontier.

Even though we understand string theory better than we did in decades past, there is still no clear fundamental description of the theory that works in all situations, and the set of four-dimensional solutions, or string vacua, is still poorly understood.  The work of Washington Taylor and Barton Zwiebach combines physical understanding with modern mathematical methods to address these questions, and has led to new insights into how observed physics fits into the framework of string theory as well as the development of new mathematical results and ideas. Alan Guth ‘s foundational work on inflationary cosmology has led him to focus on basic questions about the physics of the multiverse that arises naturally in the context of the many string theory vacua, and which provides the only current natural explanation for the observed small but positive cosmological constant.

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Gravity: We might have been getting it wrong this whole time

by Motoko Kakubayashi, Kavli Institute for the Physics and Mathematics of the Universe

Symmetry has been one of the guiding principles in physicists' search for fundamental laws of nature. What does it mean that laws of nature have symmetry? It means that laws look the same before and after an operation, similar to a mirror reflection, the same but right is now left in the reflection.

Physicists have been looking for laws that explain both the microscopic world of elementary particles and the macroscopic world of the universe and the Big Bang at its beginning, expecting that such fundamental laws should have symmetry in all circumstances. However, last year, two physicists found a theoretical proof that, at the most fundamental level, nature does not respect symmetry.

How did they do it? Gravity and hologram

There are four fundamental forces in the physical world: electromagnetism, strong force, weak force, and gravity . Gravity is the only force still unexplainable at the quantum level . Its effects on big objects, such as planets or stars, are relatively easy to see, but things get complicated when one tries to understand gravity in the small world of elementary particles.

To try to understand gravity on the quantum level, Hirosi Ooguri, the director of the Kavli Institute for the Physics and Mathematics of the Universe in Tokyo, and Daniel Harlow, an assistant professor at the Massachusetts Institute of Technology, started with the holographic principle. This principle explains three-dimensional phenomena influenced by gravity on a two-dimensional flat space that is not influenced by gravity. This is not a real representation of our universe, but it is close enough to help researchers study its basic aspects.

The pair then showed how quantum error correcting codes, which explain how three-dimensional gravitational phenomena pop out from two dimensions, like holograms, are not compatible with any symmetry; meaning such symmetry cannot be possible in quantum gravity.

They published their conclusion in 2019, garnering high praise from journal editors and significant media attention. But how did such an idea come to be?

It started well over four years ago, when Ooguri came across a paper about holography and its relation to quantum error correcting codes by Harlow, who was then a post doc at Harvard University. Soon after, the two met at the Institute for Advanced Study in Princeton when Ooguri was there on sabbatical and Harlow came to give a seminar.

"I went to his seminar prepared with questions," Ooguri says. "We discussed a lot afterwards, and then we started thinking maybe this idea he had can be used to explain one of the fundamental properties of quantum gravity, about the lack of symmetry ."

New research collaborations and ideas are often born from such conversations, says Ooguri, who is also a professor at the California Institute of Technology in the U.S.. Ooguri travels at least once a fortnight to give lectures, attend conferences, workshops and other events. While some might wonder if all that travel detracts from concentrating on research, Ooguri believes quite the opposite.

"Scientific progress is serendipitous," he says. "It often happens in a way that you don't expect. That kind of development is still very hard to achieve by remote exchange.

"Yes, nowadays it's easier with e-mails and video conferences," he continues, "but when you write an e-mail you have to have something to write about. When someone is in the same building, I can walk across the hallway and ask silly questions."

These silly questions are key to progress in fundamental sciences. Unlike other fields, such as applied science where researchers work towards a specific goal, the first question or idea a theoretical physicist comes up with is usually not the right one, Ooguri says. But, through discussion, other researchers ask questions derived from their curiosity, taking the research in a new direction, landing on a very interesting question, which has an even more interesting answer.

Provided by Kavli Institute for the Physics and Mathematics of the Universe

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General Relativity and Quantum Cosmology

Title: a short review of loop quantum gravity.

Abstract: An outstanding open issue in our quest for physics beyond Einstein is the unification of general relativity (GR) and quantum physics. Loop quantum gravity (LQG) is a leading approach toward this goal. At its heart is the central lesson of GR: Gravity is a manifestation of spacetime geometry. Thus, the approach emphasizes the quantum nature of geometry and focuses on its implications in extreme regimes -- near the big bang and inside black holes -- where Einstein's smooth continuum breaks down. We present a brief overview of the main ideas underlying LQG and highlight a few recent advances. This report is addressed to non-experts.

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Stochastic Gravity: Theory and Applications

1 Department of Physics, University of Maryland, College Park, 20742-4111 Maryland USA

Enric Verdaguer

2 Departament de Fisica Fonamental and C.E.R. in Astrophysics, Particles and Cosmology Universitat de Barcelona, Av. Diagonal 647, 08028 Barcelona, Spain

Whereas semiclassical gravity is based on the semiclassical Einstein equation with sources given by the expectation value of the stress-energy tensor of quantum fields, stochastic semiclassical gravity is based on the Einstein-Langevin equation, which has in addition sources due to the noise kernel. The noise kernel is the vacuum expectation value of the (operatorvalued) stress-energy bi-tensor which describes the fluctuations of quantum matter fields in curved spacetimes. In the first part, we describe the fundamentals of this new theory via two approaches: the axiomatic and the functional. The axiomatic approach is useful to see the structure of the theory from the framework of semiclassical gravity, showing the link from the mean value of the stress-energy tensor to their correlation functions. The functional approach uses the Feynman-Vernon influence functional and the Schwinger-Keldysh closed-time-path effective action methods which are convenient for computations. It also brings out the open systems concepts and the statistical and stochastic contents of the theory such as dissipation, fluctuations, noise, and decoherence. We then focus on the properties of the stress-energy bi-tensor. We obtain a general expression for the noise kernel of a quantum field defined at two distinct points in an arbitrary curved spacetime as products of covariant derivatives of the quantum field’s Green function. In the second part, we describe three applications of stochastic gravity theory. First, we consider metric perturbations in a Minkowski spacetime. We offer an analytical solution of the Einstein-Langevin equation and compute the two-point correlation functions for the linearized Einstein tensor and for the metric perturbations. Second, we discuss structure formation from the stochastic gravity viewpoint, which can go beyond the standard treatment by incorporating the full quantum effect of the inflaton fluctuations. Third, we discuss the backreaction of Hawking radiation in the gravitational background of a quasi-static black hole (enclosed in a box). We derive a fluctuation-dissipation relation between the fluctuations in the radiation and the dissipative dynamics of metric fluctuations.

Stochastic semiclassical gravity is a theory developed in the 1990s using semiclassical gravity (quantum fields in classical spacetimes, solved self-consistently) as the starting point and aiming at a theory of quantum gravity as the goal. While semiclassical gravity is based on the semiclassical Einstein equation with the source given by the expectation value of the stress-energy tensor of quantum fields, stochastic gravity includes also its fluctuations in a new stochastic semiclassical or the Einstein-Langevin equation. If the centerpiece in semiclassical gravity theory is the vacuum expectation value of the stress-energy tensor of a quantum field, and the central issues being how well the vacuum is defined and how the divergences can be controlled by regularization and renormalization, the centerpiece in stochastic semiclassical gravity theory is the stress-energy bitensor and its expectation value known as the noise kernel. The mathematical properties of this quantity and its physical content in relation to the behavior of fluctuations of quantum fields in curved spacetimes are the central issues of this new theory. How they induce metric fluctuations and seed the structures of the universe, how they affect the black hole horizons and the backreaction of Hawking radiance in black hole dynamics, including implications on trans-Planckian physics, are new horizons to explore. On the theoretical issues, stochastic gravity is the necessary foundation to investigate the validity of semiclassical gravity and the viability of inflationary cosmology based on the appearance and sustenance of a vacuum energy-dominated phase. It is also a useful beachhead supported by well-established low energy (sub-Planckian) physics to explore the connection with high energy (Planckian) physics in the realm of quantum gravity.

In this review we summarize major work on and results of this theory since 1998. It is in the nature of a progress report rather than a review. In fact we will have room only to discuss a handful of topics of basic importance. A review of ideas leading to stochastic gravity and further developments originating from it can be found in [ 149 , 154 ]; a set of lectures which include a discussion of the issue of the validity of semiclassical gravity in [ 168 ]; a pedagogical introduction of stochastic gravity theory with a more complete treatment of backreaction problems in cosmology and black holes in [ 169 ]. A comprehensive formal description of the fundamentals is given in [ 207 , 208 ] while that of the noise kernel in arbitrary spacetimes in [ 208 , 244 , 245 ]. Here we will try to mention all related work so the reader can at least trace out the parallel and sequential developments. The references at the end of each topic below are representative work where one can seek out further treatments.

Stochastic gravity theory is built on three pillars: general relativity, quantum fields, and nonequilibrium statistical mechanics. The first two uphold semiclassical gravity, the last two span statistical field theory. Strictly speaking one can understand a great deal without appealing to statistical mechanics, and we will try to do so here. But concepts such as quantum open systems [ 71 , 200 , 291 ] and techniques such as the influence functional [ 89 , 88 ] (which is related to the closed-time-path effective action [ 257 , 11 , 184 , 66 , 272 , 41 , 70 , 76 , 181 , 39 , 182 , 236 ]) were a great help in our understanding of the physical meaning of issues involved toward the construction of this new theory, foremost because quantum fluctuations and correlation have become the focus. Quantum statistical field theory and the statistical mechanics of quantum field theory [ 40 , 42 , 44 , 46 ] also aided us in searching for the connection with quantum gravity through the retrieval of correlations and coherence. We show the scope of stochastic gravity as follows:

• From general relativity [ 215 , 285 ] to semiclassical gravity.
• i. Stress-energy tensor: Regularization and renormalization.
• ii. Self-consistent solution: Backreaction problems [ 203 , 115 , 158 , 159 , 124 , 3 , 4 ].
• iii. Effective action: Closed time path, initial value formulation [ 257 , 11 , 184 , 66 , 272 , 41 , 70 , 76 , 181 , 39 , 182 , 236 ].
• iv. Equation of motion: Real and causal.
• i. Open quantum systems [ 71 , 200 , 291 ].
• ii. Influence functional: Stochastic equations [ 89 ].
• iii. Noise and decoherence: Quantum to classical transition [ 303 , 304 , 305 , 306 , 180 , 33 , 279 , 307 , 109 , 114 , 221 , 222 , 223 , 224 , 225 , 226 , 105 , 125 , 83 , 120 , 122 , 30 , 239 , 278 , 170 , 171 , 172 , 121 , 81 , 82 , 185 , 186 , 187 , 173 ].
• Decoherence in quantum cosmology and emergence of classical spacetimes [ 188 , 119 , 228 , 150 , 36 , 37 , 160 ].
• Dissipation from particle creation [ 76 , 181 , 39 , 182 , 236 , 57 ]; backreaction as fluctuation-dissipation relation (FDR) [ 167 ].
• Noise from fluctuations of quantum fields [ 149 , 151 , 43 ].
• Einstein-Langevin equations [ 43 , 157 , 167 , 58 , 59 , 38 , 202 , 207 , 208 , 206 ].
• Metric fluctuations in Minkowski spacetime [ 209 ].
• Validity of semiclassical gravity [ 163 , 243 ].
• Viability of vacuum dominance and inflationary cosmology.
• Stress-energy bi-tensor and noise kernel: Regularization reassessed [ 244 , 245 ].
• Wave propagation in stochastic geometry [ 166 ].
• Black hole horizon fluctuations: Spontaneous/active versus induced/passive [ 94 , 294 , 267 , 268 , 14 , 15 , 211 , 232 , 245 ].
• Noise induced inflation [ 50 ].
• Structure formation [ 45 , 213 , 212 , 51 , 254 ]; trace anomaly-driven inflation [269, 280, 132].
• Black hole backreaction as FDR [ 60 , 258 , 259 , 217 , 164 , 54 , 55 , 264 ].
• Metric fluctuations and trans-Planckian problem [ 14 , 15 , 211 , 232 , 219 ].
• Spacetime fam [ 62 , 63 , 101 , 102 , 103 ].
• Universal ‘metric conductance’ fluctuations [ 261 ].
• General relativity as geometro-hydrodynamics [ 146 ].
• Semiclassical gravity as mesoscopic physics [ 153 ].
• i. Via correlation hierarchy of interacting quantum fields [ 154 , 42 , 46 , 155 ].
• ii. Possible relation to string theory and matrix theory.

We list only the latest work in the respective topics above describing ongoing research. The reader should consult the references therein for earlier work and the background material. We do not seek a complete coverage here, but will discuss only the selected topics in theory, issues, and applications. We use the (+, +, +) sign conventions of [ 215 , 285 ], and units in which c = ħ =1.

From Semiclassical to Stochastic Gravity

There are three main steps that lead to the recent development of stochastic gravity. The first step begins with quantum field theory in curved spacetime [ 75 , 25 , 100 , 286 , 113 ], which describes the behavior of quantum matter fields propagating in a specified (not dynamically determined by the quantum matter field as source) background gravitational field. In this theory the gravitational field is given by the classical spacetime metric determined from classical sources by the classical Einstein equations, and the quantum fields propagate as test fields in such a spacetime. An important process described by quantum field theory in curved spacetime is indeed particle creation from the vacuum, and effects of vacuum fluctuations and polarizations, in the early universe [ 234 , 260 , 300 , 301 , 147 , 21 , 22 , 23 , 75 , 96 , 65 ], and Hawking radiation in black holes [ 130 , 131 , 174 , 235 , 282 ].

The second step in the description of the interaction of gravity with quantum fields is backreaction, i.e., the effect of the quantum fields on the spacetime geometry. The source here is the expectation value of the stress-energy operator for the matter fields in some quantum state in the spacetime, a classical observable. However, since this object is quadratic in the field operators, which are only well defined as distributions on the spacetime, it involves ill defined quantities. It contains ultraviolet divergences, the removal of which requires a renormalization procedure [ 75 , 67 , 68 ]. The final expectation value of the stress-energy operator using a reasonable regularization technique is essentially unique, modulo some terms which depend on the spacetime curvature and which are independent of the quantum state. This uniqueness was proved by Wald [ 283 , 284 ] who investigated the criteria that a physically meaningful expectation value of the stress-energy tensor ought to satisfy.

The theory obtained from a self-consistent solution of the geometry of the spacetime and the quantum field is known as semiclassical gravity . Incorporating the backreaction of the quantum matter field on the spacetime is thus the central task in semiclassical gravity. One assumes a general class of spacetime where the quantum fields live in and act on, and seek a solution which satisfies simultaneously the Einstein equation for the spacetime and the field equations for the quantum fields. The Einstein equation which has the expectation value of the stress-energy operator of the quantum matter field as the source is known as the semiclassical Einstein equation . Semiclassical gravity was first investigated in cosmological backreaction problems [ 203 , 115 , 158 , 159 , 124 , 3 , 4 , 123 , 90 , 129 ]; an example is the damping of anisotropy in Bianchi universes by the backreaction of vacuum particle creation. Using the effect of quantum field processes such as particle creation to explain why the universe is so isotropic at the present was investigated in the context of chaotic cosmology [ 214 , 19 , 20 ] in the late 1970s prior to the inflationary cosmology proposal of the 1980s [ 117 , 2 , 197 , 198 ], which assumes the vacuum expectation value of an inflaton field as the source, another, perhaps more well-known, example of semiclassical gravity.

The importance of quantum fluctuations

For a free quantum field, semiclassical gravity is fairly well understood. The theory is in some sense unique, since the only reasonable c-number stress-energy tensor that one may construct [ 283 , 284 ] with the stress-energy operator is a renormalized expectation value. However, the scope and limitations of the theory are not so well understood. It is expected that the semiclassical theory would break down at the Planck scale. One can conceivably assume that it would also break down when the fluctuations of the stress-energy operator are large [ 92 , 194 ]. Calculations of the fluctuations of the energy density for Minkowski, Casimir and hot flat spaces as well as Einstein and de Sitter universes are available [ 194 , 242 , 163 , 243 , 244 , 245 , 241 , 208 , 209 , 251 , 254 , 227 , 69 ]. It is less clear, however, how to quantify what a large fluctuation is, and different criteria have been proposed [ 194 , 93 , 95 , 163 , 243 , 9 , 10 ]. The issue of the validity of the semiclassical gravity viewed in the light of quantum fluctuations is summarized in our Erice lectures [ 168 ]. One can see the essence of the problem by the following example inspired by Ford [ 92 ].

Let us assume a quantum state formed by an isolated system which consists of a superposition with equal amplitude of one configuration of mass M with the center of mass at X 1 , and another configuration of the same mass with the center of mass at X 2 . The semiclassical theory as described by the semiclassical Einstein equation predicts that the center of mass of the gravitational field of the system is centered at 1/2 ( X 1 + X 2 ). However, one would expect that if we send a succession of test particles to probe the gravitational field of the above system, half of the time they would react to a gravitational field of mass M centered at X 1 and half of the time to the field centered at X 2 . The two predictions are clearly different; note that the fluctuation in the position of the center of masses is of the order of ( X 1 − X 2 ) 2 . Although this example raises the issue of how to place the importance of fluctuations to the mean, a word of caution should be added to the effect that it should not be taken too literally. In fact, if the previous masses are macroscopic, the quantum system decoheres very quickly [ 306 , 307 ] and instead of being described by a pure quantum state it is described by a density matrix which diagonalizes in a certain pointer basis. For observables associated to such a pointer basis, the density matrix description is equivalent to that provided by a statistical ensemble. The results will differ, in any case, from the semiclassical prediction.

In other words, one would expect that a stochastic source that describes the quantum fluctuations should enter into the semiclassical equations. A significant step in this direction was made in [ 149 ], where it was proposed to view the backreaction problem in the framework of an open quantum system: the quantum fields seen as the “environment” and the gravitational field as the “system”. Following this proposal a systematic study of the connection between semiclassical gravity and open quantum systems resulted in the development of a new conceptual and technical framework where (semiclassical) Einstein-Langevin equations were derived [ 43 , 157 , 167 , 58 , 59 , 38 , 202 ]. The key technical factor to most of these results was the use of the influence functional method of Feynman and Vernon [ 89 ], when only the coarse-grained effect of the environment on the system is of interest. Note that the word semiclassical put in parentheses refers to the fact that the noise source in the Einstein-Langevin equation arises from the quantum field, while the background spacetime is classical; generally we will not carry this word since there is no confusion that the source which contributes to the stochastic features of this theory comes from quantum fields.

In the language of the consistent histories formulation of quantum mechanics [ 114 , 221 , 222 , 223 , 224 , 225 , 226 , 105 , 125 , 83 , 120 , 122 , 30 , 239 , 278 , 170 , 171 , 172 , 121 , 81 , 82 , 185 , 186 , 187 , 173 ] for the existence of a semiclassical regime for the dynamics of the system, one needs two requirements: The first is decoherence, which guarantees that probabilities can be consistently assigned to histories describing the evolution of the system, and the second is that these probabilities should peak near histories which correspond to solutions of classical equations of motion. The effect of the environment is crucial, on the one hand, to provide decoherence and, on the other hand, to produce both dissipation and noise to the system through backreaction, thus inducing a semiclassical stochastic dynamics on the system. As shown by different authors [ 106 , 303 , 304 , 305 , 306 , 180 , 33 , 279 , 307 , 109 ], indeed over a long history predating the current revival of decoherence, stochastic semiclassical equations are obtained in an open quantum system after a coarse graining of the environmental degrees of freedom and a further coarse graining in the system variables. It is expected but has not yet been shown that this mechanism could also work for decoherence and classicalization of the metric field. Thus far, the analogy could be made formally [ 206 ] or under certain assumptions, such as adopting the Born-Oppenheimer approximation in quantum cosmology [ 237 , 238 ].

An alternative axiomatic approach to the Einstein-Langevin equation without invoking the open system paradigm was later suggested, based on the formulation of a self-consistent dynamical equation for a perturbative extension of semiclassical gravity able to account for the lowest order stress-energy fluctuations of matter fields [ 207 ]. It was shown that the same equation could be derived, in this general case, from the influence functional of Feynman and Vernon [ 208 ]. The field equation is deduced via an effective action which is computed assuming that the gravitational field is a c-number. The important new element in the derivation of the Einstein-Langevin equation, and of the stochastic gravity theory, is the physical observable that measures the stress-energy fluctuations, namely, the expectation value of the symmetrized bi-tensor constructed with the stress-energy tensor operator: the noise kernel . It is interesting to note that the Einstein-Langevin equation can also be understood as a useful intermediary tool to compute symmetrized two-point correlations of the quantum metric perturbations on the semiclassical background, independent of a suitable classicalization mechanism [ 255 ].

The Einstein-Langevin Equation: Axiomatic Approach

In this section we introduce stochastic semiclassical gravity , or stochastic gravity for short, in an axiomatic way. It is introduced as an extension of semiclassical gravity motivated by the search of self-consistent equations which describe the backreaction of the quantum stress-energy fluctuations on the gravitational field [ 207 ].

Semiclassical gravity

Semiclassical gravity describes the interaction of a classical gravitational field with quantum matter fields. This theory can be formally derived as the leading 1/ N approximation of quantum gravity interacting with N independent and identical free quantum fields [ 142 , 143 , 128 , 277 ] which interact with gravity only. By keeping the value of NG finite, where G is Newton’s gravitational constant, one arrives at a theory in which formally the gravitational field can be treated as a c-number field (i.e. quantized at tree level) while matter fields are fully quantized. The semiclassical theory may be summarized as follows.

where ∇ a is the covariant derivative associated to the metric g ab , ξ is a coupling parameter between the field and the scalar curvature of the underlying spacetime R , and g =det g ab .

The field may be quantized in the manifold using the standard canonical quantization formalism [ 25 , 100 , 286 ]. The field operator in the Heisenberg representation φ ̂ is an operator-valued distribution solution of the Klein-Gordon equation, the field equation derived from Equation (1) ,

We may write the field operator as φ ̂ [ g ; x ) to indicate that it is a functional of the metric g ab and a function of the spacetime point x . This notation will be used also for other operators and tensors.

where □=∇ a ∇ a , and G ab is the Einstein tensor. With the notation T ab [ g , φ ] we explicitly indicate that the stress-energy tensor is a functional of the metric g ab and the field φ .

The next step is to define a stress-energy tensor operator T ̂ ab [ g ; x ). Naively one would replace the classical field φ [ g ; x ) in the above functional by the quantum operator φ ̂ [ g ; x ), but this procedure involves taking the product of two distributions at the same spacetime point. This is ill-defined and we need a regularization procedure. There are several regularization methods which one may use; one is the point-splitting or point-separation regularization method [ 67 , 68 ], in which one introduces a point y in a neighborhood of the point x and then uses as the regulator the vector tangent at the point x of the geodesic joining x and y ; this method is discussed for instance in [ 243 , 244 , 245 ] and in Section 5 . Another well known method is dimensional regularization in which one works in arbitrary n dimensions, where n is not necessarily an integer, and then uses as the regulator the parameter ∊ = n −4; this method is implicitly used in this section. The regularized stress-energy operator using the Weyl ordering prescription, i.e. symmetrical ordering, can be written as

Note that if dimensional regularization is used, the field operator φ ̂ [ g ; x ) propagates in an n -dimensional spacetime. Once the regularization prescription has been introduced, a regularized and renormalized stress-energy operator T ̂ ab R [ g ; x ) may be defined as

which differs from the regularized T ̂ ab [ g ; x ) by the identity operator times some tensor counterterms F ab C [ g ; x ), which depend on the regulator and are local functionals of the metric (see [ 208 ] for details). The field states can be chosen in such a way that for any pair of physically acceptable states (i.e., Hadamard states in the sense of [ 286 ]), | ψ 〉 and | ϕ 〉, the matrix element 〈 ψ | T ab R | ϕ 〉, defined as the limit when the regulator takes the physical value is finite and satisfies Wald’s axioms [ 100 , 283 ]. These counterterms can be extracted from the singular part of a Schwinger-DeWitt series [ 100 , 67 , 68 , 31 ]. The choice of these counterterms is not unique, but this ambiguity can be absorbed into the renormalized coupling constants which appear in the equations of motion for the gravitational field.

The semiclassical Einstein equation for the metric g ab can then be written as

where C abcd is the Weyl tensor. The tensors A ab and B ab come from the functional derivatives with respect to the metric of the terms quadratic in the curvature in Equation (8) ; they are explicitly given by

where R abcd and R ab are the Riemann and Ricci tensors, respectively. These two tensors are, like the Einstein and metric tensors, symmetric and divergenceless: ∇ a A ab =0=∇ a B ab .

For a free quantum field this theory is robust in the sense that it is self-consistent and fairly well understood. As long as the gravitational field is assumed to be described by a classical metric, the above semiclassical Einstein equations seems to be the only plausible dynamical equation for this metric: The metric couples to matter fields via the stress-energy tensor, and for a given quantum state the only physically observable c-number stress-energy tensor that one can construct is the above renormalized expectation value. However, lacking a full quantum gravity theory, the scope and limits of the theory are not so well understood. It is assumed that the semiclassical theory should break down at Planck scales, which is when simple order of magnitude estimates suggest that the quantum effects of gravity should not be ignored, because the energy of a quantum fluctuation in a Planck size region, as determined by the Heisenberg uncertainty principle, is comparable to the gravitational energy of that fluctuation.

The theory is expected to break down when the fluctuations of the stress-energy operator are large [ 92 ]. A criterion based on the ratio of the fluctuations to the mean was proposed by Kuo and Ford [ 194 ] (see also work via zeta-function methods [ 242 , 69 ]). This proposal was questioned by Phillips and Hu [ 163 , 243 , 244 ] because it does not contain a scale at which the theory is probed or how accurately the theory can be resolved. They suggested the use of a smearing scale or pointseparation distance for integrating over the bi-tensor quantities, equivalent to a stipulation of the resolution level of measurements; see also the response by Ford [ 93 , 95 ]. A different criterion is recently suggested by Anderson et al. [ 9 , 10 ] based on linear response theory. A partial summary of this issue can be found in our Erice Lectures [ 168 ].

Stochastic gravity

The purpose of stochastic gravity is to extend the semiclassical theory to account for these fluctuations in a self-consistent way. A physical observable that describes these fluctuations to lowest order is the noise kernel bi-tensor, which is defined through the two point correlation of the stress-energy operator as

where the curly brackets mean anticommutator, and where

This bi-tensor can also be written as N ab , c ′ d ⌂ [ g ; x , y ), or N ab , c ′ d ′ ( x , y ) as we do in Section 5 , to emphasize that it is a tensor with respect to the first two indices at the point x and a tensor with respect to the last two indices at the point y , but we shall not follow this notation here. The noise kernel is defined in terms of the unrenormalized stress-tensor operator T ̂ ab [ g ; x ) on a given background metric g ab , thus a regulator is implicitly assumed on the right-hand side of Equation (11) . However, for a linear quantum field the above kernel — the expectation function of a bi-tensor — is free of ultraviolet divergences because the regularized T ab [ g ; x ) differs from the renormalized T ab R [ g ; x ) by the identity operator times some tensor counterterms (see Equation (6) ), so that in the subtraction (12) the counterterms cancel. Consequently the ultraviolet behavior of 〈 T ̂ ab ( x ) T ̂ cd ( y )〉 is the same as that of 〈 T ̂ ab (x)〉〈 T ̂ cd ( y )〉, and T ̂ ab can be replaced by the renormalized operator T ̂ ab R in Equation (11) ; an alternative proof of this result is given in [ 244 , 245 ]. The noise kernel should be thought of as a distribution function; the limit of coincidence points has meaning only in the sense of distributions. The bi-tensor N abcd [ g ; x , y ), or N abcd ( x , y ) for short, is real and positive semi-definite, as a consequence of T ̂ ab R being self-adjoint. A simple proof is given in [ 169 ].

Once the fluctuations of the stress-energy operator have been characterized, we can perturbatively extend the semiclassical theory to account for such fluctuations. Thus we will assume that the background spacetime metric g ab is a solution of the semiclassical Einstein Equations (7) , and we will write the new metric for the extended theory as g ab + h ab , where we will assume that h ab is a perturbation to the background solution. The renormalized stress-energy operator and the state of the quantum field may now be denoted by T ̂ ab R [ g + h ] and | ψ [ g + h ]〉, respectively, and 〈 T ̂ ab R [ g + h ]〉 will be the corresponding expectation value.

Let us now introduce a Gaussian stochastic tensor field ξ ab [ g ; x ) defined by the following correlators:

where 〈…〉 is means statistical average. The symmetry and positive semi-definite property of the noise kernel guarantees that the stochastic field tensor ξ ab[ g , x ), or ξ ab ( x ) for short, just introduced is well defined. Note that this stochastic tensor captures only partially the quantum nature of the fluctuations of the stress-energy operator since it assumes that cumulants of higher order are zero.

All these properties make it quite natural to incorporate into the Einstein equations the stress-energy fluctuations by using the stochastic tensor ξ ab [ g ; x ) as the source of the metric perturbations. Thus we will write the following equation:

This equation is in the form of a (semiclassical) Einstein-Langevin equation ; it is a dynamical equation for the metric perturbation h ab to linear order. It describes the backreaction of the metric to the quantum fluctuations of the stress-energy tensor of matter fields, and gives a first order extension to semiclassical gravity as described by the semiclassical Einstein equation (7) .

Note that we refer to the Einstein-Langevin equation as a first order extension to the semiclassical Einstein equation of semiclassical gravity and the lowest level representation of stochastic gravity. However, stochastic gravity has a much broader meaning, as it refers to the range of theories based on second and higher order correlation functions. Noise can be defined in effectively open systems (e.g., correlation noise [ 46 ] in the Schwinger-Dyson equation hierarchy) to some degree, but one should not expect the Langevin form to prevail. In this sense we say that stochastic gravity is the intermediate theory between semiclassical gravity (a mean field theory based on the expectation values of the energy-momentum tensor of quantum fields) and quantum gravity (the full hierarchy of correlation functions retaining complete quantum coherence [ 154 , 155 ]).

The renormalization of the operator T ̂ ab [ g + h ] is carried out exactly as in the previous case, now in the perturbed metric g ab + h ab . Note that the stochastic source ξ ab[ g ; x ) is not dynamical; it is independent of h ab since it describes the fluctuations of the stress tensor on the semiclassical background g ab .

We see that whereas the semiclassical theory depends on the expectation value of the pointdefined value of the stress-energy operator, the stochastic theory carries information also on the two point correlation of the stress-energy operator. We should also emphasize that, even if the metric fluctuations appears classical and stochastic, their origin is quantum not only because they are induced by the fluctuations of quantum matter, but also because they are the suitably coarse-grained variables left over from the quantum gravity fluctuations after some mechanism for decoherence and classicalization of the metric field [ 106 , 126 , 83 , 120 , 122 , 293 ]. One may, in fact, derive the stochastic semiclassical theory from a full quantum theory. This was done via the world-line influence functional method for a moving charged particle in an electromagnetic field in quantum electrodynamics [ 178 ]. From another viewpoint, quite independent of whether a classicalization mechanism is mandatory or implementable, the Einstein-Langevin equation proves to be a useful tool to compute the symmetrized two point correlations of the quantum metric perturbations [ 255 ]. This is illustrated in the linear toy model discussed in [ 169 ], which has features of some quantum Brownian models [ 49 , 47 , 48 ].

The Einstein-Langevin Equation: Functional Approach

The Einstein-Langevin equation (14) may also be derived by a method based on functional techniques [ 208 ]. Here we will summarize these techniques starting with semiclassical gravity.

In semiclassical gravity functional methods were used to study the backreaction of quantum fields in cosmological models [ 123 , 90 , 129 ]. The primary advantage of the effective action approach is, in addition to the well-known fact that it is easy to introduce perturbation schemes like loop expansion and nPI formalisms, that it yields a fully self-consistent solution. For a general discussion in the semiclassical context of these two approaches, equation of motion versus effective action, see, e.g., the work of Hu and Parker (1978) versus Hartle and Hu (1979) in [ 203 , 115 , 158 , 159 , 124 , 3 , 4 ]. See also comments in Sec. 5.6 of [ 169 ] on the black hole backreaction problem comparing the approach by York et al. [ 297 , 298 , 299 ] versus that of Sinha, Raval, and Hu [ 264 ].

The well known in-out effective action method treated in textbooks, however, led to equations of motion which were not real because they were tailored to compute transition elements of quantum operators rather than expectation values. The correct technique to use for the backreaction problem is the Schwinger-Keldysh closed-time-path (CTP) or ‘in-in’ effective action [ 257 , 11 , 184 , 66 , 272 , 41 , 70 ]. These techniques were adapted to the gravitational context [ 76 , 181 , 39 , 182 , 236 , 57 ] and applied to different problems in cosmology. One could deduce the semiclassical Einstein equation from the CTP effective action for the gravitational field (at tree level) with quantum matter fields.

Furthermore, in this case the CTP functional formalism turns out to be related [ 272 , 43 , 58 , 201 , 112 , 54 , 55 , 216 , 196 , 208 , 206 ] to the influence functional formalism of Feynman and Vernon [ 89 ], since the full quantum system may be understood as consisting of a distinguished subsystem (the “system” of interest) interacting with the remaining degrees of freedom (the environment). Integrating out the environment variables in a CTP path integral yields the influence functional, from which one can define an effective action for the dynamics of the system [ 43 , 167 , 156 , 112 ]. This approach to semiclassical gravity is motivated by the observation [ 149 ] that in some open quantum systems classicalization and decoherence [ 303 , 304 , 305 , 306 , 180 , 33 , 279 , 307 , 109 ] on the system may be brought about by interaction with an environment, the environment being in this case the matter fields and some “high-momentum” gravitational modes [ 188 , 119 , 228 , 150 , 36 , 37 , 160 , 293 ]. Unfortunately, since the form of a complete quantum theory of gravity interacting with matter is unknown, we do not know what these “high-momentum” gravitational modes are. Such a fundamental quantum theory might not even be a field theory, in which case the metric and scalar fields would not be fundamental objects [ 154 ]. Thus, in this case, we cannot attempt to evaluate the influence action of Feynman and Vernon starting from the fundamental quantum theory and performing the path integrations in the environment variables. Instead, we introduce the influence action for an effective quantum field theory of gravity and matter [ 79 , 78 , 77 , 80 , 263 , 237 , 238 ], in which such “high-momentum” gravitational modes are assumed to have already been “integrated out.”

Influence action for semiclassical gravity

Alternatively, the above double path integral can be rewritten as a CTP integral, namely, as a single path integral in a complex time contour with two different time branches, one going forward in time from t i to t f , and the other going backward in time from t f to t i (in practice one usually takes t i →-∞). From this influence functional, the influence action S IF [ g + , g - ], or S IF [ g ±] for short, defined by

carries all the information about the environment (the matter fields) relevant to the system (the gravitational field). Then we can define the CTP effective action for the gravitational field, S eff [ g ± ], as

This is the effective action for the classical gravitational field in the CTP formalism. However, since the gravitational field is treated only at the tree level, this is also the effective classical action from which the classical equations of motion can be derived.

Influence action for stochastic gravity

In the spirit of the previous derivation of the Einstein-Langevin equation, we now seek a dynamical equation for a linear perturbation h ab to the semiclassical metric g ab , solution of Equation (7) . Strictly speaking, if we use dimensional regularization we must consider the n-dimensional version of that equation. From the results just described, if such an equation were simply a linearized semiclassical Einstein equation, it could be obtained from an expansion of the effective action S eff [ g + h α ]. In particular, since, from Equation (18) , we have that

the expansion of 〈 T ̂ ab [ g + h ]〉 to linear order in h ab can be obtained from an expansion of the influence action S IF [ g + h ± ] up to second order in h ab ± .

with t ̂ ab defined in Equation (12) ; [ , ] denotes the commutator and { , } the anti-commutator. Here we use a Weyl ordering prescription for the operators. The symbol T* denotes the following ordered operations: First, time order the field operators φ ̂ and then apply the derivative operators which appear in each term of the product T ab ( x ) T cd ( y ), where T ab is the functional (3) . This T* “time ordering” arises because we have path integrals containing products of derivatives of the field, which can be expressed as derivatives of the path integrals which do not contain such derivatives. Notice, from their definitions, that all the kernels which appear in expressions (20) are real and also H A abcd is free of ultraviolet divergences in the limit n →4.

From Equation (18) and (20) , since S IF [ g , g ]=0 and S IF [ g - , g + ]=- S IF * [ g + , g - ], we can writethe expansion for the influence action S IF [ g + h ± ] around a background metric g ab in terms of the previous kernels. Taking into account that these kernels satisfy the symmetry relations

and introducing the new kernel

the expansion of S IF can be finally written as

where we have used the notation

From Equations (23) and (19) it is clear that the imaginary part of the influence action does not contribute to the perturbed semiclassical Einstein equation (the expectation value of the stressenergy tensor is real), however, as it depends on the noise kernel, it contains information on the fluctuations of the operator T ̂ ab [ g ].

We are now in a position to carry out the derivation of the semiclassical Einstein-Langevin equation. The procedure is well known [ 43 , 167 , 58 , 110 , 26 , 296 , 246 ]: It consists of deriving a new “stochastic” effective action from the observation that the effect of the imaginary part of the influence action (23) on the corresponding influence functional is equivalent to the averaged effect of the stochastic source ξ ab coupled linearly to the perturbations h ab ± . This observation follows from the identity first invoked by Feynman and Vernon for such purpose:

We may now introduce the stochastic effective action as

where the “stochastic” influence action is defined as

Note that, in fact, the influence functional can now be written as a statistical average over ξ ab :

The stochastic equation of motion for hab reads

Explicit form of the Einstein-Langevin equation

We can write the Einstein-Langevin equation in a more explicit form by working out the expansion of 〈 T ̂ ab [ g + h ]〉 up to linear order in the perturbation h ab . From Equation (19) , we see that this expansion can be easily obtained from Equation (23) . The result is

Here we use a subscript n on a given tensor to indicate that we are explicitly working in n dimensions, as we use dimensional regularization, and we also use the superindex (1) to generally indicate that the tensor is the first order correction, linear in h ab , in a perturbative expansion around the background g ab .

Using the Klein-Gordon equation (2) , and expressions (3) for the stress-energy tensor and the corresponding operator, we can write

It is understood that indices are raised with the background inverse metric g ab , and that all the covariant derivatives are associated to the metric g ab .

Substituting Equation (31) into the n -dimensional version of the Einstein-Langevin Equation (14) , taking into account that g ab satisfies the semiclassical Einstein equation (7) , and substituting expression (32) , we can write the Einstein-Langevin equation in dimensional regularization as

where t ̂ ab is defined by Equation (12) , then the kernels N and H A can be written as

where we have used that

and the fact that the first term on the right-hand side of this identity is real, whereas the second one is pure imaginary. Once we perform the renormalization procedure in Equation (34) , setting n =4 will yield the physical Einstein-Langevin equation. Due to the presence of the kernel H n abcd ( x , y ), this equation will be usually non-local in the metric perturbation. In Section 6 we will carry out an explicit evaluation of the physical Einstein-Langevin equation which will illustrate the procedure.

The kernels for the vacuum state

When the expectation values in the Einstein-Langevin equation are taken in a vacuum state |0〉, such as, for instance, an “in” vacuum, we can be more explicit, since we can write the expectation values in terms of the Wightman and Feynman functions, defined as

These expressions for the kernels in the Einstein-Langevin equation will be very useful for explicit calculations. To simplify the notation, we omit the functional dependence on the semiclassical metric g ab , which will be understood in all the expressions below.

Noise Kernel and Point-Separation

In this section we explore further the properties of the noise kernel and the stress-energy bi-tensor. Similar to what was done for the stress-energy tensor it is desirable to relate the noise kernel defined at separated points to the Green function of a quantum field. We pointed out earlier [ 154 ] that field quantities defined at two separated points may possess important information which could be the starting point for probes into possible extended structures of spacetime. Of more practical concern is how one can define a finite quantity at one point or in some small region around it from the noise kernel defined at two separated points. When we refer to, say, the fluctuations of energy density in ordinary (point-wise) quantum field theory, we are in actuality asking such a question. This is essential for addressing fundamental issues like

• the validity of semiclassical gravity [ 194 ] — e.g., whether the fluctuations to mean ratio is a correct criterion [ 163 , 243 , 93 , 95 , 9 , 10 ];
• whether the fluctuations in the vacuum energy density which drives some models of inflationary cosmology violates the positive energy condition;
• physical effects of black hole horizon fluctuations and Hawking radiation backreaction — to begin with, is the fluctuations finite or infinite?
• general relativity as a low energy effective theory in the geometro-hydrodynamic limit towards a kinetic theory approach to quantum gravity [ 146 , 154 , 155 ].

Thus, for comparison with ordinary phenomena at low energy we need to find a reasonable prescription for obtaining a finite quantity of the noise kernel in the limit of ordinary (pointdefined) quantum field theory. Regularization schemes used in obtaining a finite expression for the stress-energy tensor have been applied to the noise kernel . This includes the simple normal ordering [ 194 , 295 ] and smeared field operator [ 243 ] methods applied to the Minkowski and Casimir spaces, zeta-function [ 87 , 189 , 53 ] for spacetimes with an Euclidean section, applied to the Casimir effect [ 69 ] and the Einstein Universe [ 242 ], or the covariant point-separation methods applied to the Minkowski [ 243 ], hot flat space and the Schwarzschild spacetime [ 245 ]. There are differences and deliberations on whether it is meaningful to seek a point-wise expression for the noise kernel, and if so what is the correct way to proceed — e.g., regularization by a subtraction scheme or by integrating over a test-field. Intuitively the smear field method [ 243 ] may better preserve the integrity of the noise kernel as it provides a sampling of the two point function rather than using a subtraction scheme which alters its innate properties by forcing a nonlocal quantity into a local one. More investigation is needed to clarify these points, which bear on important issues like the validity of semiclassical gravity. We shall set a more modest goal here, to derive a general expression for the noise kernel for quantum fields in an arbitrary curved spacetime in terms of Green functions and leave the discussion of point-wise limit to a later date. For this purpose the covariant point-separation method which highlights the bi-tensor features, when used not as a regularization scheme, is perhaps closest to the spirit of stochastic gravity.

The task of finding a general expression of the noise-kernel for quantum fields in curved spacetimes was carried out by Phillips and Hu in two papers using the “modified” point separation scheme [ 282 , 1 , 284 ]. Their first paper [ 244 ] begins with a discussion of the procedures for dealing with the quantum stress tensor bi-operator at two separated points, and ends with a general expression of the noise kernel defined at separated points expressed as products of covariant derivatives up to the fourth order of the quantum field’s Green function. (The stress tensor involves up to two covariant derivatives.) This result holds for x ≠ y without recourse to renormalization of the Green function, showing that N abc ′ d ′ ( x , y ) is always finite for x ≠ y (and off the light cone for massless theories). In particular, for a massless conformally coupled free scalar field on a four dimensional manifold, they computed the trace of the noise kernel at both points and found this double trace vanishes identically. This implies that there is no stochastic correction to the trace anomaly for massless conformal fields, in agreement with results arrived at in [ 43 , 58 , 208 ] (see also Section 3 ). In their second paper [ 245 ] a Gaussian approximation for the Green function (which is what limits the accuracy of the results) is used to derive finite expressions for two specific classes of spacetimes, ultrastatic spacetimes, such as the hot flat space, and the conformally- ultrastatic spacetimes, such as the Schwarzschild spacetime. Again, the validity of these results may depend on how we view the relevance and meaning of regularization. We will only report the result of their first paper here.

Point separation

Thus the first order of business is the construction of the stress tensor and then to derive the symmetric stress-energy tensor two point function, the noise kernel, in terms of the Wightman Green function. In this section we will use the traditional notation for index tensors in the point-separation context.

n -tensors and end-point expansions

To simplify notation, henceforth we will eliminate the semicolons after the first one for multiple covariant derivatives at multiple points.

Having objects defined at different points, the coincident limit is defined as evaluation “on the diagonal”, in the sense of the spacetime support of the function or tensor, and the usual shorthand [ G ( x , y )]≡ G ( x , x ) is used. This extends to n -tensors as

i.e., this becomes a rank ( n 1 + n 2 + n 3 + n 4 ) tensor at x . The multi-variable chain rule relates covariant derivatives acting at different points, when we are interested in the coincident limit:

This result is referred to as Synge’s theorem in this context; we follow Fulling’s discussion [ 100 ].

The main bi-scalar we need is the world function σ ( x , y ). This is defined as a half of the square of the geodesic distance between the points x and y . It satisfies the equation

Often in the literature, a covariant derivative is implied when the world function appears with indices, σ a ≡ σ ; a i.e., taking the covariant derivative at x , while σ a ′ means the covariant derivative at y . This is done since the vector - σ a is the tangent vector to the geodesic with length equal to the distance between x and y . As σ a records information about distance and direction for the two points, this makes it ideal for constructing a series expansion of a bi-scalar. The end point expansion of a bi-scalar S ( x , y ) is of the form

satisfies the equation

Stress-energy bi-tensor operator and noise kernel

Even though we believe that the point-separated results are more basic in the sense that it reflects a deeper structure of the quantum theory of spacetime, we will nevertheless start with quantities defined at one point, because they are what enter in conventional quantum field theory. We will use point separation to introduce the bi-quantities. The key issue here is thus the distinction between point-defined ( pt ) and point-separated ( bi ) quantities.

For a free classical scalar field φ with the action S m [ g , φ ] defined in Equation (1) , the classical stress-energy tensor is

Let us first seek a point-separated extension of these classical quantities and then consider the quantum field operators. Point separation is symmetrically extended to products of covariant derivatives of the field according to

To carry out point separation on Equation (49) , we first define the differential operator

from which we obtain the classical stress tensor as

That the classical tensor field no longer appears as a product of scalar fields at a single point allows a smooth transition to the quantum tensor field. From the viewpoint of the stress tensor, the separation of points is an artificial construct, so when promoting the classical field to a quantum one, neither point should be favored. The product of field configurations is taken to be the symmetrized operator product, denoted by curly brackets:

With this, the point separated stress-energy tensor operator is defined as

While the classical stress tensor was defined at the coincidence limit x ′→ x , we cannot attach any physical meaning to the quantum stress tensor at one point until the issue of regularization is dealt with, which will happen in the next section. For now, we will maintain point separation so as to have a mathematically meaningful operator.

The expectation value of the point-separated stress tensor can now be taken. This amounts to replacing the field operators by their expectation value, which is given by the Hadamard (or Schwinger) function

and the point-separated stress tensor is defined as

Finiteness of the noise kernel

We now turn our attention to the noise kernel introduced in Equation (11) , which is the symmetrized product of the (mean subtracted) stress tensor operator:

Since T ̂ ab ( x ) defined at one point can be ill-behaved as it is generally divergent, one can question the soundness of these quantities. But as will be shown later, the noise kernel is finite for y ≠ x . All field operator products present in the first expectation value that could be divergent, are canceled by similar products in the second term. We will replace each of the stress tensor operators in the above expression for the noise kernel by their point separated versions, effectively separating the two points ( x , y ) into the four points ( x , x ′, y , y ′). This will allow us to express the noise kernel in terms of a pair of differential operators acting on a combination of four and two point functions. Wick’s theorem will allow the four point functions to be re-expressed in terms of two point functions. From this we see that all possible divergences for y ≠ x will cancel. When the coincidence limit is taken, divergences do occur. The above procedure will allow us to isolate the divergences and to obtain a finite result.

where the four point function is

We assume that the pairs ( x , x ′) and ( y , y ′) are each within their respective Riemann normal coordinate neighborhoods so as to avoid problems that possible geodesic caustics might be present. When we later turn our attention to computing the limit y → x , after issues of regularization are addressed, we will want to assume that all four points are within the same Riemann normal coordinate neighborhood.

Expanding out the anti-commutators in Equation (59) and applyingWick’s theorem, the four point function becomes

We can now easily see that the noise kernel defined via this function is indeed well defined for the limit ( x ′, y ′)→( x , y ):

From this we can see that the noise kernel is also well defined for y ≠ x ; any divergence present in the first expectation value of Equation (59) have been cancelled by those present in the pair of Green functions in the second term, in agreement with the results of Section 3 .

Explicit form of the noise kernel

(Our notation is that ∇ a acts at x , ∇ c ′ at y , ∇ b ″ at x ′, and ∇ d ‴ at y ′.) Expanding out the differential operator above, we can determine which derivatives act on which Wightman function:

If we now let x ′→ x and y ′→ y , the contribution to the noise kernel is (including the factor of 1/8 present in the definition of the noise kernel):

That this term can be written as the sum of a part involving G xy and one involving G yx is a general property of the entire noise kernel. It thus takes the form

This can be connected with the zeta function approach to this problem [ 242 ] as follows: Recall when the quantum stress tensor fluctuations determined in the Euclidean section is analytically continued back to Lorentzian signature ( τ → it ), the time ordered product results. On the other hand, if the continuation is τ →- it , the anti-time ordered product results. With this in mind, the noise kernel is seen to be related to the quantum stress tensor fluctuations derived via the effective action as

Trace of the noise kernel

One of the most interesting and surprising results to come out of the investigations of the quantum stress tensor undertaken in the 1970s was the discovery of the trace anomaly [ 61 , 84 ]. When the trace of the stress tensor T = g ab T ab is evaluated for a field configuration that satisties the field equation (2) , the trace is seen to vanish for massless conformally coupled fields. When this analysis is carried over to the renormalized expectation value of the quantum stress tensor, the trace no longer vanishes. Wald [ 284 ] showed that this was due to the failure of the renormalized Hadamard function G ren ( x , x ′) to be symmetric in x and x ′, implying that it does not necessarily satisfy the field equation (2) in the variable x ′. (The definition of G ren ( x , x ′) in the context of point separation will come next.)

With this in mind, we can now determine the noise associated with the trace. Taking the trace at both points x and y of the noise kernel functional (67) yields

For the massless conformal case, this reduces to

which holds for any function G ( x , y ). For G being the Green function, it satisfies the field equation (2) :

We will only assume that the Green function satisfies the field equation in its first variable. Using the fact □ R =0 (because the covariant derivatives act at a different point than at which R is supported), it follows that

With these results, the noise kernel trace becomes

which vanishes for the massless conformal case. We have thus shown, based solely on the definition of the point separated noise kernel, that there is no noise associated with the trace anomaly. This result obtained in [ 245 ] is completely general since it is assumed that the Green function is only satisfying the field equations in its first variable; an alternative proof of this result was given in [ 208 ]. This condition holds not just for the classical field case, but also for the regularized quantum case, where one does not expect the Green function to satisfy the field equation in both variables. One can see this result from the simple observation used in Section 3 : Since the trace anomaly is known to be locally determined and quantum state independent, whereas the noise present in the quantum field is non-local, it is hard to find a noise associated with it. This general result is in agreement with previous findings [ 43 , 167 , 58 ], derived from the Feynman-Vernon influence functional formalism [ 89 , 88 ] for some particular cases.

Metric Fluctuations in Minkowski Spacetime

Although the Minkowski vacuum is an eigenstate of the total four-momentum operator of a field in Minkowski spacetime, it is not an eigenstate of the stress-energy operator. Hence, even for those solutions of semiclassical gravity such as the Minkowski metric, for which the expectation value of the stress-energy operator can always be chosen to be zero, the fluctuations of this operator are non-vanishing. This fact leads to consider the stochastic metric perturbations induced by these fluctuations.

Here we derive the Einstein-Langevin equation for the metric perturbations in a Minkowski background. We solve this equation for the linearized Einstein tensor and compute the associated two-point correlation functions, as well as, the two-point correlation functions for the metric perturbations. Even though, in this case, we expect to have negligibly small values for these correlation functions for points separated by lengths larger than the Planck length, there are several reasons why it is worth carrying out this calculation.

On the one hand, these are the first backreaction solutions of the full Einstein-Langevin equation. There are analogous solutions to a “reduced” version of this equation inspired in a “minisuperspace” model [ 59 , 38 ], and there is also a previous attempt to obtain a solution to the Einstein-Langevin equation in [ 58 ], but there the non-local terms in the Einstein-Langevin equation were neglected.

On the other hand, the results of this calculation, which confirm our expectations that gravitational fluctuations are negligible at length scales larger than the Planck length, but also predict that the fluctuations are strongly suppressed on small scales, can be considered a first test of stochastic semiclassical gravity. In addition, these results reveal an important connection between stochastic gravity and the large N expansion of quantum gravity. We can also extract conclusions on the possible qualitative behavior of the solutions to the Einstein-Langevin equation. Thus, it is interesting to note that the correlation functions at short scales are characterized by correlation lengths of the order of the Planck length; furthermore, such correlation lengths enter in a non-analytic way in the correlation functions.

We advise the reader that his section is rather technical since it deals with an explicit non-trivial backreaction computation in stochastic gravity. We have tried to make it reasonable self-contained and detailed, however a more detailed exposition can be found in [ 209 ].

Perturbations around Minkowski spacetime

As we have already mentioned the vacuum |0〉 is an eigenstate of the total four-momentum operator in Minkowski spacetime, but not an eigenstate of T ̂ ab R [ η ]. Hence, even in the Minkowski background, there are quantum fluctuations in the stress-energy tensor and, as a result, the noise kernel does not vanish. This fact leads to consider the stochastic corrections to this class of trivial solutions of semiclassical gravity. Since, in this case, the Wightman and Feynman functions (37) , their values in the two-point coincidence limit, and the products of derivatives of two of such functions appearing in expressions (38) and (39) are known in dimensional regularization, we can compute the Einstein-Langevin equation using the methods outlined in Sections 3 and 4 .

where the integrals in dimensional regularization have been computed in the standard way [ 209 ], and where Γ( z ) is Euler’s gamma function. The semiclassical Einstein equation (7) in n dimensions before renormalization reduces now to

which corresponds to a renormalization of the cosmological constant

The kernels in the Minkowski background

where I S (- p )= I S ( p ) and I A (- p )=- I A ( p ). These two functions are explicitly given by

After some manipulations, we find

and we finally get

Notice that the noise and dissipation kernels defined in Equation (94) are actually real because, for the noise kernels, only the cos px terms of the exponentials e ipx contribute to the integrals, and, for the dissipation kernels, the only contribution of such exponentials comes from the i sin px terms.

Let us define the integrals

Finally after a rather long but straightforward calculation, and after expanding around n =4, we get,

where κ n has been defined in Equation (86) , and φ ̂ ( p 2 ) and Δ n ( x ) are given by

The Einstein-Langevin equation

With the previous results for the kernels we can now write the n-dimensional Einstein-Langevin equation (87) , previous to the renormalization. Taking also into account Equations (82) and (83) , we may finally write:

Notice that the terms containing the bare cosmological constant have cancelled. These equations can now be renormalized, that is, we can now write the bare coupling constants as renormalized coupling constants plus some suitably chosen counterterms, and take the limit n →4. In order to carry out such a procedure, it is convenient to distinguish between massive and massless scalar fields. The details of the calculation can be found in [ 209 ]. It is convenient to introduce the two new kernels

and use that, from Equation (102) , we have

The coupling constants are then easily renormalized. We note that in the massless limit, the Newtonian gravitational constant is not renormalized and, in the conformal coupling case, Δ ξ =0, we have that β B = β . Note also that, by making m =0 in Equation (94) , the noise and dissipation kernels can be written as

It is also convenient to introduce the new kernel

Correlation functions for gravitational perturbations

The Fourier transform of the Einstein-Langevin Equation (111) now reads

which in the massless case reduces to

Correlation functions for the linearized Einstein tensor

Substituting this ansatz into Equations (122) , it is easy to see that it solves these equations if we take

and where the notation [ ] r means that the zeros of the denominators are regulated with appropriate prescriptions in such a way that d 1 ( p ) and d 2 ( p ) are well defined Lorentz invariant scalar distributions. This yields a particular solution to the Einstein-Langevin equations,

which, since the stochastic source is conserved, satisfies the conservation condition. Note that, in the case of a massless scalar field ( m =0), the above solution has a functional form analogous to that of the solutions of linearized semiclassical gravity found in the appendix of [ 91 ]. Notice also that, for a massless conformally coupled field ( m =0 and Δ ξ = 0), the second term on the right-hand side of Equation (123) will not contribute in the correlation functions (120) , since in this case the stochastic source is traceless.

This last expression is well defined as a bi-distribution and can be easily evaluated using Equation (118) . The final explicit result for the Fourier transformed correlation function for the Einstein tensor is thus

To obtain the correlation functions in coordinate space, Equation (120) , we take the inverse Fourier transform. The final result is

where F l ( p ), l =1, 2, 3, are given in Equations (116) and (125) . Notice that, for a massless field ( m =0), we have

Correlation functions for the metric perturbations

Conformally coupled field

For a conformally coupled field, i.e., when m =0 and Δ ξ =0, the previous correlation functions are greatly simplified and can be approximated explicitly in terms of analytic functions. The detailed results are given in [ 209 ]; here we outline the main features.

In case (b) when the denominator has zeros, a detailed calculation carried out in [ 209 ] shows that

which indicates an exponential decay at distances around the Planck scale. Thus short scale fluctuations are strongly suppressed.

For the two-point metric correlation the results are similar. In this case we have

The integrand has the same behavior of the correlation function of Equation (137) , thus matter fields tends to suppress the short scale metric perturbations. In this case we find, as for the correlation of the Einstein tensor, that for case (a) above we have

and for case (b) we have

This expression allows a direct comparison with the graviton propagator for linearized quantum gravity in the 1/ N expansion found by Tomboulis [ 277 ]. One can see that the imaginary part of the graviton propagator leads, in fact, to Equation (139) . In [ 255 ] it is shown that the two-point correlation functions for the metric perturbations derived from the Einstein-Langevin equation are equivalent to the symmetrized quantum two-point correlation functions for the metric fluctuations in the large N expansion of quantum gravity interacting with N matter fields.

For background solutions of semiclassical gravity with other scales present apart from the Planck scales (for instance, for matter fields in a thermal state), stress-energy fluctuations may be important at larger scales. For such backgrounds, stochastic semiclassical gravity might predict correlation functions with characteristic correlation lengths larger than the Planck scales. It seems quite plausible, nevertheless, that these correlation functions would remain non-analytic in their characteristic correlation lengths. This would imply that these correlation functions could not be obtained from a calculation involving a perturbative expansion in the characteristic correlation lengths. In particular, if these correlation lengths are proportional to the Planck constant ħ , the gravitational correlation functions could not be obtained from an expansion in ħ . Hence, stochastic semiclassical gravity might predict a behavior for gravitational correlation functions different from that of the analogous functions in perturbative quantum gravity [ 79 , 78 , 77 , 80 ]. This is not necessarily inconsistent with having neglected action terms of higher order in ~ when considering semiclassical gravity as an effective theory [ 91 ]. It is, in fact, consistent with the closed connection of stochastic gravity with the large N expansion of quantum gravity interacting with N matter fields.

Structure Formation

Cosmological structure formation is a key problem in modern cosmology [ 190 , 229 ] and inflation offers a natural solution to this problem. If an inflationary period is present, the initial seeds for the generation of the primordial inhomogeneities that lead to the large scale structure have their source in the quantum fluctuations of the inflaton field, the field which is generally responsible for driving inflation. Stochastic gravity provides a sound and natural formalism for the derivation of the cosmological perturbations generated during inflation.

In [ 254 ] it was shown that the correlation functions that follow from the Einstein-Langevin equation which emerges in the framework of stochastic gravity coincide with that obtained with the usual quantization procedures [ 218 ], when both the metric perturbations and the inflaton fluctuations are both linearized. Stochastic gravity, however, can naturally deal with the fluctuations of the inflaton field even beyond the linear approximation.

Here we will illustrate the equivalence with the usual formalism, based on the quantization of the linear cosmological and inflaton perturbations, with one of the simplest chaotic inflationary models in which the background spacetime is a quasi-de Sitter universe [ 253 , 254 ].

In this chaotic inflationary model [ 199 ] the inflaton field φ of mass m is described by the following Lagrangian density:

The conditions for the existence of an inflationary period, which is characterized by an accelerated cosmological expansion, is that the value of the field over a region with the typical size of the Hubble radius is higher than the Planck mass m P . This is because in order to solve the cosmological horizon and flatness problem more than 60 e-folds of expansion are needed; to achieve this the scalar field should begin with a value higher than 3 m P . The inflaton mass is small: As we will see, the large scale anisotropies measured in the cosmic background radiation [ 265 ] restrict the inflaton mass to be of the order of 10 -6 m P . We will not discuss the naturalness of this inflationary model and we will simply assume that if one such region is found (inside a much larger universe) it will inflate to become our observable universe.

We want to study the metric perturbations produced by the stress-energy tensor fluctuations of the inflaton field on the homogeneous background of a flat Friedmann-Robertson-Walker model, described by the cosmological scale factor a ( η ), where η is the conformal time, which is driven by the homogeneous inflaton field φ ( η )=〈 φ ̂ 〉. Thus we write the inflaton field in the following form:

where the scalar metric perturbations Φ( x ) and Ψ( x ) correspond to Bardeen’s gauge invariant variables [ 12 ].

The Einstein-Langevin equation for scalar metric perturbations

The Einstein-Langevin equation as described in Section 3 is gauge invariant, and thus we can work in a desired gauge and then extract the gauge invariant quantities. The Einstein-Langevin equation (14) reads now

where the subindices indicate the degree of dependence on the homogeneous field φ , and its perturbation ϕ . The first term in this equation depends only on the homogeneous field and it is given by the classical expression. The second term is proportional to 〈 ϕ̂ [ g ̃ ]〉 which is not zero because the field dynamics is considered on the perturbed spacetime, i.e., this term includes the coupling of the field with h ab and may be obtained from the expectation value of the linearized Klein-Gordon equation,

The last term in Equation (145) corresponds to the expectation value to the stress tensor for a free scalar field on the spacetime of the perturbed metric.

After using the previous decomposition, the noise kernel N abcd [ g ; x , y ) defined in Equation (11) can be written as

Correlation functions for scalar metric perturbations

The the two-point correlation function for the scalar metric perturbations induced by the inflaton fluctuations is thus given by

Here two-point correlation function for the stochastic source, which is connected to the stressenergy tensor fluctuations through the noise kernel, is given by

In practice, to make the explicit computation of the Hadamard function, we will assume that the field state is in the Euclidean vacuum and the background spacetime is de Sitter. Furthermore we will compute the Hadamard function for a massless field, and will make a perturbative expansion in terms of the dimensionless parameter m / m P . Thus we consider

are the positive frequency k-modes for a massless minimally coupled scalar field on a de Sitter background, which define the Euclidean vacuum state, â k |0〉=0 [ 25 ].

The assumption of a massless field for the computation of the Hadamard function is made because massless modes in de Sitter are much simpler to deal with than massive modes. We can see that this is, however, a reasonable approximation as follows. For a given mode the m =0 approximation is reasonable when its wavelength λ is shorter that the Compton wavelength, λ c =1/ m . In our case we have a very small mass m , and the horizon size H -1 , where H is the Hubble constant H = ȧ / a (here a ( t ) with t the physical time dt = adη ), satisfies that H -1 <λ c . Thus, for modes inside the horizon, λ<λ c and m =0 is a reasonable approximation. Outside the horizon massive modes decay in amplitude as ∼exp(-m 2 t / H ), whereas massless modes remain constant, thus when modes leave the horizon the approximation will eventually break down. However, we only need to ensure that the approximation is still valid after 60 e-folds, i.e., Ht ∼60, but this is the case since 60 m 2 < H 2 given that m ∼10 -6 m P , and m ≪ H as in most inflationary models [ 190 , 229 ].

The background geometry is not exactly that of de Sitter spacetime, for which a ( η )=-( Hη ) -1 with -∞1< η <0. One can expand in terms of the “slow-roll” parameters and assume that to first order φ̇ ( t )≃ m P 2 ( m / m P ), where t is the physical time. The correlation function for the metric perturbation (150) can then be easily computed; see [ 253 , 254 ] for details. The final result, however, is very weakly dependent on the initial conditions, as one may understand from the fact that the accelerated expansion of de quasi-de Sitter spacetime during inflation erases the information about the initial conditions. Thus one may take the initial time to be η 0 =-∞, and obtain to lowest order in m / m P the expression

From this result two main conclusions are derived. First, the prediction of an almost Harrison-Zel’dovich scale-invariant spectrum for large scales, i.e., small values of k . Second, since the correlation function is of order of ( m / m P ) 2 , a severe bound to the mass m is imposed by the gravitational fluctuations derived from the small values of the Cosmic Microwave Background (CMB) anisotropies detected by COBE. This bound is of the order of ( m / m P )∼10 -6 [ 265 , 218 ].

We should now comment on some differences with those works in [ 45 , 213 , 212 , 51 ] which used a self-interacting scalar field or a scalar field interacting nonlinearly with other fields. In those works an important relaxation of the ratio m / m P was found. The long wavelength modes of the inflaton field were regarded as an open system in an environment made out of the shorter wavelength modes. Then, Langevin type equations were used to compute the correlations of the long wavelength modes driven by the fluctuations of the shorter wavelength modes. In order to get a significant relaxation on the above ratio, however, one had to assume that the correlations of the free long wavelength modes, which correspond to the dispersion of the system initial state, had to be very small. Otherwise they dominate by several orders of magnitude those fluctuations that come from the noise of the environment. This would require a great amount of fine-tuning for the initial quantum state of each mode [ 254 ]. We should remark that in the model discussed here there is no environment for the inflaton fluctuations. The inflaton fluctuations, however, are responsible for the noise that induces the metric perturbations.

One important advantage of the Einstein-Langevin approach to the gravitational fluctuations in inflaton over the approach based on the quantization of the linear perturbations of both the metric and the inflaton field [ 218 ], is that an exact treatment of the inflaton quantum fluctuations is possible. This leads to corrections to the almost scale invariant spectrum for scalar metric perturbations at large scales, and has implications for the spectrum of the cosmic microwave background anisotropies. However, in the standard inflationary models these corrections are subdominant. Furthermore when the full non linear effect of the quantum field is considered, tensorial metric perturbations are also induced by the inflaton fluctuations. An estimation of this effect, presumably subdominant over the free tensorial fluctuations, has not been performed.

We should remark that although the gravitational fluctuations are here assumed to be classical, the correlation functions obtained correspond to the expectation values of the symmetrized quantum metric perturbations [ 49 , 254 ]. This means that even in the absence of decoherence the fluctuations predicted by the Einstein-Langevin equation, whose solutions do not describe the actual dynamics of the gravitational field any longer, still give the correct symmetrized quantum two-point functions.

Another important advantage of the stochastic gravity approach is that one may also compute the gravitational fluctuations in inflationary models which are not driven by an inflaton field, such as Starobinsky inflation which is driven by the trace anomaly due to conformally coupled quantum fields. In fact, Einstein’s semiclassical equation (7) for a massless quantum field which is conformally coupled to the gravitational field admits an inflationary solution which is almost de Sitter initially and ends up in a matter-dominated-like regime [ 269 , 280 ]. In these models the standard approach based on the quantization of the gravitational and the matter fields to linear order cannot be used. This is because the calculation of the metric perturbations correspond to having only the last term in the noise kernel in Equation (147) , since there is no homogeneous field φ ( η ) as the expectation value 〈 φ ̂ 〉=0, and linearization becomes trivial.

In the trace anomaly induced inflation framework, Hawking et al. [ 132 ] were able to compute the two-point quantum correlation function for scalar and tensorial metric perturbations in a spatially closed de Sitter universe, making use of the anti-de Sitter conformal field theory correspondence. They find that short scale metric perturbations are strongly suppressed by the conformal matter fields. This is similar to what we obtained in Section 6 for the induced metric fluctuations in Minkowski spacetime. In the stochastic gravity context, the noise kernel in a spatially closed de Sitter background was derived in [ 252 ]. However, in a spatially flat arbitrary Friedmann-Robertson-Walker model the Einstein-Langevin equation describing the metric perturbations was first obtained in [ 58 ] (see also [ 169 ]). The two-point correlation functions for the metric perturbations can be derived from its solutions, but this is work still in progress.

Black Hole Backreaction

As another illustration of the application of stochastic gravity we consider fluctuations and backreaction in black hole spacetimes. The celebrated Hawking effect of particle creation from black holes is constructed from a quantum field theory in a curved spacetime framework. The oftmentioned ‘black hole evaporation’ referring to the reduction of the mass of a black hole due to particle creation must entail backreaction considerations. Backreaction of Hawking radiation [ 118 , 13 , 297 , 298 , 299 , 135 , 136 , 8 ] could alter the evolution of the background spacetime and change the nature of its end state, more drastically so for Planck size black holes. Because of the higher symmetry in cosmological spacetimes, backreaction studies of processes therein have progressed further than the corresponding black hole problems, which to a large degree is still concerned with finding the right approximations for the regularized energy-momentum tensor [ 177 , 233 , 210 , 6 , 7 , 5 , 134 ] for even the simplest spacetimes such as the spherically symmetric family including the important Schwarzschild metric (for a summary of the cosmological backreaction problem treated in the stochastic gravity theory, see [ 169 ]). The latest important work is that of Hiscock, Larson, and Anderson [ 134 ] on backreaction in the interior of a black hole, where one can find a concise summary of earlier work. To name a few of the important landmarks in this endeavor (this is adopted from [ 134 ]), Howard and Candelas [ 145 , 144 ] have computed the stress-energy of a conformally invariant scalar field in the Schwarzschild geometry. Jensen and Ottewill [ 176 ] have computed the vacuum stress-energy of a massless vector field in Schwarzschild spacetime. Approximation methods have been developed by Page, Brown, and Ottewill [ 231 , 28 , 29 ] for conformally invariant fields in Schwarzschild spacetime, Frolov and Zel’nikov [ 99 ] for conformally invariant fields in a general static spacetime, and Anderson, Hiscock, and Samuel [ 6 , 7 ] for massless arbitrarily coupled scalar fields in a general static spherically symmetric spacetime. Furthermore the DeWitt-Schwinger approximation has been derived by Frolov and Zel’nikov [ 97 , 98 ] for massive fields in Kerr spacetime, by Anderson, Hiscock, and Samuel [ 6 , 7 ] for a general (arbitrary curvature coupling and mass) scalar field in a general static spherically symmetric spacetime and Anderson, Hiscock, and Samuel have applied their method to the Reissner-Nordström geometry [ 5 ]. Though arduous and demanding, the effort continues on because of the importance of backreaction effects of Hawing radiation in black holes. They are expected to address some of the most basic issues such as black hole thermodynamics [ 174 , 235 , 282 , 16 , 17 , 18 , 266 , 175 , 287 , 281 , 275 , 183 , 271 , 139 , 140 , 205 , 204 ] and the black hole end state and information loss puzzles [ 230 ].

Here we wish to address the black hole backreaction problem with new insights provided by stochastic semiclassical gravity. (For the latest developments see, e.g., the reviews [ 151 , 154 , 168 , 169 ]). It is not our intention to seek better approximations for the regularized energy-momentum tensor, but to point out new ingredients lacking in the existing framework based on semiclassical gravity. In particular one needs to consider both the dissipation and the fluctuations aspects in the backreaction of particle creation or vacuum polarization.

In a short note [ 164 ] Hu, Raval, and Sinha discussed the formulation of the problem in this new light, commented on some shortcomings of existing works, and sketched the strategy [ 264 ] behind the stochastic gravity theory approach to the black hole fluctuations and backreaction problem. Here we follow their treatment with focus on the class of quasi-static black holes.

From the new perspective provided by statistical field theory and stochastic gravity, it is not difficult to postulate that the backreaction effect is the manifestation of a fluctuation-dissipation relation [ 85 , 86 , 220 , 35 , 34 , 288 ]. This was first conjectured by Candelas and Sciama [ 60 , 258 , 259 ] for a dynamic Kerr black hole emitting Hawking radiation, and by Mottola [ 217 ] for a static black hole (in a box) in quasi-equilibrium with its radiation via linear response theory [ 191 , 24 , 192 , 195 , 193 ]. However, these proposals as originally formulated do not capture the full spirit and content of the self-consistent dynamical backreaction problem. Generally speaking (paraphrasing Mottola), linear response theory is not designed for tackling backreaction problems. More specifically, if one assumes a specified background spacetime (static in this case) and state (thermal) of the matter field(s) as done in [ 217 ], one would get a specific self-consistent solution. But in the most general situation which a full backreaction program demands of, the spacetime and the state of matter should be determined by their dynamics under mutual influence on an equal footing, and the solutions checked to be physically sound by some criteria like stability consideration. A recent work of Anderson, Molina-Paris, and Mottola [ 9 , 10 ] on linear response theory does not make these restrictions. They addressed the issue of the validity of semiclassical gravity (SCG) based on an analysis of the stability of solutions to the semiclassical Einstein equation (SEE). However, on this issue, Hu, Roura, and Verdaguer [ 165 ] pointed out the importance of including both the intrinsic and induced fluctuations in the stability analysis, the latter being given by the noise kernel. The fluctuation part represented by the noise kernel is amiss in the fluctuation-dissipation relation proposed by Candelas and Sciama [ 60 , 258 , 259 ] (see below). As will be shown in an explicit example later, the backreaction is an intrinsically dynamic process. The Einstein-Langevin equation in stochastic gravity overcomes both of these deficiencies.

For Candelas and Sciama [ 60 , 258 , 259 ], the classical formula they showed relating the dissipation in area linearly to the squared absolute value of the shear amplitude is suggestive of a fluctuation-dissipation relation. When the gravitational perturbations are quantized (they choose the quantum state to be the Unruh vacuum) they argue that it approximates a flux of radiation from the hole at large radii. Thus the dissipation in area due to the Hawking flux of gravitational radiation is allegedly related to the quantum fluctuations of gravitons. The criticism in [ 164 ] is that their’s is not a fluctuation-dissipation relation in the truly statistical mechanical sense, because it does not relate dissipation of a certain quantity (in this case, horizon area) to the fluctuations of the same quantity . To do so would require one to compute the two point function of the area, which, being a four-point function of the graviton field, is related to a two-point function of the stress tensor. The stress tensor is the true “generalized force” acting on the spacetime via the equations of motion, and the dissipation in the metric must eventually be related to the fluctuations of this generalized force for the relation to qualify as a fluctuation-dissipation relation.

From this reasoning, we see that the stress-energy bi-tensor and its vacuum expectation value known as the noise kernel are the new ingredients in backreaction considerations. But these are exactly the centerpieces in stochastic gravity. Therefore the correct framework to address semiclassical backreaction problems is stochastic gravity theory, where fluctuations and dissipation are the equally essential components. The noise kernel for quantum fields in Minkowski and de Sitter spacetime has been carried out by Martin, Roura, and Verdaguer [ 207 , 209 , 254 ], and for thermal fields in black hole spacetimes and scalar fields in general spacetimes by Campos, Hu, and Phillips [ 54 , 55 , 244 , 245 , 241 ]. Earlier, for cosmological backreaction problems Hu and Sinha [ 167 ] derived a generalized expression relating dissipation (of anisotropy in Bianchi Type I universes) and fluctuations (measured by particle numbers created in neighboring histories). This example shows that one can understand the backreaction of particle creation as a manifestation of a (generalized) fluctuation-dissipation relation.

As an illustration of the application of stochastic gravity theory we outline the steps in a black hole backreaction calculation, focusing on the manageable quasi-static class. We adopt the Hartle-Hawking picture [ 127 ] where the black hole is bathed eternally — actually in quasi-thermal equilibrium — in the Hawking radiance it emits. It is described here by a massless scalar quantum field at the Hawking temperature. As is well-known, this quasi-equilibrium condition is possible only if the black hole is enclosed in a box of size suitably larger than the event horizon. We can divide our consideration into the far field case and the near horizon case. Campos and Hu [ 54 , 55 ] have treated a relativistic thermal plasma in a weak gravitational field. Since the far field limit of a Schwarzschild metric is just the perturbed Minkowski spacetime, one can perform a perturbation expansion off hot flat space using the thermal Green functions [ 108 ]. Strictly speaking the location of the box holding the black hole in equilibrium with its thermal radiation is as far as one can go, thus the metric may not reach the perturbed Minkowski form. But one can also put the black hole and its radiation in an anti-de Sitter space [ 133 ], which contains such a region. Hot flat space has been studied before for various purposes (see, e.g., [ 116 , 249 , 250 , 72 , 27 ]). Campos and Hu derived a stochastic CTP effective action and from it an equation of motion, the Einstein-Langevin equation, for the dynamical effect of a scalar quantum field on a background spacetime. To perform calculations leading to the Einstein-Langevin equation, one needs to begin with a self-consistent solution of the semiclassical Einstein equation for the thermal field and the perturbed background spacetime. For a black hole background, a semiclassical gravity solution is provided by York [ 297 , 298 , 299 ]. For a Robertson-Walker background with thermal fields, it is given by Hu [ 148 ]. Recently, Sinha, Raval, and Hu [ 264 ] outlined a strategy for treating the near horizon case, following the same scheme of Campos and Hu. In both cases two new terms appear which are absent in semiclassical gravity considerations: a nonlocal dissipation and a (generally colored) noise kernel. When one takes the noise average, one recovers York’s [ 297 , 298 , 299 ] semiclassical equations for radially perturbed quasi-static black holes. For the near horizon case one cannot obtain the full details yet, because the Green function for a scalar field in the Schwarzschild metric comes only in an approximate form (e.g., Page approximation [ 231 ]), which, though reasonably accurate for the stress tensor, fails poorly for the noise kernel [ 245 , 241 ]. In addition a formula is derived in [ 264 ] expressing the CTP effective action in terms of the Bogolyubov coefficients. Since it measures not only the number of particles created, but also the difference of particle creation in alternative histories, this provides a useful avenue to explore the wider set of issues in black hole physics related to noise and fluctuations.

Since backreaction calculations in semiclassical gravity have been under study for a much longer time than in stochastic gravity, we will concentrate on explaining how the new stochastic features arise from the framework of semiclassical gravity, i.e., noise and fluctuations and their consequences. Technically the goal is to obtain an influence action for this model of a black hole coupled to a scalar field and to derive an Einstein-Langevin equation from it. As a by-product, from the fluctuation-dissipation relation, one can derive the vacuum susceptibility function and the isothermal compressibility function for black holes, two quantities of fundamental interest in characterizing the nonequilibrium thermodynamic properties of black holes.

In this model the black hole spacetime is described by a spherically symmetric static metric with a line element of the following general form written in advanced time Eddington-Finkelstein coordinates,

with standard line element

We look for this class of perturbed metrics in the form given by Equation (153) (thus restricting our consideration only to spherically symmetric perturbations),

To facilitate later comparisons with our program we briefly recall York’s work [ 297 , 298 , 299 ]. (See also the work by Hochberg and Kephart [ 135 ] for a massless vector field, by Hochberg, Kephart, and York [ 136 ] for a massless spinor field, and by Anderson, Hiscock, Whitesell, and York [ 8 ] for a quantized massless scalar field with arbitrary coupling to spacetime curvature.) York considered the semiclassical Einstein equation,

CTP effective action for the black hole

We first derive the CTP effective action for the model described in Sec. ( 7.1 ). Using the metric (156) (and neglecting the surface terms that appear in an integration by parts) we have the action for the scalar field written perturbatively as

where the first and second order perturbative operators V (1) and V (2) are given by

From quantum field theory in curved spacetime considerations discussed above we take the following action for the gravitational field:

We are interested in computing the CTP effective action (163) for the matter action and when the field φ is initially in the Hartle-Hawking vacuum. This is equivalent to saying that the initial state of the field is described by a thermal density matrix at a finite temperature T = T H . The CTP effective action at finite temperature T ≡1/ β for this model is given by (for details see [ 54 , 55 ])

In computing the traces, some terms containing divergences are canceled using counterterms introduced in the classical gravitational action after dimensional regularization.

Near flat case

One can express the Fourier transforms of these kernels as

respectively.

The physical meanings of these kernels can be extracted if we write the renormalized CTP effective action at finite temperature (169) in an influence functional form [ 32 , 111 , 161 , 162 ]. N, the imaginary part of the CTP effective action can be identified with the noise kernel and D, the antisymmetric piece of the real part with the dissipation kernel. Campos and Hu [ 54 , 55 ] have shown that these kernels identified as such indeed satisfy a thermal fluctuation-dissipation relation.

and at high temperatures the leading term ( β -4 ) may be written as

Near horizon case

In this section we show how a semiclassical Einstein-Langevin equation can be derived from the previous thermal CTP effective action. This equation depicts the stochastic evolution of the perturbations of the black hole under the influence of the fluctuations of the thermal scalar field.

As we have seen before and here, the Einstein.Langevin equation is a dynamical equation governing the dissipative evolution of the gravitational field under the influence of the fluctuations of the quantum field, which, in the case of black holes, takes the form of thermal radiance. From its form we can see that even for the quasi-static case under study the backreaction of Hawking radiation on the black hole spacetime has an innate dynamical nature.

For the far field case, making use of the explicit forms available for the noise and dissipation kernels, Campos and Hu [ 54 , 55 ] formally proved the existence of a fluctuation-dissipation relation at all temperatures between the quantum fluctuations of the thermal radiance and the dissipation of the gravitational field. They also showed the formal equivalence of this method with linear response theory for lowest order perturbations of a near-equilibrium system, and how the response functions such as the contribution of the quantum scalar field to the thermal graviton polarization tensor can be derived. An important quantity not usually obtained in linear response theory, but of equal importance, manifest in the CTP stochastic approach is the noise term arising from the quantum and statistical fluctuations in the thermal field. The example given in this section shows that the backreaction is intrinsically a dynamic process described (at this level of sophistication) by the Einstein-Langevin equation.

Discussions

We make a few remarks here and draw some connection with related work on black hole fluctuations.

Black hole backreaction

The other important conceptual point that comes to light from this approach is that related to the fluctuation-dissipation relation. In the quantum Brownian motion analog (see, e.g., [ 32 , 111 , 161 , 162 ] and references therein), the dissipation of the energy of the Brownian particle as it approaches equilibrium and the fluctuations at equilibrium are connected by the fluctuation-dissipation relation. Here the backreaction of quantum fields on black holes also consists of two forms — dissipation and fluctuation or noise — corresponding to the real and imaginary parts of the influence functional as embodied in the dissipation and noise kernels. A fluctuation-dissipation relation has been shown to exist for the near flat case by Campos and Hu [ 54 , 55 ] and we anticipate that it should also exist between the noise and dissipation kernels for the general case, as it is a categorical relation [ 32 , 111 , 161 , 162 , 151 ]. Martin and Verdaguer have also proved the existence of a fluctuation-dissipation relation when the semiclassical background is a stationary spacetime and the quantum field is in thermal equilibrium. Their result was then extended to a conformal field in a conformally stationary background [ 207 ]. The existence of a fluctuation-dissipation relation for the black hole case has been discussed by some authors previously [ 60 , 258 , 259 , 217 ]. In [ 164 ], Hu, Raval, and Sinha have described how this approach and its results differ from those of previous authors. The fluctuation-dissipation relation reveals an interesting connection between black holes interacting with quantum fields and non-equilibrium statistical mechanics. Even in its restricted quasi-static form, this relation will allow us to study nonequilibrium thermodynamic properties of the black hole under the influence of stochastic fluctuations of the energy-momentum tensor dictated by the noise terms.

There are limitations of a technical nature in the specific example invoked here. For one we have to confine ourselves to small perturbations about a background metric. For another, as mentioned above, there is no reliable approximation to the Schwarzschild thermal Green’s function to explicitly compute the noise and dissipation kernels. This limits our ability to present explicit analytical expressions for these kernels. One can try to improve on Page’s approximation by retaining terms to higher order. A less ambitious first step could be to confine attention to the horizon and using approximations that are restricted to near the horizon and work out the Influence Functional in this regime.

Yet another technical limitation of the specific example is the following. Although we have allowed for backreaction effects to modify the initial state in the sense that the temperature of the Hartle-Hawking state gets affected by the backreaction, we have essentially confined our analysis to a Hartle-Hawking thermal state of the field. This analysis does not directly extend to a more general class of states, for example to the case where the initial state of the field is in the Unruh vacuum. Thus, we will not be able to comment on issues of the stability of an isolated radiating black hole under the influence of stochastic fluctuations.

Metric fluctuations in black holes

In addition to the work described above by Campos, Hu, Raval, and Sinha [ 54 , 55 , 164 , 264 ] and earlier work quoted therein, we mention also some recent work on black hole metric fluctuations and their effect on Hawking radiation. For example, Casher et al. [ 64 ] and Sorkin [ 267 , 268 ] have concentrated on the issue of fluctuations of the horizon induced by a fluctuating metric. Casher et al. [ 64 ] consider the fluctuations of the horizon induced by the “atmosphere” of high angular momentum particles near the horizon, while Sorkin [ 267 , 268 ] calculates fluctuations of the shape of the horizon induced by the quantum field fluctuations under a Newtonian approximation. Both group of authors come to the conclusion that horizon fluctuations become large at scales much larger than the Planck scale (note that Ford and Svaiter [ 94 ] later presented results contrary to this claim). However, though these works do deal with backreaction, the fluctuations considered do not arise as an explicit stochastic noise term as in our treatment. It may be worthwhile exploring the horizon fluctuations induced by the stochastic metric in our model and comparing the conclusions with the above authors. Barrabes et al. [ 14 , 15 ] have considered the propagation of null rays and massless fields in a black hole fluctuating geometry, and have shown that the stochastic nature of the metric leads to a modified dispersion relation and helps to confront the trans-Planckian frequency problem. However, in this case the stochastic noise is put in by hand and does not naturally arise from coarse graining as in the quantum open systems approach. It also does not take backreaction into account. It will be interesting to explore how a stochastic black hole metric, arising as a solution to the Einstein-Langevin equation, hence fully incorporating backreaction, would affect the trans-Planckian problem.

Ford and his collaborators [ 94 , 95 , 294 ] have also explored the issue of metric fluctuations in detail and in particular have studied the fluctuations of the black hole horizon induced by metric fluctuations. However, the fluctuations they have considered are in the context of a fixed background and do not relate to the backreaction.

Another work originating from the same vein of stochastic gravity but not complying with the backreaction spirit is that of Hu and Shiokawa [ 166 ], who study effects associated with electromagnetic wave propagation in a Robertson-Walker universe and the Schwarzschild spacetime with a small amount of given metric stochasticity. They find that time-independent randomness can decrease the total luminosity of Hawking radiation due to multiple scattering of waves outside the black hole and gives rise to event horizon fluctuations and fluctuations in the Hawking temperature. The stochasticity in a background metric in their work is assumed rather than derived (from quantum field fluctuations, as in this work), and so is not in the same spirit of backreaction. But it is interesting to compare their results with that of backreaction, so one can begin to get a sense of the different sources of stochasticity and their weights (see, e.g., [ 154 ] for a list of possible sources of stochasticity).

In a subsequent paper Shiokawa [ 261 ] showed that the scalar and spinor waves in a stochastic spacetime behave similarly to the electrons in a disordered system. Viewing this as a quantum transport problem, he expressed the conductance and its fluctuations in terms of a nonlinear sigma model in the closed time path formalism and showed that the conductance fluctuations are universal, independent of the volume of the stochastic region and the amount of stochasticity. This result can have significant importance in characterizing the mesoscopic behavior of spacetimes resting between the semiclassical and the quantum regimes.

Concluding Remarks

In the first part of this review on the fundamentals of theory we have given two routes to the establishment of stochastic gravity and derived a general (finite) expression for the noise kernel. In the second part we gave three applications, the correlation functions of gravitons in a perturbed Minkowski metric, structure formation in stochastic gravity theory, and the outline of a program for the study of black hole fluctuations and backreaction. A central issue which stochastic gravity can perhaps best address is the validity of semiclassical gravity as measured by the fluctuations of stress-energy compared to the mean. We will include a review of this topic in a future update.

There is ongoing research related to the topics discussed in this review. On the theory side, Roura and Verdaguer [ 255 ] have recently shown how stochastic gravity can be related to the large N limit of quantum metric fluctuations. Given N free matter fields interacting with the gravitational field, Hartle and Horowitz [ 128 ], and Tomboulis [ 277 ] have shown that semiclassical gravity can be obtained as the leading order large N limit (while keeping N times the gravitational coupling constant fixed). It is of interest to find out where in this setting can one place the fluctuations of the quantum fields and the metric fluctuations they induce; specifically, whether the inclusion of these sources will lead to an Einstein-Langevin equation [ 43 , 157 , 167 , 58 , 202 ], as it was derived historically in other ways, as described in the first part of this review. This is useful because it may provide another pathway or angle in connecting semiclassical to quantum gravity (a related idea is the kinetic theory approach to quantum gravity described in [ 155 ]).

Theoretically, stochastic gravity is at the frontline of the ‘bottom-up’ approach to quantum gravity [ 146 , 154 , 155 ]. Structurally, as can be seen from the issues discussed and the applications given, stochastic gravity has a very rich constituency because it is based on quantum field theory and nonequilibrium statistical mechanics in a curved spacetime context. The open systems concepts and the closed-time-path/influence functional methods constitute an extended framework suitable for treating the backreaction and fluctuations problems of dynamical spacetimes interacting with quantum fields. We have seen it applied to cosmological backreaction problems. It can also be applied to treat the backreaction of Hawking radiation in a fully dynamical black hole collapse situation. One can then address related issues such as the black hole end state and information loss puzzles (see, e.g., [ 230 , 152 ] and references therein). The main reason why this program has not progressed as swiftly as desired is due more to technical rather than programatic difficulties (such as finding reasonable analytic approximations for the Green function or numerical evaluation of mode-sums near the black hole horizon). Finally, the multiplex structure of this theory could be used to explore new lines of inquiry and launch new programs of research, such as nonequilibrium black hole thermodynamics and statistical mechanics.

Acknowledgements

The materials presented here originated from research work of BLH with Antonio Campos, Nicholas Phillips, Alpan Raval, and Sukanya Sinha, and of EV with Rosario Martin and Albert Roura. We thank them as well as Daniel Arteaga, Andrew Matacz, Tom Shiokawa, and Yuhong Zhang for fruitful collaboration and their cordial friendship since their Ph.D. days. We enjoy lively discussions with our friends and colleagues Esteban Calzetta, Diego Mazzitelli, and Juan Pablo Paz whose work in the early years contributed toward the establishment of this field. We acknowledge useful discussions with Paul Anderson, Larry Ford, Ted Jacobson, Renaud Parentani, and Raphael Sorkin. This work is supported in part by NSF grant PHY98-00967, the MICYT Research Project No. FPA-2001-3598, and the European project HPRN-CT-2000-00131.

We will often use the shortened term stochastic gravity as there is no confusion as to the nature and source of stochasticity in gravity being induced from the quantum fields and not a priori from the classical spacetime.

It is well-known that several regularization methods can work equally well for the removal of ultraviolet divergences in the stress-energy tensor of quantum fields in curved spacetime. Their mutual relations are known, and discrepancies explained. This formal structure of regularization schemes for quantum fields in curved spacetime should remain intact when applied to the regularization of the noise kernel in general curved spacetimes; it is the meaning and relevance of regularization of the noise kernel which is more of a concern (see comments below). Specific considerations will of course enter for each method. But for the methods employed so far, such as zeta-function, point separation, dimensional, smeared-field, applied to simple cases (Casimir, Einstein, thermal fields) there is no new inconsistency or discrepancy.

By “Lorentz invariant” we mean invariant under the transformations of the orthochronous Lorentz subgroup; see [ 137 ] for more details on the definition and properties of these tensor distributions.

Contributor Information

Bei Lok Hu, Email: ude.dmu.scisyhp@buh , http://www.physics.umd.edu/people/faculty/hu.html .

Enric Verdaguer, Email: se.bu.nff@eugadrev .

Review of the gravity model: origins and critical analysis of its theoretical development

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• Published: 26 April 2023
• Volume 3 , article number  95 , ( 2023 )

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This article presents a bibliographic review of the gravitational model in international trade from when it was first associated with Newton's law of universal gravitation. Firstly, I will introduce the concept of gravity in commerce as originally intended by Isard (Q J Econ 68(2):305–320, 1954) in relation to Tinbergen and Tobler’s approaches. Secondly, I will analyse the theoretical roots of international economics according to several authors, including McCallum, Anderson and van Wincoop, Krugman, Tranos, and Nijkamp. Thirdly, attention will be drawn to the evolution of the ideas of the authors mentioned above upon Isard’s initial approach rooted in physics. Furthermore, I will focus on how the tool rooted in physics can be applied or adapted to international trade. Throughout this article, I will try to keep the econophysical nature of this model.

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Introduction

This paper aims to examine the gravitational model and the usefulness of this theoretical approach in international economics. The methodology used follows a detailed historical analysis of the literature on the origin and chronological development of this vital instrument, beginning with Isaac Newton's initial formula in physics and concluding with its most current applications. Therefore, the paragraphs follow a chronological order, which is the most appropriate for describing the interdisciplinary development of the physical model. It is worth noting that the element of novelty consists in the suitable combination of the sequential scheme, where the paragraphs are organized thematically and chronologically. To sum up, the organization of the paragraphs follows a historical evolution and a thematic division, that provides a better comprehension and facilitates the gradual discovery of the topics covered. Whilst, in the last 30 years, the topic has been extensively investigated and reviewed, by authors such as Philbrick ( 1973 ), Anderson ( 1979a , b , 2011 ), Yotov et al. ( 2016 ), Head ( 2003 ), Carrothers ( 1956 ), the present study represents a step forward. Indeed, this research offers an updated and detailed analysis with a broader range of the existing applications of the model. While filling a void in the academic panorama, the study is suggesting an improved thematic division of the gravity model applications to economics and its econophysical evolution.

Contributions applying the concept of gravity are examined, ranging from those describing the movement of people (Ravenstein 1885 ; Stewart 1947 ; among others) and those that determine the trade flows (Isard 1954 ; Tinbergen 1962 ) to more modern applications. An in-depth analysis of the consistency of gravity of trade with the main theories of international economics is carried out. More specifically, the author investigates the notion of gravity as initially defined by Isard ( 1954 ) in relation to the various approaches of Tinbergen's conventional gravity model and Tobler's first law of geography. It takes up the contributions of Isard ( 1954 ) and Tinbergen ( 1962 ) and argues that Isard’s is more analytically based on the concept of gravity borrowed from physics. On the contrary, Tinbergen, generally considered by literature the forerunner of the gravitational model of international trade, proposes a formula which is a simple basis for the econometric estimation of bilateral trade flows between two countries; an empirical tool that, at most, metaphorically echoes Newton's law without representing a true analytical transposition to the level of ecophysics. Therefore, pursuing its object of research, this work provides a comparison between a simple spatial framework of three variables (Tobler), of which the most important is distance, a simple bilateral trade flows model (Tinbergen) and a structured scheme using the gravity metaphor that incorporates the concepts of attractions’ forces, as well as spatial concentration (the proper gravitational model) as outlined by Isard himself. The issue of the priority of Isard ( 1954 ) over Tinbergen ( 1962 ), which unfortunately most economists and articles of the field have erred or ignored, is then resolved. Therefore, the key evolutions of the model were studied based on their usefulness, adjusting the deviations according to an econophysical perspective. The latter is more consistent with Newton's theory and its initial applications to demography and economics made, respectively, by Stewart ( 1947 ) and Isard ( 1954 ). Although the gravity model has been extensively used in numerous empirical research and other economic models, its theoretical underpinnings are still econophysical. The author seems to be particularly sympathetic to the last-mentioned scholars and, more specifically, to Isard's ambition to focus on a multilateral model of the interaction between economic forces and the attempt to merge theories of location choice with theories of trade. It should be highlighted that the gravitational model, as will be made clear throughout this study, is not just a model of flow but also of the interaction of trade forces and spatial concentration. On the other hand, Tobler's model has been used as a reference point for models that depart beyond physics and it is essential to understand the theoretical development of the model in social sciences. In fact, despite being inspired by gravity models, Tobler’s Law is more general and represents purely a distance decay law in geography.

The paper goes on according to the question of research by analyzing the theoretical roots of international economics as defined by various authors following Tinbergen up to the most recent ones. It will be investigated how the physics-based tools can be applied to the social sciences, trying to maintain the econophysical nature of the model. According to this critical approach, the paper will explore and take into account the main developments of the gravitational model as described by authors such as McCallum, Anderson and Van Wincoop, Krugman, Tranos, and Nijkamp. Hence, it can be stated that the main contribution of the article is to the historical field.

Following Isard's viewpoint, the most convincing attempts in the economic literature to develop a multilateral gravitational model are analyzed. The study of the original model lays the foundation for an econophysical critique of the multilateral developments that took place in the trade model. The theoretical underpinnings of the international economy as defined by numerous authors will then be examined. Among these, greater emphasis will be placed on the concept of remoteness as well as the multilateral resistance factors of Anderson and Van Wincoop ( 2003 ). Although the notion of gravity in physics is multilateral, it has mostly been employed in economics in accordance with Tinbergen's ( 1962 ) standard formula, which describes bilateral trade between two countries. The multilateral approach was later adopted by Krugman ( 1991 ), who formulated the New Economic Geography, and McCallum's study of border effects (1995). As we will see Krugman's model recalls many elements of the Isard’s one. These attempts, however, do not constitute a direct evolution of Isard's original multilateral formula, but rather a multilateral implementation of Tinbergen's standard bilateral formula. In short, there is definitely little literature on econophysics in the prevailing econometric method that various authors have followed after Tinbergen's work. Moreover, this produces a variety of challenges, both new and old, even in the international economic model.

General concept and focus on distance

Among the applications of gravity law to social sciences, those related to trade have undoubtedly been the most prolific. Hundreds of economics research studies have employed the gravity equation and its solid theoretical foundations to investigate the determinant of international trade. The framework known in economics as the “gravity model” is a theoretical approach that may explain and predict trade flows based on two components: the gross domestic product (GDP) and the distance between countries. As will be seen, the gravitational model has a flexible structure that encompasses multiple market forms and can be used to assess the impact of various commercial policies. It is commonly applied in empirical work on bilateral exchanges between countries because of its predictive power in estimating exchange flows between countries (Head and Mayer 2014 ).

According to Isard ( 1954 : 308), “the distance variables act in much the same manner with respect to the social world as to the natural world”. As a result, any model that takes distance as a direct component that suggests costs must incorporate a gravitational link. Carrothers ( 1955 : 99) drew a parallel between human interaction and Newtonian physics of matter. Even though “it may not be possible to describe the actions and reactions of the individual human in mathematical terms” the behaviour of a group of people was foreseeable due to the study of probability in mathematics. As stated by the author, this phenomenon has been noticed in all social sciences because people behave differently in groups than they do as individuals. Likewise, in physics, it was impossible to describe the behaviour of individual molecules, but it was conceivable to predict the activity of a set of them. The logical considerations just mentioned, particularly those of Comte ( 1854 ), Isard ( 1954 ), and Carrothes ( 1955 ), are required premises for developing physical reasoning in social sciences, and without which a gravitational sociophysical structure would make no sense due to a lack of logical coherence. In some ways, one can choose whether to believe that human actions are regulated by universal rules or not. Nevertheless, beyond the solution to this complex dilemma, to use a gravity model in social sciences, one must accept a certain level of determinism in human life.

The gravitational trade model naturally predicts commerce between partners using a metaphorical interpretation of Newton's universal law (countries, regions, or companies). In physics, any particle of matter in the universe attracts any other particle via a gravitational force that is directly proportional to the product of their masses and inversely proportional to the square of their distance apart.

Newton's Law, when applied to international trade, implies that, just as particles attract each other in proportion to their size and mutual distance, partners’ trade in proportion to their respective economic size and mutual distance (“frictions of the distance”).

In physics, the formula of Gravity from ground up is the following (Schutz 2003 , pp. 13–18):

where \(F\) is the attractive force, and \(M\) are the masses (mass i and mass j ), \(D\) is the distance between the centres of the two objects, \(G\) is a universal gravitational constant.

Based on the gravity model, international trade flows positively correlate with market sizes. For this reason, when applying the formula to international trade, the economic dimension must be considered first. For states, GDP provides this function. Besides increasing GDP, a nation's sales of goods and services increase the number of services its citizens can import. Further, distance is determined by geographical and spatial factors, including trade barriers. The second factor is distance, which takes into account the geographical and spatial aspects, including trade barriers. Distance affects travel costs and the ease of establishing contacts, which in turn affects communications, which in turn affects trade. Economic size is generally considered a push, while distance is a pull.

On the basis of the considerations above, the application of the gravity model to international economics can be summarized as follows:

where \({F}_{ij}\) indicates the value of trade between two countries (country \(i\) and country \(j\) ), \(C\) is a constant, Footnote 1 \({GDP}_{i}\) is the GDP of country \(i\) , \({GDP}_{j}\) is the GDP of country \(j\) (sometimes even the GNP is used). Footnote 2 Finally, \({D}_{ij}\) is the distance between the two countries. Using the gravity model as an estimation of the flow of international trade also implies the application of geographical factors.

The major problem with the gravity model is estimating the resistance component. Of course, numerous variables other than distance influence transit costs. Using distance as a crow's route may result in the loss of vital information. To address this issue, Hummels ( 1999 ) replaced crows' paths with real distance, while Limão and Venables ( 2001 ) assessed the effect of infrastructure quality on transportation costs. Based on the approach created by Mayer and Zignago ( 2005 ), the distance between countries is measured in kilometers and reflects the distance between pairs of cities, weighted by the proportion of the country's population that resides in each city. Geographical distance is used as a starting point, and it is enhanced by a variety of modifications designed to account more economic, cultural, social, technological, linguistic, and other characteristics. As the opposite of distance friction, these methods occasionally use proximity instead of distance (Torre and Wallet 2014 ). The choice to consider a straight-line distance or an alternative method of transportation has an impact on the geographic distance measurement, as do other variables that may relate to the locations of the origin and destination. Considering the potentially vast physical region that many nations occupy, as well as the fact that commerce and economic activity occur across a multitude of their boundaries. The most popular databases, USITIC and CEPII, take geographic distance into account. This includes population-weighted distances between countries and distances that represent the spatial distribution of activities inside a country. But also, many proxies for cultural proximity as language, religion, origins of the legal system, colonial ties, etc. This historically ties with the fact that Tinbergen's study group was the first to use colonial, postcolonial, religion, and other cultural and social parameters in a trade model, which is related to the fact that Tinbergen ( 1962 ) developed the best-known and most widely applied formula for the gravity model, which was formalized later by his student Linnemann ( 1966 ) in his thesis. This is done in an effort to more precisely measure the distance that international trade must cover. The estimated coefficient for the distance variable is significant and has a negative sign. Similar to Herrera's conclusion (Gómez-Herrera 2013 ), which has depicted any transport cost or trade obstacles that might prevent trade flows. Therefore, if we build a gravitational field across this non-geographic distance, it will spread across these same economic distances in a virtual and not real space. In this sense, in these cases, the term “distance” does not refer to an actual distance measured in meters but rather to a dimensionless quantity (also known as a “pure quantity”) that measures the resistance of distance in the gravitational model of trade. Or rather it should be measured in economic distance meters. Therefore, in these empirical studies and in some gravitational model’s databases, we are not considering the absolute distance between countries expressed in meters, but a relative economic distance whose value depends on the comparison between the distances of other countries. At this point, the use of a weighted or network graph with vertices and edges, which considers the economic distances between states or regions, could be more appropriate. This solution could be effective, since state to state or state to region distances would be more of interest to economists than the distance of any other point in space. On the other hand, the exclusive use of geographical distance would allow an easier representation on a bi-dimensional map. The Newtonian gravitational approach directly implies the specification of the power function, which suggests elasticity at constant distances. However, under a multilateral system, the relative impact of other countries or regions on a trade system could be omitted. Therefore, there are many applications outside trade analysis that use exponential distance decay functions where the elasticity could be affected by distance. The essential difference is that an exponential function has its variable in its exponent, but a power function has its variable in its base. These alternative forms are particularly used in geography and regional sciences (Östh et al. 2016 ). Due to the function's power nature, changes in distance friction also affect trade volume in a proportional manner. Through model fitting and posterior contrast with dimensional extinction on current assertions, various ways lead to varying power constants. Most studies in economics ignore the square of the distance, which in physics implies a precise verifiable empirical relationship between different measuring systems (system of units) for masses, distances, and forces. The square can be omitted in economics since the value of distance, which is calculated through estimates, is not geographic but economic. The principle is theoretical in social sciences, and its empirical application is dependent on the proportionality between the members of mathematical equality (in the formula), and thus on the practical estimation of distances.

Indeed, the type and number of variables considered in the distance estimation approach are arbitrary in comparison to GDP and exchange flow (F), which are clearly characterized based on a certain currency.

After developing a method (measurement system) for estimating distances in trade, it may be important to determine the accuracy of the square of the distance. The square of distance, however, is still present in some authors.

It is worth noting, for instance, that the square of the distance has a consistent value in Stewart's applications in social sciences, from which Isard took inspiration for his model. In fact, the scholar distinguished between a demographic force ( \({F}_{ij}=k \left(\frac{{P}_{i}{P}_{j}}{{D}_{ij}^{2}}\right)\) ) and a demographic energy ( \({E}_{ij}=k \left(\frac{{P}_{i}{P}_{j}}{{D}_{ij}}\right)\) ), thus employing the square of the distance. In social sciences, this principle is theoretical, and its empirical application is influenced on the practical assessment of the distances, which in turn are given by the proportionality between the members of mathematical equality (in the formula). In fact, while for the distance determination method, the type and number of variables considered is not systematic, the measurement of GDP and exchange flow ( F ) is measurable through a given currency and thus, univocal. When an evaluation method (measurement system) has been established for trade distances, I recommend evaluating their square of distance accuracy. In Anderson's view ( 2004 : 334), the theory that human behavior varies non-linearly, but with the square of a given variable, has proved “strong and resilient” over time. The square is used in a variety of works in social sciences. For example, Malthus forecasts ongoing problems with the geometric growth of the human population in relation to the arithmetic growth of necessities. In 1890, the father of regression, statistician and sociologist Francis Galton understood that for the mathematical use of the square of the deviation, “there seems to be a wide field for the application of these methods to social problems” (Galton 1890 ).

Nonetheless, since the application of the law of gravity to the social sciences, the concept of a squared exponent of distance has only sometimes been applied. It was rather maintained a gravitational structure that takes into account the attraction force (often referred to as 'magnetism') of a considered variable mass and a more generalized use of friction or distance resistance based on the specific phenomenon in question (Sen and Smith 1955 ). However, this is only true for some researchers, as shown by Stewart's ( 1941 ) approach, which places theoretical emphasis on the square of distance.

To sum up, the gravity equation can be conceived as a representation of the main forces that affect international commerce demand and supply. If the country is the origin, then it represents the total amount that can be provided to all customers and represents the total destination of the demand. As a result, distance, which is inversely proportional to the commercial flow between countries, acts as a sort of ‘wedge’ tax imposing commercial costs and influencing choices. Therefore, GDP plays the role of physical mass in attracting trade flows between two countries. It follows that the greater the GDP, the greater the volume of trade that occurs. The resistance factor includes all variables acting as trade barriers that negatively affect trade between countries, including distance. This is used as a measure of transportation costs in most studies.

Shift from applications concerning the flow of people to the trade model based on the exchange of goods and services

Van Bergeijk and Brakman point out that the gravity model and its earliest systematic and scientific applications to social sciences dates back to the nineteenth century. First, Carey ( 1858 , p. 42) applied Newton’s universal law of gravity to “railway traffic and migration” by tracing a tendency for the individual to “gravitate to his fellow man”. According to the scholar, the individual, as a component of society (‘molecule of society’), is subject to a force of social attraction to other social agents, just as matter in physics is subject to a force of gravitational attraction with other matter. An early cogent formulation of the gravity narrative is Ravenstein’s ( 1885 , pp. 198–99), who explains how "currents" of migration are driven by the "absorption of centers of commerce and industry" but "grow less with the distance proportionately". Above all social interactions, the gravity research for immigrant flows was the first to be conceptualized. Early research focused on applications to flows of people rather than movements of things. In urban geography, the gravity model is used to assess traffic patterns, migration between two places, and the attraction of people to a center or to several centers. Many factors, including politics, language, culture, taxation, and others, affect how far individuals perceive a distance or the attractiveness of an area. These characteristics are influenced by free trade agreements and, more broadly, by population movements.

Ravenstein ( 1885 ) defined this notion in migration law , stating that a “populated center” draws migrants from other “populated centers” in proportion to population size and inversely linked to distance between centres. The mathematical function is as follows:

where \({M}_{ji}\) is the migration flow from centre j to centre i , \(f\left({P}_{i}\right)\) is a function of population size i and \({D}_{ij}\) is the distance between the two population centres.

Later, Reilly ( 1929 ) extended the concept of social gravitation to consumers and cities. The author illustrates how the city’s force of attraction exerted on nearby customers is directly proportional to the size of the population and inversely proportional to the square of the distance between the two. Therefore, ceteris paribus , larger cities will be more appealing to customers than smaller ones. Nonetheless, in accordance with Newton’s law, at the specific ‘breaking point’, consumers will be indifferent to either of the two competing cities, as demonstrated in the following formula (Reilly 1931 ):

where \({P}_{i}\) and \({P}_{j}\) are, respectively, the size of the centre i and the centre j , \({D}_{i}\) is the distance of the indifference point from the centre i and \({D}_{j}\) is its distance from the centre j .

The astrophysicist John Q. Stewart ( 1941 ) applied the laws of physics to the study of social sciences, laying the theoretical groundwork for further research in social physics. The researcher realized that the law of gravity might also be used to describe demographic phenomena when studying empirical regularities linked to distance in social sciences.

To that purpose, the author employs the notion of demographic gravitation , which states that many people, such as the population of a big city, act as a force of attraction for other individuals who eventually choose to migrate there. The following is the formula:

where \({F}_{ij}\) is the interaction force between the demographic centres i and j , \(k\) is a constant, \({P}_{i}\) is the population of the area i , \({P}_{j}\) is the population of area j , and \({D}_{ij}\) is the distance between i and j. Footnote 3 More specifically, Stewart proposed to adopt different values for the population by nationality, considering the ‘molecular weight’ of each member of the population. The author picked the molecular mass of the average American as a unit to standardize the measurements. The latter, for example, will be distinguishable from an Australian aborigine, whose molecular weight will presumably be less than one.

Taking into consideration the vector property already mentioned, this can also be expressed at point i as (Philbrick 1973 , p. 42):

where \({V}_{i}\) is the population's potential for attraction to centre i , \(K\) is a constant, \({P}_{j}\) is the population of all other areas in the particular universe under consideration, and \({D}_{ij}\) is the distance.

In several studies, the astrophysicist proved how the outcome of his formula might be utilized to draw a map of the surface using the device of ‘contours of equipotential’. The latter is akin to a ‘synoptic weather chart’, as demonstrated in Fig. 1 produced by Stewart. This is also clear from the fact that formula ( 15 ) is analogous to and derives from the formulas of gravitational fields ( 6 ) in physics. The graphic depiction takes on the features of a gravitational or magnetic field, complete with its equipotential contours.

Stewart’s contours of equal population potential for the United States, 1940, Source: Isard ( 1954 )

Gravity is used in the study of migration, demographics, and journeys, as well as in the study of international trade. Stewart's model has had a direct impact on the economic model, which replicates its logic and dynamics, but it transposes gravity to a flow of commerce rather than to a stream of people. The application to the study of international trade traces back to the work of Walter Isard ( 1954 ), but it was popularized in Tinbergen’s ( 1962 ) standard bilateral form, which was directly related to Newton’s Law of Universal Gravitation. Subsequently, several economists revisited it. Isard was inspired by Stewart’s model and introduced the concept of income potential . Thus, the author applies a gravitational field equation in physics to the study of international trade.

Isard and Tinbergen

We will now be moving on to the analysis of the gravitational model in international economics, starting from Isard’s and Tinbergen’s studies, which have been mentioned in the first section. Contrary to what has been erroneously reported and inferred in most research papers, Tinbergen ( 1962 ) was not de facto the first one to use the gravitational laws of physics in international trade. Footnote 4 Indeed, the application of the gravity model to social sciences was quite common in the second half of the last century (Stewart, Zipf, Vining, Ullman and others), and Isard was the first to suggest its application to international economics in 1954.

Although he has never made a comment regarding authorship, Tinbergen's name is regularly associated with numerous laws and models. For some reason, the Dutch scholar was quoted. These quotes were mentioned again, and whenever something he said was linked to a theme, it became a law or a rule.

As part of his academic activities, Isard focuses on multiregional input–output frameworks, the gravity model, potential models, game theory, interregional programming, complex industrial analysis, conflict resolution strategies, environmental and ecological analysis, and more.

This multiregional approach led Isard to become one of the main founders of regional science and peace science in economics. His handbook Methods of Regional Analysis: an introduction to Regional Science , which was the first book in use in Regional Science, contains empirical methodologies he elaborated through the years, either alone or with his collaborators. Since he was a student at Harvard, under the supervision of Alvin Hansen and Abbott Usher, Isard was interested in location issues. After his employment at Harvard, together with Capron, he published a classic paper on the location of the U.S. iron and steel industry in 1949, followed by Location and Space Economy (Isard 1956 ). So, it can be said that he pursued his interest in regional science and localization theories from an early age (Boyce 2003 ) and aimed to develop, in his words, a "superior set of tools" (Isard 1954 : 305–320), culminating in the dream of developing an improved theory of trade and regional phenomena.

This single general theory summarized three broad theoretical perspectives, starting with international trade, the Walrasian’s notions of general equilibrium and the concepts of location, in which the key elements were distance and transport inputs. Isard believed that “a general and comprehensive location theory and a general and comprehensive trade theory are one and the same ”. This integration affects both the long-run and short-run analyses. In the long-run analysis it does so by combining the notions of “distance inputs transport orientation” and of opportunity cost to yield a superior set of tools (Isard et al. 1969 : 319).

In the short-run analysis a considerable development of the two theories of trade and localization is necessary to perform a fusion. For this purpose, Isard proposes a multilateral analysis through the gravitational instrument (Isard et al. 1969 ). According to the scholar (1954), distance and transport factors were still being undervalued in international trade studies: he assessed that since transport inputs are not flows from a stock like capital inputs, they must not be treated as a factor of production. Referring to Weber's previous theories, he acknowledged that their treatment as intermediaries was more appropriate. According to Weber, the classical theory of commerce fails to take into account the significant proportion of industry that is transportation-oriented, and he aspires to a more comprehensive theory that incorporates transportation costs (Weber 1911 ), and Ohlin, who endorsed Weber’s approach and thought that the theory of international trade was only part of the localization theory, he argued that excluding the distance component would mean restricting oneself to an unrealistic world (Isard and Peck 1954 : 114). Only by merging trade with geographical space through localization theories, it would be possible to develop a greater theory in the field of international economics. Consistent with these assumptions and inspired by Newton’s universal gravity, Isard introduced a functional law to analyse international trade flows. At the heart of his theories lies the idea that geographical proximity promotes trade, because of the several underlying reasons, ranging from low transport costs to institutional and linguistic similarities between states. Therefore, he concluded that a gravitational relationship must arise in any model that considers distance to be a direct factor in increasing costs. Footnote 5 According to Isard ( 1954 ), the resistance effect of distance acts in a similar way in the social world, if compared to the natural world. Indeed, the economist himself stated that “the distance variables act in much the same manner with respect to the social world as to the natural world” (Isard 1954 : 308). Indeed, Isard, inspired by the socio-physics approach of Carey ( 1858 ) and Stewart ( 1941 ), thought the human being as a “molecule of society” that “gravitated to his fellow man”; thereby allowing scholars to use newton's laws of physics also in social sciences. He drew on Stewart’s notion of demographic gravitation (1941), a concept that, along with “demographic energy” and “potential of the population”, forms the foundation of his theories, creating the structural basis for later studies in social physics. The key concept is that human beings behave as a whole of molecules of society: the population of a city, for instance, acts as a force of attraction for other people, who will be attracted by its potential and will decide to migrate there). Isard took it up from Stewart’s earlier study, as highlighted by Fig.  1 , to re-propose it in his 1954 book to explain the concept (1947) of ‘income potential’ in international economics, according to the following formula (Isard 1954 : 308):

\({Y}_{j}\) is the income of nation (region) \(j\) , \({d}_{ij}\) is the average effective distance (that is distance adjusted for level of transport rates) between nations \(i\) and \(j\) , Footnote 6 \(a\) is a constant power to which \({d}_{ij}\) is raised, \(k\) is a constant similar to the gravitational constant, \(iV\) is the income potential produced by nation \(j\) upon nation \(i\) , and \(iV\) is the income potential produced by all nations upon nation \(i\) .

In Isard’s formula, the income potential , produced by a given nation on nation \(i\) , varies inversely with the intervening distance. Between two nations similar in resources and technological development, the closest to nation \(i\) will have a greater income potential on \(i\) .

Isard's formula ( 7 ) is multilateral , since it considers the effect on the income of nation \(i\) caused by the change in wealth of a sum of countries, considering the friction of their distances. Footnote 7 Furthermore, according to the scholar (Isard 1954 : 318–320), this formula would become over time an instrument for a complete and superior synthesis between the theory of localization and international trade. Before Isard, Ohlin ( 1933 ) faced the same problem: he stated it was necessary to demonstrate that the theory of international trade is only part of a general theory of localization. In order for spatial and price aspects to be considered simultaneously, economic geographers and economists should collaborate. According to economists, it is also necessary to analyse the movements of goods in relation to the movements of factors of production at national and international levels. It should be noted that Isard and Ohlin had been inspired by Weber's ideas ( 1911 ), according to which the classical theory of commerce overlooked "the significant amount of industry that is transport-oriented".

In 1962, Tinbergen and his students developed the traditional mathematical equation of the gravity model used for trade (2), and applied it empirically for the first time to the study of international trade in the work Shaping the World Economy . Moreover, in his original estimate, he predicted positive signs for the coefficients of economic attractors and negative signs for the distance. The in theory predicted a coefficient that could also be negative in the case of self-sufficiency. However, he was more interested in the empirical aspect, and every country he analysed a positive coefficient. We discussed "expected signs" in the gravitational model's regression because later studies confirmed the sign of the coefficients: positive for economic size and negative for distance. He also argued that the size of the economy of the importing country plays a dual role, indicating both total demand (internal and external) and the degree of diversity of production.

Tinbergen obtained a doctorate in Physics from the University of Leiden in Netherlands, and he was the first Nobel Laureate in Economics with Ragnar Frisch in 1969. His passion for these two disciplines led him to study international trade and to apply Newton’s gravitational law to economics studies. His model appeared in the appendix of a semi-popular book that discusses a new economic ordering for the world, without references, and this could explain why there is none related to Isard’s work.

While Isard was the first scholar to bring gravity into the analysis of trade flows, Tinbergen is considered the pioneer of the standard gravitational formula ( 2 ) which, stemming from Newton's gravitational law, (1) creates a simple and intuitive bilateral model for the empirical analysis of trade between pairs of countries. While Isard was looking for a complete theoretical approach for an all-inclusive analysis, Tinbergen was more interested in the empirical aspect. Footnote 8 Isard's gravitational instrument was created as part of a more exhaustive analysis of the international economy. As previously seen, whatever trade theory is considered, progress can only be achieved by combining international trade with economic geography via space and localization theories. The goal of this fusion is to create a superior set of tools and a conceptual framework for empirical and theoretical regional analysis. Isard develops the concept of income potential to avoid the customary but unrealistic two-country analysis and to consider the impact of the distance variable on trade and income in an aggregate and multi-country framework.

Tinbergen supervised Linnemann's thesis, which is now considered the canonical reference for the early gravity equation formalization. Probably, the model was chosen for describing the worldwide network of bilateral commerce, especially in terms of political ties. Tinbergen was motivated to move into economics by development issues and social policies. Indeed, throughout time, he started to stress the need for science to have an impact on society. He considered this crucial in an era where humans had failed to see the inevitable consequences of technology, of the psychosocial pressures to which they were subjected, of their exploitation of natural resources, and of the environment’s pollution. This was due to “our one-sided appreciation and our complacent acceptance of the blessings of our civilization, of reduced infant mortality, of increased affluence, of our “spiritual life” and last but not least of science itself” (Hinde 1990 ). Due to the fact that the model credited to Tinbergen is more econometric than econophysical like Isard's, a physics scholar may not often notice his contribution. Not only did Tinbergen use econometrics but he is also considered one of its founders. This is quite fascinating and brings up the duality that was thoroughly covered in this study. In fact, Tinbergen and Ragnar Frisch are regarded as two of the fathers of econometrics. In 1969, together with Frisch, he became the first recipient of the Nobel Prize in economics for having developed and applied dynamic models to the analysis of the economic process. Footnote 9 Isard, on the other hand, is legitimately recognized as one of the founders of econophysics. Therefore, it is worth noting the differences between these two fields of study. The econophysical method is based on the application of the rational logic and structures that characterize the world of physics, whereas econometric solely uses mathematical and statistical tools to construct models to assess the validity of economics hypotheses. In summary, econometrics can be defined as the process of developing economic models using statistical and mathematical techniques. The application of physics-related methods and tools to economics is known as econophysics, on the other hand. The econophysics approach does include econometric tools, but it is not this one that characterizes it.

As it will be seen in the next paragraphs, considering the simplicity of application of the standard gravitational formula and its popularity, this empirical approach has been further developed by subsequent authors. As a matter of fact, these new studies followed Tinbergen’s model, constituting a mathematical improvement and an econometric refinement of the formula. Tinbergen’s model was even applied to different locations, markets, and time frames, ensuring its success in literature, whilst over time moving away from Isard’s econophysics theoretical model and losing value and relevance in terms of the relationships of attraction, magnetism, and differing levels of economic power between various trading partners.

Tobler’s Law and the comprehensive econophysics concept of gravity in trade

Tobler’s economic model has been applied to the study of international economics as previously mentioned. The Tobler model in economics will be useful here in clarifying the difference between a simple distance decay trade model and a true gravitational trade model. By definition, the latter will cross the boundaries of trade into other fields of economics such as geopolitics and the economic analysis of spatial concentration. In this sense, it provides a valid comparison between a simple spatial framework based on three variables, with distance being the most significant, and a structural framework based on the metaphor of gravity in terms of ideas of attraction forces and spatial concentration. These forces modify the surrounding reality which, therefore, cannot be static. This confirms that the gravity trade model has a theoretical econophysical structure based on a strong bond between psychics and the economic model. Regarding this link between natural and social sciences, another point to take into consideration is that human interactions are considered through deterministic premises. Footnote 10

As previously mentioned, the Swiss-American geographer Tobler ( 1970 ) was also inspired by the use of the gravitational model in social sciences. He formulated his first law of geography in the Journal Economic Geography and applied it to the study of international trade. Footnote 11 (Tobler 1970 : 234–240). In formal terms, the formula is (Rey 2001 , pp. 9393–9399):

where \({T}_{ij}\) is the degree of spatial interaction (migration, trade flows, air travel, et cetera) between place \(i\) and \(j\) . \({T}_{ij}\) is proportionate to the size of the populations and inversely correlated to the distance. \({P}_{i}\) is the population of the origin and \({P}_{j}\) of the destination. \({D}_{ij}\) is the distance between these locations. The strength of the decline in interaction with increasing separation is estimated by \(\beta\) .

The similarity between Tinbergen's standard equation of gravity (2) and Tobler’s first law of geography (8) is evident. However, unlike the idea of gravity, Tobler’s law is more closely related to geography, to distance decay, and to a geometric approach of similarity. Gravity also entails relationships of attraction and economic force between various trading partners, as we have already seen. Furthermore, Tobler’s law was about ‘everything’ and not only measures masses as for the gravity, thereby leaving appropriate room for studying non-material concepts such as knowledge, culture, ideas, et cetera.

As previously noted, Tinbergen’s interest lies primarily in the empirical aspects and in the study of bilateral trade between countries. Further studies were inspired by this approach, as they left aside physical aspects which were originally present in Isard. It is worth recalling that Tinbergen was a physicist and that he was the first Nobel Laureate in Economics, prize won alongside Ragnar Frisch in 1969. This suggests that, besides emphasizing the intense connection between physics and economics and the usefulness of the application of the former to the latter, many economists yet overlook this aspect and the scholar was aware of such link when introducing the equation of gravity in international trade. Although Tinbergen was also aware of the interdisciplinary nature of the model, as his education was more related to physics than to economics, the gravitational model he developed (and later generalized by Tobler in geography) lost its original econophysic connotation over time. Starting with Tinbergen’s formula for estimating bilateral flows, the focus of economics was mainly on the econometric development of the model and its empirical refinement to adapt it to the study of different cases in space and time. Nevertheless, Isard’s original model is not, as previously mentioned, a simple empirical exercise that uses three independent variables to calculate a bilateral flow. This seems to go against the view of many economists with respect to Tinbergen’s equation. Such perspective implies a diminished value of the word gravity itself, which should theoretically characterize this model. It would not make sense to apply the law of universal gravitation to explain only a theory of distance decay that could be sufficiently interpreted by Tobler’s first law of geography, according to which “everything is connected to everything else, but things near are more connected than things far” . Footnote 12 Certainly, this principle is not exclusive to the gravity model: the latter could be read on a spatial dimension and through the principle of contiguity and spatial correlation. However, Tobler's approach, based on this spatial dimension, results less specific since it focuses more on the concept of space rather than on the gravitational model. In fact, strictly related the gravity models is the concept of attractive forces, even in direct competition with each other where economic weight provides attractive commercial power. In short, Tobler's approach describes how a certain effect is diluted as space increases, whereas strictly related the gravity model is the specific concept of commercial force and commercial attraction, which fades as distance increases. Although this law was more related to geography than gravity itself, Tobler ( 1975 ) has later demonstrated his awareness of the dimension of physics. Through a detailed analysis of spatial models and flow implicit in a geographic interaction, the scholar used the gravitational model in relation to the gravitational fields generated by it, considering its magnetic potential field. He used a field of vectors computed from the related net exchanges, which explains the flows and approximates the gradient of a scalar potential. The introduction of a flow field, which is typical of physics, could be cartographically represented as a “wind” for the geographer. The concept of field and the equipotential lines already analysed by Isard ( 1954 ) and Stewart ( 1941 ) before him are then taken up again. In addition, Tobler analyses in a geometric and detailed way how to calculate the latitude and longitude coordinate of positions in this field.

The application of gravity to trade therefore encompasses a more complex structure and logic, composed of economic forces acting in space and principles of attraction, which in this case are commercial and spatial concentration. Footnote 13

Further developments of gravity model in international trade

As already mentioned, Isard ( 1954 ) was the first to apply the laws of gravity to international economics. Nonetheless, in 1962, the physicist Tingerben developed a gravitational model on the influence of distance in international trade with a spatial-level perspective, which became the most influential and famous work in this field. The gravity equation measured trade costs or barriers that were not yet fully considered in the classical theory of international trade. Starting with Tinbergen, the model began to be applied extensively to empirically test different markets. However, this large-scale application was progressively drifting from Isard’s original model based on theoretical econophysics. Based on these considerations, the following paragraph aims at examining the most important authors following a chronological order. This approach is deemed the most appropriate to analyse the evolutions that the gravity model has undergone over time from an interdisciplinary perspective.

The contribution of Pullainen, Pöyhönen and Linnemann

Pullainen, Pöyhönen and Linneman’s contribution has been critical since they further developed the gravity equation with an econometric and empirical prospect. The Finnish economists Pulliainen ( 1963 ) and Pöyhönen ( 1963 ), who were a competing research group with the one working at the Netherlands Economic Institute where Tinbergen was the director, used the gravity equation in their studies. Linneman was part of that group as well. He later became a professor in trade and development, first at the Institute of Social Studies in The Hague and later at VU Amsterdam, where he also followed in the gravity tradition. While Pulliainen considered a gravity model without commodity prices, Pöyhönen developed a gravitational model similar to Tinbergen’s one using a matrix form for the exchange of goods. The latter considered the GNP and the amount of population in a state as “masses”. His model can be summarized as follows (Zhang and Kristensen 1995 :309):

where \({EX}_{ij}\) is the size of a bilateral trade flow, the two \(GNP\) are the two countries’ gross national product, \({D}_{ij}\) is the distance, \({\alpha }_{1}, {\alpha }_{2}\) are income elasticities of exports, \({\alpha }_{3}\) is a transportation cost coefficient, \(\alpha 4\) is an “isolation parameter”, \({c}_{i}\) is an export parameter for the \({i}^{th}\) exporting country, \({c}_{j}\) is an export parameter for the \({j}{ \rm th}\) importing country, \(c\) is a constant.

As for Linnemann ( 1966 ), he studied the role of distance in international trade, looking for a quantitative explanation of commodity flows. Tinbergen at the University of Rotterdam oversaw his PHD thesis. He extended the gravity model using the Walrasian model, and he classified three categories of costs associated with trade: transportation costs, time-related costs (perishability, et cetera), and costs linked to cultural unfamiliarity (lack of familiarity with law, cultures, customs, and languages, et cetera). Linnemann elaborated the data coming from 80 countries and 6300 bilateral trade exchanges. He found that both distance, which was treated as a proxy, and population size negatively affected trade relations. This outcome was due to the fact that the most populated countries had a larger market and, hence, they were more self-sufficient. Therefore, the scholar's approach was econometrics like his teacher’s, and referring to Isard’s work he stated, “Some authors emphasize the analogy with the gravitation law in physics (…) we fail to see any justification for this” (Linnemann 1966 : 34–35). Indeed, he intended to emphasize the lack of empirical evidence in the theoretical physical model, and for this reason, he had calculated that the elasticity of trade flows to distance was not equal to 2. Professor Linnemann creates his own empirical trade flow model. In essence, he develops a model of the interplay between prospective supply and demand for traded products, together with what he refers to as trade-resisting factors. Both natural and artificial barriers to commerce, such as the significant element of distance, tariffs and other commercial arrangements, are included in the latter. It is presumable that country \(j\) 's prospective demand for country \(i\) 's goods will rely on country \(j\) 's GDP and population size. His international trade model can be elaborated as follows (Zhang and Kristensen 1995 : 310):

where \(EX\) is the bilateral flow, interaction between potential demand and potential exporter’ supply of traded goods, which is dependent to “resistance” to a trade flow as well, i.e. their cost of doing business. In his equation, \(GNP\) is countries' potential demand which is assumed to depend on that country's income or GDP. The two \(POP\) represent the two population size, \({\alpha }_{4}, {\alpha }_{5}\) are population elasticities of exports, \({\alpha }_{1}, {\alpha }_{2}\) are income elasticities of exports, \({\alpha }_{6}\) is an “isolation parameter”, \(D\) is the distance which coincides with the resistance to trade. The latter includes both natural and artificial barriers to commerce, such as the significant element of distance. Artificial barriers to trade include tariffs and other trade agreements. and \(PR\) is the potential preferential-trade factor which aims to strengthen the trade link between \(i\) e \(j\) and it equals to \(\left(PR\right)\) . Population is a crucial element here with respect to formula ( 10 ) above. Indeed, unlike GNP, population is included by Linneman at the denominator as a factor of resistance: a larger population reduces trade. This is because GDP per capita is taken into account (the higher the GDP per capita the more trade is promoted) and because an increase in population increases a country's internal trade magnetism. This is because larger countries are better able to be self-sufficient. It is worth noting that it has become clear, especially after World War II, that a country's international trade tendes to grow faster than its GDP. According to Linneman, this is partly due to a growth path in a dynamic equilibrium since new countries are accepted as members almost continuously.

Leamer ( 1974 ) used the gravity equation and the Heckscher–Ohlin’s model separately, in a regression analysis to explain variables of trade flows. He added factor-endowment variables, finding that they performed less well than the standard income and population variables. Leamer and Levinsohn ( 1995 ) integrated distance and comparative advantage into the H–O model. They argued that, as in the gravitational model, the majority of goods are produced close to the most important economic centres, which are also their destinations, whereas small countries, far from the central development zones, gain a comparative advantage by focusing on goods (such as shoes and clothes) easy to transport.

Anderson's theoretical derivation

As previously mentioned, Tinbergen’s bilateral gravitational model became the standard of the gravitational law’s econophysics application to international trade; indeed, Walter Isard's name is not even mentioned in large sections of international literature. This is probably due to the simplicity of Tinbergen's model, its resemblance to Newton's law, and its empirical usefulness in estimating bilateral trade between pairs of countries. For this reason, many scholars have evaluated the nature of the model merely based on the empirical and practical approach that Tinbergen adopted in his model. Thus, the strong stability of the standard gravitational model and its power to explain bilateral trade flows stimulated a ‘new’ theoretical exploration which, however, no longer included Isard's theoretical development. Anyway, it is worth noting that Head and Mayer (Head and Mayer 2014 ) individuate in Tinbergen’s model a lack of theoretical foundations which represented one of the main weaknesses that led to the limited acceptance of the gravitational model as a central element in the theory of international trade by the majority of scholars. Anderson (Anderson 2011 : 134) called the early gravity model “an intellectual orphan, unconnected to the rich family of economic theory. Shahriar et al. ( 2019 : 29), however, made a good point by observing that: “As a result of the previous studies the model, in no way, is an intellectual ‘orphan’ but rather is now connected to the rich family of economic theory”. Indeed, the criticism of the lack of theoretical roots does not consider Isard's theory but limits itself to considering only the developments of the empirical model starting from Tinbergen. The focus is now on guaranteeing that every empirical test of the gravity equation is clearly defined on theoretical grounds and that it can be linked to one of the existing theoretical frameworks given the abundance of models available. As a result, the most recent methodological advancements pointed out the importance of precisely defining the structural form of the gravity equation as well as the consequences of inaccuracy. In this context, the multilateral dimension of the gravity model is a first significant area of contributions.

Anderson ( 1979a , b ) was considered the first to attempt a theoretical derivation of the gravity equation and explain the role of income variables. Footnote 14 Before him, most scholars saw the gravitational model as a fortunate application of physics to economics. Footnote 15 In International trade theory, the gravity equation is one of the most stable empirical relationships, allowing it to accurately estimate bilateral trade flows. Anderson interpreted the gravity model envisioning an approach conducting cross-section budget investigations. There are efficiency improvements to trade off against the bias issues that have recently been shown, especially given how much transit costs vary. According to Anderson's research, due to the complexity of modeling trade flows, the gravity model may deserve more development and use, therefore any potentially effective technique must be respected. Anderson for instance, attempted its application to policy which resulted significantly hampered by its “unidentified” features. There is no theoretical reason for including policy instruments like border taxes in the equation and drawing conclusions about the impact of taxes based on how the equation has evolved over time without accounting for tax changes provides no assurance of correctness.

The price equality of traded goods model by Samuelson-Heckscher-Ohlin had been proven unrealistic for the differentiation effects arising from the borders of countries. Anderson assumed that preferences for traded goods are homothetic and homogeneous across countries. Furthermore, he assumed that goods are to be differentiated based on the country of origin. According to the gravity model, consumers spending on tradable goods is a steady reduced-form function of population and income. Transit cost variables determine the percentage of each tradable good category among areas. The scholar also assumed perfect competition, that consumer utility preference function is a Cobb–Douglas type, the consumer’s homothetic utility function, constant elasticity of substation (CES), a labour only production factor, the absence of trade costs and that each country produces only one particular good that can be substituted with another nation’s goods (Anderson 1979 : 106–16). Each country is assigned a particular GDP, avoiding the GDP being the result of an underlying production function. Since trade was separated from traditional causal factors such as technology or factor endowment, Anderson and van Wincoop ( 2004 ) used the term “separable trade theory” to describe this approach (Anderson and van Wincoop 2003 : 691–751). In their model, economists explain why only a fraction of goods shipped arrive at their destination by including iceberg costs. The rest has metaphorically melted during transport. If imports are measured at the CIF value, of course, transport costs reduce trade flows. As a result, in a condition of equilibrium and regardless of price, every country will consume commodities from any other country and every type of product will be traded internationally. National income will be equal to the sum of domestic and foreign demand for the single commodity produced by each country. Every good other than the one produced in that country will be imported. For this reason, as in the gravitational model, larger countries will have larger trade flows (Bacchetta et al. 2012 ).

Krugman resumes Isard's dream of synthesis between the theory of localization and international trade

Krugman, ( 1991 ) first with Fujita ( 1999 ), and then with Fujita et al. ( 1999 ), took his previous study with Helpman and Krugman ( 1985 ), indeed, through their theory of New Economic Geography, they began to study the effects of economies of scale, business clusters, technological innovation, and the digitalization of innovations. This new theory sought to explain how the variables of economic and geographical distance in international trade influence the phenomena of concentration and distribution of countries’ economic activity. In a world of rising returns and high transportation costs, there will be an incentive to concentrate production of a good close to its largest market. It may be more effective to locate manufacturing in a single place as close as possible to a bigger market to minimize transportation costs. That is why nations would tend to export the products for which they have a sizable local market as it is frequently stressed in location theory rather than trade theory. In this context, high domestic demand for an item will tend to result in an import rather than an export, which would always be more advantageous since it reduces the need for transportation by equalizing the manufacturing costs in both nations. This advantage needs to be counterbalanced by a salary difference to safeguard employee’s employment in both nations. Therefore, considering the transportation expenses, it can be demonstrated that the nations with more significant domestic markets will have higher pay rates. To sum up, nations will tend to export items for which they have sizable domestic markets as legal justification. On the other hand, smaller nations with far smaller marketplaces, must make up for this disadvantage by paying lower wages. The Heckscher-Ohlin theory and what has been explained so far have a link. According to Mundell ( 1957 ), trade restrictions like tariffs or shipping costs would prevent factor movements in an H–O environment where factor mobility would be similar.

Therefore, a general equilibrium analysis in trade is provided where factor movements, spatial dimension, and location problems are considered. Since in international economies distance and commercial costs, which affect either the flow of trade or the position of firms, have been mainly studied through the gravity equation, it is not surprising that the empirical studies related to Krugman's theory of New Economic Geography have made extensive use of the calculation of data and of the gravitational instrument. Therefore, the two models have frequently influenced each other indirectly. I do not intend to minimize the attribute’new’ in the proper name New Economic Geography . Krugman's theory certainly enriches the economic subject matter. However, in 1954 Isard, albeit in a different way, had evidently already addressed these topics. This scientific heritage is recognized by Krugman himself, who, with Fujita, quotes Isard ( 1956 ). Krugman recognizes his ideas as “a continuation, perhaps even a validation, of Isard's dream of returning space to the core of economics” (Fujita and Krugman 2004 : 153). As mentioned, Ohlin ( 1933 ) had already faced these problems with the following intentions before Isard. First, he presented the issue of overcoming the need for more attention of economic theory to the localization problems through a collaboration between economists and economic geographers to demonstrate that the theory of international trade is only part of a general theory of localization. Second, through this new perspective, the need to build a theory of international trade in harmony with the theory of prices and independent of the classical theory of the value of labour was highlighted, in which the differences in transport cost and in the supply of factors of production are considered. Finally, Ohlin identified how challenging it was to use this theory to simultaneously analyse the relationship between national and international movements of goods and factors of production. At that time, however, the lack of modelling techniques to combine increasing returns to scale with the analysis of general equilibrium did not allow Ohlin to satisfy these three objectives, which instead have been partially accomplished by the New Economic Geography . Moreover, this purpose of returning space to the centre of the economy is familiar for the gravitational model. Indeed, we have already seen Isard’s attempts to measure the intensity of the magnetism of a centre or a region through gravitational models in the previous discussion on the model’s origins. Furthermore, the economist had argued extensively about the need to arrive at a “superior synthesis between the theory of localization and International trade”. Krugman, for his part, examined business and space by recognizing two forces: centrifugal forces that push companies to compete to satisfy the demand in any market and region of the world, and centripetal forces which, due to increasing returns to scale, account for the agglomeration of companies. The wealthiest and most productive economic regions can attract more companies and workers thanks to economies of scale, knowledge, and migratory phenomena. Hence, the result will be a concentration of production in certain areas. There is a tendency on the part of industries belonging to the same sector to come together, thereby creating a more efficient labour market for companies and workers in that sector. This would benefit specialisation and, consequently, the positive spillover effects.

The study also analysed distances and transportation costs, arguing that these force companies towards locations close to large outlet markets. Moreover, it identified the “home market effect” as the tendency of businesses to operate in the country with the highest consumption of a particular good.

McCallum border puzzle: border effects

McCallum ( 1995 ) used the gravity equation to estimate the influence of national borders as a discontinuity of distance, and he pointed out that the border’s effect is an obstacle to trade. This topic was familiar to the gravitational model in economics or, more generally, to its applications in social sciences. In 1948, astronomer Stewart published Demographic Gravitation: Evidence and Applications in which he brilliantly developed a personal interpretation of physics and gravity applied to the study of demography, human behavior and relationships. His research was one of the most important theoretical basis for later studies on the use of gravity in all social sciences, to the extent that it inspired, as mentioned before, Walter Isard himself, to develop a gravitational model applied to trade. The astronomer, in his work, individuated the influence of borders and distance on gravitational models, which reduced the “mutual relationships” of flow in gravity models in many evidences. In this respect, Steward stated that “we might take this sort of thing as evidence that frontiers can produce the equivalent of a large increase in all the international distances, reducing the energies and potentials accordingly” (Stewart 1948 , pp. 55–56). McCallum's study has been followed by further research, which applied the gravity model to international economics to compare domestic and international trade. In his analysis of Canadian and U.S. trade, Footnote 16 McCallum ( 1995 ) empirically demonstrated that the inclination to trade is higher between neighbouring regions of the same nation as among bordering areas of different states. Although the latter is similar in cultural, legal, and linguistic terms without barriers at the border, it was found that the national border, ceteris paribus , had a negative effect on the exchanges between them. More specifically, McCallum’s study proved that the estimated interprovincial trade between Canadian provinces was 20 times larger (2, 2%) than between the U.S. and Canadian Provinces. The national border effect, which has become known in the literature as the “McCullum Border Puzzle”, has been one of the most discussed topics in international economics. Footnote 17 As a matter of fact, commercial costs increase at the border. There are transaction costs due to customs and other formalities, differences in legal systems, languages and monetary regimes, discrimination through tariffs on foreign products, and other non-quantifiable factors.

Remoteness: return to multilateral gravity

In physics, gravity is by definition a multilateral phenomenon, making it suitable for analysing trade relations between many countries, as Isard ( 1954 ) initially demonstrated. Nonetheless, the gravity model was initially used as a framework for studying only bilateral trade between two countries, as intended by the theories of Tinbergen ( 1962 ) in their standard version. However, the New Economic Geography by Krugman ( 1991 ) and the Study of the Effects of Borders by McCallum ( 1995 ) brought economists’ attention back to a multilateral approach towards international trade studies. Deardorff ( 1998a , b , c ) believed, for example, that the relative distance of trading partners would influence the volume of trade. Wei ( 1996 ), Helliwell ( 1997 ), Nitsch ( 2000 ) and Chen ( 2004 ) developed this approach and defined a variable capable of considering the third country effects. These forces, which are external to two trading partners, are otherwise known as a “third-country effect” and they have been repeatedly studied over time by other authors such as Bang ( 2006 ), Blonigen et al. ( 2007 ), and Baltagi ( 2008 ). Thus, for example, Australia and New Zealand were supposed to trade more with each other at the same distance compared to another pair of countries such as Austria and Portugal, which are close to numerous other countries and markets. This variable was therefore called remoteness . In a pair of partner countries \(i\) and \(j\) , it is defined as follows:

where \(z\) is, for country \(i\) , all trading partners other than \(j\) . \({Y}_{z}\) is country \(i\) ’s GDP and \({D}_{iz}\) is the distance between country \(i\) from any commercial partner other than \(j\) .

In the standard bilateral gravitational form, remoteness substitutes Footnote 18 of the constant K of the Eq. ( 7 ) which translated in econophysics terms the gravitational constant \(G\) of the formula ( 2 ), as follows:

Such transposition is theoretically adequate since in physics \(G\) is a constant value throughout the universe. Similarly, in economics, it would be an exogenous variable, given that the whole economic context would influence it. Unlike the cosmic system in the economic system, G can change over time because it is influenced by technological development and historical and social factors of its time. Some agents may influence the division of work, such as the use of the productivity of capital and labour and how they bring process or market innovations. Consequently, the impact of \(C\) would always be multilateral and tend to be constant in the short-term. At the same time, the factors of distance and countries’ GDP would be subject to asymmetric changes, as in the case of new bilateral trade agreements changing the distance between two regions. Unlike in physics, in economics the constant might change with symmetrical adjustments of the entire market taken into consideration. For instance, using digital currencies in the countries involved could positively impact trade by promoting internet purchases. Also, because of this solidity, A is assumed to be constant. Due to this characteristic of universality and consideration of the general context of a market, many economists tend to remove it for simplification or instead lead other researchers to consider it as an element of multilateralism present for a pair of countries, as in the previous formulae. In the economic model, however, remoteness is arbitrary since it describes only the effect of third countries and no other context elements. It represents an attempt to move away from the bilateral model of trade in formula ( 2 ), which had historically become the standard gravity model in the study of international trade from a multilateral perspective. Nevertheless, this approach indirectly resumes, logically and practically, the model put forward by Isard ( 1954 ). Indeed, by isolating the constant \(k\) in the Eq. ( 7 ), the outcome will be similar to the one obtained in formula ( 11 ). Footnote 19 Finally, the substitution of remoteness in constant \(C\) , formula ( 2 ), represents an effort aimed at introducing a multilateral dimension in a bilateral trade model.

Anderson and Van Wincoop’s multilateral model

Anderson and van Wincoop ( 2003 ) have criticized the remoteness indexes, whose theoretical bases were not deemed correct since they assumed distance as the only factor of resistance to trade. The authors argued that the underlying problem stemmed from physics in the case where a multitude of bodies, forces, and directions are considered. As in the case of remoteness, the scholars studied the multilateral dimension by means of a bilateral trade model, albeit with the promotion of an economic perspective rather than from the point of view of econophysics. Therefore, they focus on territorial boundaries and proceed to assess multilateralism through multilateral resistance. Footnote 20 To this end, they used a gravity model with the consumer’s utility function of the CES type (Constant Elasticity of Substitution) and iceberg-costs. They questioned whether only bilateral transaction costs influenced trade flow, or adjustments due to other countries’ influence were appropriate. Intuitively, the higher the resistance to trade with other countries outside the pair, the more likely the pair is to trade. In this regard, the two economists defined:

Outward multilateral resistance as the set of barriers and factors that hinder all exports of a given country in all target markets.

Inward multilateral resistance as all the barriers and factors that hinder a country's imports from another country.

According to the two scholars, leaving these factors out of an empirical gravity model makes the econometric estimates inconsistent and biased since it would amount to omitting variables. Multilateral resistance denotes the average resistance to imports and exports trade between a country and any possible trade partner. In practice, focusing only on the bilateral dimension between two countries is no longer possible. It is necessary to move to a multilateral analysis by examining multilateral resistance factors . The latter shows that comparing relative trading costs is far preferable to simply comparing absolute trading costs, as already partially seen in the gravitational model and in the connection that this one has with the comparative advantages in Ricardo ( 1817 ). Relative trade costs determine trade, that is, a country's propensity to import from another country based on two ‘resistance’ factors: firstly, with respect to the partner’s imports and, secondly, related to the second country’s exporters. Supposing a country pair is isolated from world markets due to geographic distance, high tariff barriers or other trade costs. In that case, this will result in low multilateral resistances and high trades between the two.

Anderson and van Wincoop ( 2003 ) used an empirical cyclical convergence process by trial and error to calculate multilateral resistance. The commercial costs were estimated first in an equation without resistance terms. Then these commercial costs were used to construct a series of multilateral resistances to be inserted into a new regression equation, restoring the new commercial costs. The latter was subsequently reused to build new multilateral resistances, as in the initial phase. The process would repeat itself until the gravity estimates were the same.

In summary, Anderson and van Wincoop’s gravity model can be rewritten as follows (Anderson and van Wincoop 2003 : 5–12; Yotov et al. 2016 : 15–16):

where \({X}_{ij}\) is the trade flows from exporter \(i\) to destination \(j\) ; \({y}^{W}\) is the world GDP; \({Y}_{i}\) and \({Y}_{j}\) are the GDP of country \(i\) and country \(j\) respectively; \(\sigma > 1\) is the elasticity of substitution among goods from different countries; \({t}_{ij}\) is the bilateral trade frictions between country \(i\) and \(j\) , which considers various trade variables (bilateral distance, tariffs, currency, regional trade agreements, et cetera); \({P}_{j}\) is the inward multilateral resistance and denotes importer \(j\) ’s ease of market access, and \({\pi }_{i}\) is the outward multilateral resistance and represents exporter \(i\) ’s ease of market access. The formula shows the total effects of trade costs in three components ( \({t}_{ij}\) , \({P}_{j}\) and \({\pi }_{i}\) ), which creates a wedge between actual trade and frictionless trade. According to all of our theories, bilateral trade flows are determined by more than simply bilateral trade costs and exporter and importer incomes; what also counts is the so-called “bilateral resistance”. This indicates that trade between any two countries is determined not just by their respective incomes, but also by the “cost” of dealing between those countries in comparison to trading with all other countries (Anderson and Van Wincoop 2003 )

Adam and Cobham ( 2007 ) distinguished multilateral resistance terms (MRT) from bilateral resistance terms (BRT). Looking at a pair of countries, the former would identify the barriers of each of the two countries with the rest of the world; the latter would instead indicate the barriers between the two countries. There would therefore be a compensation effect in a zero-sum game . Basically, Adam and Cobham's distinction is semantic, i.e., it classifies trade barriers between two countries through the terminology of resistance terms . For instance, in the context of the exchange between Italy and France, an increase of the United Kingdom barriers to the exchange, with the same level of BRT IT-FR , would determine a growth of the MRT IT-FR and a consequent intensification of the exchange between Italy and France to the detriment of the exchange with the United Kingdom. Footnote 21 The evaluations of Anderson and van Wincoop consider a multilateral context. In other words, it is not simply considered a bilateral model but one looking at numerous states. Anderson and Van Wincoop modify the model by adding multilateral resistance factors to the bilateral model. However, this has no logical connection either with Isard’s original model, which was already multilateral, or with physics, since to assess the effect of several masses on two masses, multilateral resistance factors are considered, which have no transposition to physics. To that end, adding new and external elements to gravity without conceptual links to the original model is controversial, though it might seem logical.

Anderson and van Wincoop’s model explains international trade from a multilateral point of view by adding multilateral resistance terms to the gravitational model. However, unlike the classic Isard model, which evokes gravity through the transposition of masses and distances to GDP and distances of the countries considered, it does not provide any theoretical transposition of multilateral resistance factors to the force of gravity in physics. There is, therefore, technically no valid motive to continue to use the term ‘gravitational’ to describe an econometric model which, although relevant from the first application of Isard, is no longer inspired by the force of gravity . Footnote 22 It is reasonable to suppose that this trend, evident in most subsequent authors, was used more for historical reasons and for the suggestive power of the term ‘gravity’, rather than to link the two fields of study, thereby losing its original evocative meaning. Furthermore, the multilateral approach can be already observed in Isard’s formula ( 7 ), Footnote 23 which summarizes not only the effect of two countries but of a multitude of them on nation \(i\) . Therefore, in a multilateral analysis of the gravitational model, there is no reason to start from the standard bilateral formula presented by Tinbergen ( 1962 ) in the equation: \({F}_{ij}= A \frac{{Y}_{i}{Y}_{J}}{{D}_{ij}}\) (Deardorff 1998, p. 9) and transform it econometrically to explain more complex multilateral phenomena without considering that the first transposition of gravity into economics was already, albeit in a theoretical way, multilateral Footnote 24 ; the roots of multilateralism of Isard, however, were not mentioned.

Anderson and van Wincoop ( 2003 ) also aimed at answering trade border issues, also known as “McCallum’s border puzzle”, by estimating the gravitational equation using multilateral resistance variables and McCallum's exact specifications and his own dataset. According to them, McCallum's idea of thick boundaries ignored the asymmetric impact of barriers on trade between small and large economies and multilateral protection levels. The results concluded that three reasons could explain the effect on borders: considering the effect of borders by comparing intra-national with international trade; the effect of borders is more remarkable for small nations; the variables not considered push the estimate of effects of the borders to increase. Overall, the authors estimate that borders in industrialized countries reduce trade by an average of 29%. Specifically, as regards Canada and the United States, the scholars found that the border from the Canadian point of view was ten times stronger than that of the United States. With Canada's economy about one-tenth that of the United States, the level of border protection appeared to be a positive function of the economic size of a state. Helliwell ( 1996 ) confirmed such results using data from the province of Quebec, which would suffer a disadvantage in the case of potential separation from Canada.

The limits of the multilateral approach of Anderson and van Wincoop, and McCallum before them, in international trade, are manifold. Firstly, as seen in Isard's formula ( 7 ), it is unnecessary to look beyond the gravity model to evaluate a multilateral context. The exact opposite is true. Once again, the analysis is affected by its standard econometric approach, overlooking the fact that the application of physics to economics is at stake. Secondly, it would be more appropriate to limit the effect of borders by considering them as part of the distance factor. While not perceiving them sometimes as such—hence the enigma—boundaries are a resistance factor which affects the distance from psychological legal, cultural, and many other points of view. Even when these elements are challenging to identify, it does not mean that they are not there. This observation is partly linked to the problems of endogeneity and reverse causality that can occur due to multilateral resistance factors, when trying to estimate the impact of regional trade agreements (RTA) on trade flows. Countries are likely to form RTAs with partners with whom they have a shorter trading distance. There is not a theory of customs unions that incorporates gravitational forces yet. The closest example is the work of Bikker ( 1987 ), who attempted to extend the gravity model to include substitution between trade flows coming from different directions or sources. This would theoretically allow for the analysis of trade production and diversion within inside a gravitational framework. However, this latter lacks a strong theoretical basis and is unable to characterize substitutions between flows. This distance is not only geographical, as previously mentioned, but representative of many proximity variables, such as the legal system, diplomatic relations, language, technological level, colonial heritage, et cetera. While a global, regional analysis requires multilateral terms by definition, it is also true that these terms are not entirely exogenous because they result from the effect of global distances being considered. Consequently, the RTA or MTR elements on the right side of the gravity equation are related to the error term because they depend, in part, on distances. Third, multilateral resistance factors do not explain at first sight which country in a pair will benefit more from a multilateral exchange, as in the case of an agreement with a third country or a new free trade area. Footnote 25 Fourth, evaluating multilateral factors in terms of their individual and combined impact in a multilateral equation is difficult. Finally, it is not logical to explain the border effect through resistance factors that depend on other countries ( third-country effect ). Borders influence bilateral trade between a pair of countries, regardless of how bilateral trade is affected by surrounding countries simultaneously, thus representing a choice of comparative advantage or opportunity cost. Footnote 26

Most recent studies

Before delving into the most recent studies in the field, what follows is a general consideration of the newer models that adopt more sophisticated statistical tools. In this section, judgements on the structure of exogenous variables are made through statistical software packages and a variety of fixed effects. Indeed, since this section aims to provide a general overview, it will not linger on the details of each study. With these premises, the first contribution that will be considered is that of David Huff, professor of Marketing and Geography. Indeed, to estimate market share and retail appeal, he proposed a model that bridges geography and business. This model became popular over the years for its ease of use and applicability to various problems. Market analysts have extensively used it to locate stores, shopping malls, standard models for the industry and other types of retail establishments. The economist and geographer David Huff ( 1963 ) re-applied gravity to consumers’ behaviour.

A subject's perceived utility of alternative retail is inversely proportional to its attractiveness. A client's space, for instance, refers to the likelihood that he or she will shop at a specific establishment compared to rivals. This likelihood is determined using distance attractiveness characteristics, as well as the competition. Huff's standard model has no differentiation of goods, no consumer feedback, and no alternative items.. The attractiveness of the retail outlet at location j on consumer demand at location i is defined as:

The probability that this consumer demand at location i is satisfied by the store j will be equal, as already seen in Reilly's Eq. ( 4 ), to the relative attractiveness of j compared to the attractiveness of other stores. As in Voorhees' model ( 1956 ), it is worth recalling that x is an exponent reflecting the effect of distance decay. Hence, total demand is distributed by multiplying expressions ( 15 ). Huff, therefore, developed a distribution model of consumer demand in space drawing on Voorhees' model. Voorhees ( 1956 ), who was attempting to forecast Baltimore traffic patterns, used gravity to trip distribution. His theory was the first to use the gravitational principle to estimate the number of trips between zones in urban settings. According to gravity formulas, the number of trips is inversely related to distance, and size variables are taken into account for both locations, such as

Voorhees created a distribution model based on such formulas, dividing the estimated number of trips from the origin area by the destination area. This estimation is inversely connected to the required journey time and proportional to the attraction each location exerts. It has the following formula (Philbrick 1973 , ibid):

where \({T}_{ij}\) is the trip between zone i and zone j , \({O}_{i}\) is the number of trips generated in zone i , \({S}_{j}\) is the attraction force of land j , \({d}_{ij}^{x}\) is the travel time between i and j , x is an exponent determined by observation, n is the total number of trips including those directed to j . Similar to the latter, who chooses a spatial distribution of trips based on the attractiveness of the soil, Huff also considers a spatial distribution, which is related to the attractiveness of the commercial area. Huff's probability model is as follows:

where P is the probability that a consumer at point i goes to the store located in j , is a measure of store attractiveness (such as store size), is the travel time between i and j , x is an exponent determined by observation, n is the total number of stores that also includes the store j . The larger the size of a store, the greater the level of attraction exerted on the consumer. The probability that customers choose that store decreases as distance increases.

However, considering a probabilistic gravity model could be wrong. Indeed, classical econophysics and gravity models follow a deterministic approach with significant simplifications, which seem to be necessary in a heuristic discipline such as economics. This approach has solid neoclassical economic foundations. This latter is based on the concept of general economic equilibrium and mathematical utility maximization, which finds its roots in Newtonian physics. The notion of general economic equilibrium, developed by engineer Léon Walras ( 1899 ) and physicist Louis Jean Baptist Bachelier ( 1900 ), is a quantitative formulation capable of balancing an economic system's supply and demand equilibrium. Mathematically, equilibrium corresponds to locating the maximum point of a suitable function.

The two scholars conceived it as a physical equilibrium between two balancing forces such as keeping the moon in a stationary orbit around the earth. The resultant of equal and opposite forces is an equilibrium situation for variables endogenous to a system that can be changed only by a change in exogenous variables. The study of gravity models through a probabilistic view poses limits in physics to the determinist approach typical of Newtonian physics and with respect to the neoclassical paradigm, the results of which, although based on simplifying premises, constitute a solid mathematical formalisation for the analysis of economics. Although, therefore, a non-determinist approach based on probability can be very stimulating and intriguing, it will not be explored further in this paper. The reasons are that one needs to find this approach robust for the gravitational tool. Gravity in physics, from Newton to Einstein, is inherently reductionist. It is still so considering that a reconciliation between gravitational interaction and quantum physics has yet to be developed, and although modern physics is leaning more towards a holistic view and has validated Bohr's Atomic Model, the questions raised by Einstein are current and still unsolved. For this reason, the term 'holistic' is inappropriate when referring to a gravitational model. Other physical tools can be more appropriately applied to economics to evaluate probability-based models.

Evenett and Keller ( 2002 ) explained that the success of the gravity equation was due to its application to the H–O model and the increasing returns to scale, with the differences in factor endowments and intra-industrial trade explaining the production levels and volumes of international trade. According to them, the data do not theoretically support the assumption of perfect specialization. The H–O model foresees the perfect specialisation of production in different countries but only for large differences in the allocation of factors. In the case of increasing returns, the goods will be more differentiated, as intra-industrial trade will be greater.

Baldwin and Taglioni ( 2006 ) identified three errors in order of importance in the theoretical economic derivation of the gravitational model. The first error, called golden , concerns the flawed estimation of the commercial costs due to the failure to consider Anderson and van Wincoop’s multilateral resistance factors or Head’s remoteness effect. This is often the case, because of the scholars’ need for directly observable multilateral resistances. The second error, called silver , refers to the fact that trade should be analysed separately according to its direction at a given moment, and considering that trade from country \(a\) to country \(b\) may not coincide with trade from \(b\) to \(a\) . Finally, the bronze error consists, in the opinion of the researchers, in considering trade flows deflated due to their evaluation based on the aggregate price in the United States.

Helpman et al. ( 2008 ), Chaney ( 2008 ) and Melitz and Ottaviano ( 2008 ) derived the gravitational equation using differentiated goods and constant firm heterogeneity, thus demonstrating the compatibility between Newton’s equation and Melitz’s model. Melitz and Rubinstein ( 2008 ) raised the problem of considering the effects of trade barriers on the value of exports and imports and on the entry of exporters. Chaney ( 2008 ) calculated this on the supply side, for exports of final goods, continuing the study of Melitz ( 2003 ) and assuming asymmetrical countries with different sizes, trade flows, labour costs, and trade barriers. The scholar demonstrated that trade flows have a large margin of adjustment to barriers and that distance effects are determined by the degree of the firm’s heterogeneity, the elasticity of substitution between goods, and the elasticity of commercial costs to distance. Therefore, the effect of distance on trade flows will not only depend on the elasticity of substitution between goods but also on the degree of the firm’s heterogeneity. Helpman et al. ( 2008 ) examined the demand side, verifying trade barriers and policies’ impact on flows. They developed a selective gravitational model, which provided both positive and zero bilateral flows between pairs of countries due to selection and business competition. The model involved two phases in two equations. First, a pair of countries decide whether it is convenient for them to trade or not ( extensive margin ); second, they decide on the volume of exchanges ( intensive margin ). According to Helpman, the recent increase in world trade is mainly due to intensifications in the exchanges of traditional trading partners. Egger and Pfaffermayr ( 2011 ), in their structural estimation of the gravity model with path dependence of country-pair level exporter status, pointed out to this persistent effect in a study, which stated that 66% of the pairs of countries that made exchanges three years earlier still exchanged with each other. Furthermore, the 20% of couples with zero exports also had no exports 3 years earlier, and only 13% showed a significant change in the average levels of trade in 3 years. According to Egger and Pfaffermayr, access to a new commercial market required entry costs ( sunk cost ), which would favour traditional exchanges. They also submitted that the third country effect on the exchange rate in a pair of countries would imply, albeit adding a certain delocalization effect on the business side, a positive flow variation in the country chosen by the company as the new location and a negative variation in the country of origin of the enterprise. This last point should be given a lot of weight. It can be placed in a geographic economy through the choice of location for an economic agent, which according to the authors depends on the comparison between the costs of exchange with one country rather than another.

Anderson and Yotov ( 2010 ) introduced a sectoral gravity model to estimate the trade of specific commodity classes and multilateral resistance variables by equating fixed effects with structural gravity counterparts. Yotov ( 2012 ) analysed the effects of globalization on the reduction of costs related to trade using the gravity model, dividing between the costs of trade within countries and costs of international trade . According to him, globalization leads to a decrease in the costs of international trade compared to the costs of domestic trade , and to a consequent decrease in the thickness of McCallum's borders. Olivero and Yotov ( 2012 ) proposed a dynamic theoretical gravity model for prediction with asset accumulation in the presence of panel data, in which exchange values were delayed as regressors. They observed a “commercial persistence ” that included a delayed decline in gravity patterns. Indeed, over time there would be a “ persistence effect of protection ” of positive trade barriers through the accumulation of capital, which would be marked by an increase in the country's production, as well as in its mass, and consequently in international trade. The researchers also showed that multilateral resistance effects should be considered as fixed effects over time.

Arkolakis et al. ( 2012 ) analysed the situation where trade is balanced, profits are a constant share of GDP and the import demand system is CES, thereby showing that a large class of quantitative trading models generate isomorphic gravity equations (equivalent in structure between them) with respect to gains from trade. Therefore, the gravitational model can calculate the welfare gains from trade in international economics.

The flat world and the death of the gravity model

Recent research has focused on the end of the gravity model, as well as on the transposition from the sphere of physics to the virtual sphere, and on the less tangible flow of goods. Footnote 27 Therefore, this branch of studies has interested the economic literature. As discussed, the gravitational model in particular has recently moved its interest in social sciences from the study of flows of things towards that of less tangible things. With regard to this, Castells ( 1996 ) has applied the concept of “space of flows”, and, Batty ( 1997 ) and Gorman and Malecki ( 2000 ) have introduced the areas of “cyber geography” and “virtual geography” with the aim of describing a new virtual space, which extends beyond the geographical one. Trade studies have also concerned complex systems and, inevitably, also digital networks. On the one hand, the interest of scholars has focused on the impact of digitalization on the different forms of distance, including the physical one (Reggiani et al. 2010 ; Tranos 2011 ). On the other hand, the consequences of economic and geographical distance on the digital arena have been at the centre of attention (Simini et al. 2012 ; Newman 2003 ; Watts 2004 ).

Nonetheless, Cairncross ( 1997 ) has questioned the application of the gravitational model to trade as he put forward the concept of “death of the distance”. Along the same lines, the notion of “flat world” by Friedman ( 2005 ) suggests the inexistence of distances, thereby implying the disappearance of income and wealth as factors of concentration. Footnote 28 Admittedly, overlooking distance would mean, from a gravitational perspective, also a diminished concentration force of the masses. This could be transposed in macroeconomics terms as a reduction of wealth since the masses can be comparable to the GDP of nations (their total income). Therefore, the absence of concentration results in perfect distribution. It follows that the gravity model would end if the “death of distance” was considered.

In this regard, Vozna ( 2016 ) examined the market through the lens of entropy to determine richness. This approach stems from Wilson ( 1970 ), who applied the notions of entropy and the “law of diminishing returns in economics”, intended as dispersion of resources and gains in productivity. Wilson’s entropy of urbanization is similar to the one introduced by Vozna with respect to dispersion of wealth. In spatial terms, the dispersion of profit can be associated in the real world with the notion of dispersion of wealth. The same applies to the concept of equality both in the field of capital among economic actors, and among subjects of different countries. It is, therefore, unsurprising that equality and dispersion dominate in the absence of distance (the latter having a single value). According to such model, the maximum of entropy generates a more competitive market. It could be argued that such analogy associating capital and distance is forward-looking. Indeed, in macroeconomics, in the long-run, profit is equal to zero in the case of perfect market competition. It follows that the “cold death of the Universe”, which is obtained with the maximum entropy in the context of physics, may be related to the already mentioned “death of distance”. Footnote 29 In both cases, there seems to be opposition to the gravitation model on the one hand, and to the economic notions of concentration of wealth and income on the other hand. Footnote 30 As a consequence, a “flat world without distance”, which entails a diminished concentration of income and wealth, originated in a perfect market equilibrium. Recently, scholars in the field of technological progress and digitalization have suggested that distance is economically relevant in the current world. This is because of the increased diversity and complexity of society and space, where access to the internet is diversified in accordance with its literacy and diffusion. However, Rietveld and Vickerman ( 2004 ) argued that the hypothesis of “death of distance” is untimely. Indeed, travel is not yet a disadvantage, and furthermore, the diversified preferences for goods and activities could be related to increasing real incomes. Moreover, the transformation of activities and transport models can influence the choice of residence, family, work, and leisure time. To conclude, McCann ( 2008 ) combined the notion of “flat world” with the one of “curved world” aiming at overcoming restricted perspectives under the aegis of Economic Geography , which brings together globalization and localization. As seen before in this paper, Tranos and Nijkamp ( 2013 ) applied digitalization to the gravity model. Tranos and Nijkamp ( 2013 ) were therefore able to overcome the new reality of cyberspatial and distance on the internet by applying digitalization to the gravity model. Disdier and Head's ( 2008 ) thorough meta-analysis of 103 studies and 1467 estimated friction elasticity values in gravity analysis with a negative "power function", showing that the distance friction, i.e. elasticity's absolute values β, and the volume of trade flows have an inverse relationship, as shown by their weighted arithmetic mean (0.855) and arithmetic mean (0.907). Moreover, the scholars reveal that the effect of distance tends to grow over time, as shown by the growth of β the as it can be seen in the Fig. 2 .

Source: Disdier and Head's ( 2008 )

Elasticity's absolute values (β) of distance effect.

These findings contradict the hypothesis that technological advancement and information and communication technology (ICT) have diminished distance’s role in spatial connections. Footnote 31 In doing so, we can conclude that the gravity models still have a “raison d’etre” in the modern world, not disappearing in the face of new circumstances but changing over time.

In economics, the question of a flat world can be viewed econophysically as in the formula ( 18 ). The commercial interchange would expand exponentially at a distance close to zero. Indeed, the concept of globalization has caused exports to increase on a global scale continuously. A challenge that may be the inverse of the death of distance is the problem of trade flows between countries being equal to zero. Zero gravity does not exist in nature. However, there are places in the cosmos that are very remote from astronomical sources where gravity is feeble, or there are points of balance between forces.

For example, we can think of three of the “three-body problem” studied by Lagrange ( 1772 ) in which two bodies with a large mass, through the interaction of the respective gravitational force, allow a third body with a much lower mass to maintain a stable position relative to them. Alternatively, we can think of a case in which a balance is generated between the gravitational and centrifugal forces which equal each other. In economics, likewise, we will have to have enormously large distances as in the case of a country embargo or the case of a remote country with no contact with the civilized world or for other reasons. There can also be a zone of equilibrium between forces, for example, considering the centrifugal one already discussed by Krugman. However, this is different from flows equal to zero. Bikker ( 1982 ) and Linders and De Groot ( 2006 ) have specifically looked at the issue of zero-value flows in the log-linear model, which cannot correctly account for the occurrence of zero-value trade flows between pairs of countries. They examine the various methods for dealing with zero flows and make the case that solution choice should be based on economic and econometric factors. The “zero problem” in trade flow analysis is traditionally overcome by excluding any trade flows with zero value, which frequently yields acceptable results. However, by excluding all zero-value flows, crucial data regarding low levels of trade are kept out of the model (Eichengreen and Irwin 1998 ). Another option is randomly adding a tiny positive number to ensure the logarithm is accurately defined for these trade flows. However, the choice of this number is often arbitrary and needs more theoretical and empirical support (Linders and De Groot 2006 ). Other options use different Tobit estimation extensions, reduced regression, and probit regression. The sample selection model better matches both factors. However, the distribution of these zero-value must be random or sample truncation may result in skewed results.

Conclusions

This paper has covered the existing theoretical formula around the gravitational model from its origins to the present time. The principal authors have been critically reviewed, from discussions around the first theoretical model in social science in trade to the multilateral and digital economy models. The implications and deep roots of the model in international trade flows have been widely discussed. The gravitational model’s strength in competitive and oligopolistic markets has been demonstrated.

What emerges is that the gravity equation represents a suitable analysis framework for multiple models and markets. This is true for bilateral and multilateral perspectives. Wide attention has been given to the study of Walter Isard, theorist of the gravitational model in economics, with a complex description of his approach and discipline. It should be noted that the models examined did not always treat Isard’s formulation with an orthodox or econophysics approach. A discussion of the fact that the scholar has been wrongly neglected in the gravity model literature was also conducted, showing how this error was the cause of several subsequent theoretical simplifications.

The major evolutions of the model has been considered, and an attempt was made to summarise their usefulness, sometimes criticising their deviations. An approach to physics has been preserved. The link between the model and economic geography was also underlined. The relevance of the theory for the area of econophysics has been considered, and discussions were made about the birth of the model in physics and its application to economics. In this respect, Tobler’s first law of geography was also illustrated for its great transversality. The latter is more general and straightforward than the gravity equation, which is an aspect well worth noting. In most of the developments in the gravitational model seen in economics, it has been found that these evolutions transcend gravity, its background, and the genesis in physics. They are, in fact, more similar to Tobler’s first law of geography applied to economics or to econometric modelling of reality, rather than to the attracting forces in trade or an econophysical link to Isard’s original model. In these studies, the word ‘gravity’ continues to be employed for its historical and suggestive power, but no longer for its real meaning, since these evolutions being in essence, advanced stand-alone models. Over time, various econometric evolutions of the gravity model have been introduced to explain variables which were not addressed in the original model. This has further refined its predictive value and broadened its scope. In this regard, therefore, these evolutions of the model go well beyond econophysics. Indeed, they are suitable frameworks of economic geography for the study of space in economics by means of the “friction of distance” or Tobler’s first law of geography.

Contrary to this background, the concept of distance in the gravity model has acquired greater relevance concerning the notions of “flat word” and “death of distance”. The focus of the recent literature on globalisation and digitalisation as the applications of gravity to economics has been analysed with closer attention to the concept of ‘entropy’. Finally, the need for future authors to expand upon Isard’s work and integrate economic geography with international economics using the gravitational model was discussed. However, it has been highlighted that this path was not entirely ‘new’ in literature but was present since its inception. Related themes and elements were new economic geography, multilateral resistance factors (MRT) and the border puzzle , whose strengths and weaknesses have been reviewed. In conclusion, Isard's theoretical analyses and his raised issues are still relevant and have not been completely resolved. From the above, it follows that no evolution of the gravitational model in literature is yet capable, in its multilinearity, of an effective synthesis between economic geography and international trade, with the ability to create a “superior theory of international trade”, as Isard had sought to elaborate. Therefore, further efforts in this direction would be desirable.

Data availability

As this article is a literature review, no new data were generated or analyzed in this study. All information sources used in this study are cited appropriately in the reference list.

Usually is a constant of proportionality, but in several situations, is used to indicate the alternative in the world market. If a country has many alternative partners, it will tend to trade little with each by splitting its flows is a constant in Feenstra ( 2016 ). According to Yotov et al. ( 2016 : 17), is the inverse of world production (1/Y world).

Economic masses are not limited to identifying only a couple of countries, but they can also be used to analyse the exchange between free trade areas, cities, regions, single markets such as the EU or economic zones such as South America or Southeast Asia. Other scholars treat it as the aggregate expenditure of a country rather than as its GDP (ibid).

Like Reilley, Stewart considers straight line distances, while providing several examples where distance should be otherwise interpreted.

A significant number of papers in the literature reports this misleading information, which I find incorrect.

Analogy with Newton's law of gravity model: just as the gravitational attraction between two objects is proportional to the product of their masses and decreases with increasing distance, trade between two countries is proportional to the product of their GDP and decreases with increasing of their distance.

Isard speaks of ‘economic distance’ , which must consider the relative movement of goods on different means of transport.

Indeed, at the time of Isard, several scholars had already tried to develop a model of multilateral international trade; among these are Machlup ( 1943 ), Frisch ( 1947 ), Metzler ( 1950 ), and Hansson ( 1952 ).

The attentive reader will not be surprised, indeed, that Tinbergen is considered to have been one of the founding fathers of econometrics (Magnus and Morgan 1987 : 117–142).

The interested reader is addressed to Tobin ( 1997 ) and to Dekker ( 2021 ).

Moreover, in gravity models, the idea of determinism in human beings’ actions must be accepted to some extent. Indeed, in an econophysical gravity model, human interactions must be governed by universal laws.

Tobler and Wineburg ( 1971 ) using the gravity model estimated the unknown locations of several Assyrian settlements for which commercial data between 1940 and 1740 BC were available. According to the first law of geography, the idea was that the more settlements traded with each other, the more similar they were in terms of position (Tobler and Wineburg 1971 ).

Beyond the proposed conceptual division between ‘structural gravity’ and ‘gravity’ as a simple tool for calculating a flow (the latter being more similar in a certain sense to a distance decay spatial model), it is worth remembering that Tobler had also drawn inspiration for his law from gravitational models using an econophysics approach contributing greatly to other works and to the development of the gravitational metaphor. Indeed, the geographer (1970) also developed a reverse gravitational model capable of deriving the position from interactions. In doing so, he was reversing his law with flows, such as data values and distance, as an unknown variable, thereby creating a tool for analysing not only flows, but also society. He stated that: “all things can be located on a map so that similar things are brought closer than less similar things ” (Tobler 1970 :537–542). In addition, a few years later in another brilliant work ( Spatial Interaction Patterns , 1975) the scholar enriched the gravitational model of structural and logical elements using mathematical and vector calculus, demonstrating a thorough understanding of physics.

This should also help understand Isard’s ambition to develop a superior theory of international economics, which integrates the theories of commerce with those of localization in economic geography. Isard, founder of the gravity model in economics, is also credited with founding regional science. It is no coincidence that the gravity model is a tool for spatial analysis, industrial location, urban development and international economics. However, few scholars have really understood this comprehensive potential.

See more in: (1) Yotov et al. ( 2016 : 12); (2) Anderson ( 2011 : 1).

The interested reader about this “fortunate empirical validity ” is referred to Reinert et al. ( 2009 : 568).

McCallum used data on interprovincial and international trade by Canadian provinces for the period 1988–1990.

Feenstra ( 2002 ), Cheong, and Kwak ( 2015 ), Magerman, Studnicka, and Van Hove ( 2016 ) and Carter, and Goemans ( 2018 ).

The interested reader can find mathematical passages in: Head, Keith ( 2003 : 2–4).

Mathematically, \(k={iV\left({\sum}_{j=1}^{n}\frac{{Y}_{j}}{{d}_{ij}^{a}}\right)}^{-1}\) .

Scholars report significant differences in the definition of remoteness and gravity obtained through multilateral factors of resistance. According to them, remoteness is a non-theoretical measure and an inadequate attempt to monitor multilateral resistance (Anderson 2011 : 3).

United Kingdom, which in this case represents the rest of the world.

As already discussed, the first application can be traced back to Tinbergen. Moreover, at this point, it would be more accurate to speak of multilateral resistance terms, for example in Tobler’s first law of geography model applied to trade.

Isard (1954: 305–320) had already uncovered the need for a multilateral study of international trade, which was already taken into account in his formula of gravity.

The economic modelling started with the subsequent simplified formula by Tinbergen (2.1).

Dai et al. ( 2014 ) considered the deviation effects of free trade agreements through the gravity equation, outcomes that FTAs divert trade within the parties of the agreement and disadvantage countries that are not members of it.

The choice made will give a precise direction to trade flows, as we have already seen in Ricardo and for some aspects in Smith.

At the end of the twentieth century, the economic debate was focused on the pivotal role played by globalisation and digitalization and, in particular, on a major consequence of these phenomena which is the lack of borders among the countries. Even though McCallum ( 1995 ) demonstrated the economic relevance of national borders by empirically using the gravity equation, the idea of a world without borders has never abandoned the economic debate.

It should be noted that Samuelson ( 1949 ) had admitted the presence of transport costs and the limits in achieving stronger economies of scale (in a restricted market with the value of transportation costs’ different from zero) as to illustrate the motives behind differences in wages across nations. Indeed, the lack of such costs allows the equality of wages in a model composed of two countries.

William Thomson, a British physicist, introduced the notion of “cold death of the Universe” in 1851. Such expression refers to the energy’s distribution in space in a uniform way, which was generated by dispersion. Therefore, from a practical point of view, the universe’s equal temperature created the conditions for maximum entropy. As a result, all the processes of energy, such as life, could not occur anymore. It may seem counterintuitive to consider a level of maximum entropy when the distance is zero. Indeed, in physics, a distance equal to zero leads to maximum concentration (big bang). However, in economics, considering Vozna’s case, entropy is measured in terms of wealth, so we will observe that a reduction in distance leads to a better distribution of wealth and a more competitive market where profits, but also average total costs, are minimized. We are going to have zero profit in the long run for individual companies, but we are also going to reach the point of maximum exchange and maximum profit for the economy as a whole, as in the formula ( 18 ).

Both in physics and in economics, the gravity originating from the mass is the force of attraction and concentration. Nonetheless, distance is the resistance factor that balances the concentration force. It results that entropy and distance are interrelated, given that the maximum dispersion and entropy of the masses occurs with greater distance among them.

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I thank Prof. Adalgiso Amendola (University of Salerno) and Prof. Sami Bensassi (University of Birmingham) for their support and helpful discussions.

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Capoani, L. Review of the gravity model: origins and critical analysis of its theoretical development. SN Bus Econ 3 , 95 (2023). https://doi.org/10.1007/s43546-023-00461-0

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Journal reference:

Nature DOI: 10.1038/s41586-024-07201-w

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This revealed that the excitation had a kind of quantum spin that has only ever been theorised to exist in gravitons. Though this isn’t a graviton per se, it is the closest thing we have seen.

Ziyu Liu at Columbia University in New York who worked on the experiment says he and his colleagues knew that graviton-like excitations could exist in their semiconductor, but it took years to make the experiment precise enough to detect them. “From the theoretical side, the story was kind of complete, but in experiments, we were really not sure,” he says.

The experiment isn’t a true analogue to space-time – electrons are confined to a flat, two-dimensional space and move more slowly than objects governed by the theory of relativity.

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Journal reference:

Nature DOI: 10.1038/s41586-024-07201-w

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• Published: 12 March 2024

Deep-sea hiatus record reveals orbital pacing by 2.4 Myr eccentricity grand cycles

• Adriana Dutkiewicz   ORCID: orcid.org/0000-0003-0198-6193 1 ,
• Slah Boulila 2 , 3 &
• R. Dietmar Müller   ORCID: orcid.org/0000-0002-3334-5764 1  

Nature Communications volume  15 , Article number:  1998 ( 2024 ) Cite this article

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• Palaeoceanography
• Palaeoclimate

Astronomical forcing of Earth’s climate is embedded in the rhythms of stratigraphic records, most famously as short-period (10 4 –10 5 year) Milankovitch cycles. Astronomical grand cycles with periods of millions of years also modulate climate variability but have been detected in relatively few proxy records. Here, we apply spectral analysis to a dataset of Cenozoic deep-sea hiatuses to reveal a ~2.4 Myr eccentricity signal, disrupted by episodes of major tectonic forcing. We propose that maxima in the hiatus cycles correspond to orbitally-forced intensification of deep-water circulation and erosive bottom current activity, linked to eccentricity maxima and peaks in insolation and seasonality. A prominent episode of cyclicity disturbance coincides with the Paleocene-Eocene Thermal Maximum (PETM) at ~56 Myr ago, and correlates with a chaotic orbital transition in the Solar System evident in several astronomical solutions. This hints at a potential intriguing coupling between the PETM and Solar System chaos.

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Introduction

In 1976 Hays et al. 1 demonstrated for the first time the presence of 10 4 –10 5 year astronomical cycles in Pleistocene deep-sea sediments, confirming Milankovitch’s theory that Earth’s climate is modulated by periodicities in perturbations of Earth’s orbit around the Sun and Earth’s spin axis 2 . Apart from the well-known astronomical cycles with periods of 19 kyr, 23 kyr, 41 kyr, 100 kyr, and 400 kyr that pace Earth’s climate 3 , the geological record also contains signals of much longer-period “grand cycles” 4 , which are predicted by astronomical theory 5 . These “grand cycles” include orbitally-forced periodicities of millions and even tens of millions of years that are similarly linked to changes in incoming solar radiation and paleoclimate 6 , 7 , 8 , 9 . The 2.4 Myr (g4–g3) eccentricity cycle related to the precession of the perihelions of Earth (g3) and Mars (g4), and the 1.2 Myr (s4–s3) obliquity cycle associated with the precession of the nodes of the two planets, are of particular interest 5 . Retrieving these signals from the geological record can provide critical information on Earth-Mars secular resonance and the timing of chaotic episodes in the inner Solar System, potentially reducing uncertainties in the computations of Earth’s orbital motion before 50 Ma, and extending astronomical calibrations of the geological timescale 6 .

Most studies of grand orbital cycles are focused on relatively short (<10 Myr), high-resolution continuous stratigraphic records that yield several eccentricity cycles in the stable 405 kyr band 10 , 11 , but very few longer-period (2.4 Myr) modulating cycles e.g., ref. 12 . The ~2.4 Myr eccentricity cycle, however, has been found embedded in the Cenozoic δ 18 O and δ 13 C isotope record 7 , 8 , 9 , 12 , 13 , in geophysical signals of stratigraphic sequences 4 , 14 , 15 , 16 , and in fossil assemblages 17 , 18 . The cycles are variably attributed to changes in temperature and ice-volume 7 , fluctuations in sediment and organic carbon accumulation linked to river runoff 8 , variations in seawater temperature 12 , and changes in ocean circulation and structure 17 — all of which are driven by astronomically forced changes in insolation and climate. Here we take a different approach by using the very aspect of stratigraphy that is the bane of cyclostratigraphy — stratigraphic discontinuities. We use a merged record of 370 deep-sea hiatuses (breaks in sedimentation) from the global ocean spanning 70–0 Ma based on the compilation of Dutkiewicz and Müller 19 from 293 scientific deep-sea drill holes (Fig.  1 ; see “Methods” section). Our analysis shows the presence of ~2.4 Myr eccentricity cycles linked to orbitally-paced erosion of the seafloor by deep-sea currents, as well as a possible astronomical chaotic transition between ~56 and 53 Ma.

Black circles and numbers denote scientific ocean drilling sites of the Deep-Sea Drilling Project, Ocean Drilling Program, Integrated Ocean Drilling Program, and the International Ocean Discovery Program used in this study. Red regions indicate major large igneous provinces (LIPs) on oceanic crust following Prokoph et al. 70 . Note ocean gateways mentioned in the text. GSR — Greenland-Scotland Ridge, OJP — Ontong-Java Plateau. Elevation from ETOPO1 global relief model ( https://www.ngdc.noaa.gov/mgg/global/ ). Winkel Tripel projection.

Results and discussion

Hiatus cyclicity during the cenozoic.

We used a weighted-average low-pass filter to remove long-term variations in the hiatus-frequency time series (Fig.  2a ), and to highlight the short-term regular oscillations (Fig.  2b ). Spectral analysis and bandpass filtering of the hiatus data illustrates that the dominant hiatus-frequency cycles in this time series have periods of 2–3 Myr (Fig.  2c ). However, the cycles are not equally pronounced in terms of their amplitude at all times (Fig.  2c ). An evolutive Fourier transform analysis reveals four periods of relatively stable unimodal grand cycle occurrence in the data, spanning ~70–56 Ma, ~50–34 Ma, ~30–22 Ma and ~15–0 Ma (Fig.  2d ). Spectral analysis per intervals (Fig.  3 ) shows dominant cycles of 2.34 Myr (70–57 Ma), 2.31 Myr (50–32 Ma), 3.15 Myr (30–20 Ma) and 2.56 Myr (15–0 Ma). The hiatus record is punctuated by three bifurcations with onsets at ~56 Ma, ~34 Ma, and ~22 Ma, during which the dominant cycle frequency splits into two modes (Fig.  2d ) with reduced amplitudes (Fig.  2c ).

Magenta boxes indicate bifurcation intervals. a Raw hiatus-frequency data from Dutkiewicz and Müller 19 sampled at 100 kyr intervals (blue), along with the smoothed data based on a 10% weighted average (wa) of the series (gray) for detrending. b Detrended hiatus-frequency data with removal of the trend shown in ( a ). c Bandpass filters of the 2.4 Myr eccentricity cycle. Note that cycle amplitudes are significantly reduced during bifurcation intervals. d Evolutive Fast Fourier Transform (FFT) amplitude spectrograms of the hiatus data. The upper panel shows a spectrogram of a larger frequency band excluding lower frequencies (<0.25 cycles/Myr), retaining higher frequencies up to 1.8 cycles/Myr (window = 6 Myr, step = 0.1 Myr). The lower panel shows a spectrogram of 0.4 ± 0.15 cycles/Myr bandpass filtered hiatus data (window = 6 Myr, step = 0.1 Myr), highlighting the frequencies of interest for the hiatus analysis. White bars labeled I–IV indicate intervals of stable unimodal signal dominated by hiatus cycles ranging from 2.31 Myr to 3.15 Myr over the last 70 Myr (see Fig.  3 ), separated by bifurcations. Changes in ocean gateways from Hovikoski et al. 45 , Eagles and Jokat 50 , Straume et al. 49 and Bialik et al. 51 . PETM—Paleocene-Eocene Thermal Maximum, AMOC—Atlantic Meridional Overturning Circulation, Cr—Cretaceous, Pl—Pliocene, Q—Quaternary.

a Amplitude (blue) and power (black) spectra for interval I (70–57 Ma) of hiatus-frequency cycle stability based respectively on the 2π-multi-taper method (MTM) and the periodogram method (see “Methods” section) of detrended data. Time windows were slightly extended to avoid edge effects. The statistical F -test is in red, and the red-noise confidence levels (CL) of 95% and 99% are in dashed, orange, and red lines, respectively (see “Methods” section). The dominant spectral peaks are highlighted in bold. b Amplitude and power spectra with annotations as in ( a ) for interval II (50–32 Ma). c Amplitude and power spectra with annotations as in ( a ) for interval III (30–20 Ma). d Amplitude and power spectra with annotations as in ( a ) for interval IV (15–0 Ma).

We first explore the relationship between the deep-sea hiatus record and Earth’s orbital eccentricity for the interval spanning 50–32 Ma using the theoretical astronomical solutions La2004 5 , La2010a–d, and La2011 6 , 20 describing Earth’s orbital motion through time. We choose this interval because it exhibits a continuous sequence of 2.31 Myr hiatus cycles (Fig.  4a ) that are very close to the 2.4 Myr (g4–g3) eccentricity cycle band 5 , 6 . This critical time interval is also within the validity limits of the orbital solutions, which are uncertain beyond ~40–50 Ma due to the chaotic behavior of the Solar System 5 , 6 , 20 . La2004 is precise back to 40 Ma 5 , while La2010a–d and La2011 extend the accuracy of La2004 orbital eccentricity model back from 40 Ma to about 50 Ma 6 , 20 . Nevertheless, the accuracy of La2004, La2010a–d, and La2011 over the interval 50–40 Ma should be potentially explored by correlations with the geological records 6 , 20 . We find that for the interval 40–32 Ma the hiatus cycles are slightly phase-shifted with the 2.4 Myr cycle band in all astronomical solutions (see “Methods” section). The phase shift significantly increases in the interval 50–40 Ma for all models except for La2004 (Fig.  4 ). La2004 cycles are intriguingly in phase with hiatus cycles at ~50–46 Ma. These correlations support the idea that the astronomical models with the exception of La2004 lose precision starting at ~40–45 Ma 5 , 6 , 20 , 21 . The match between La2004 and hiatus cycles supports the accuracy of the La2004 model for longer timescale variations 22 , and agrees with previous investigations suggesting that the La2004 solution provides a better match with some geological data 9 , 14 , 22 , 23 . We also note that the 2.4 Myr cycles in La2004 and La2010a models are in phase from 44 to 0 Ma. La2010a and La2010b start to diverge significantly at ~48 Ma. Despite La2010b including the effects from five major asteroids, in contrast to La2010c (see “Methods” section), the 2.4 Myr cycles in La2010b and La2010c are in phase over the entire 50–32 Ma time interval. Similarly, La2010d and La2011 2.4 Myr cycles are almost in phase over the whole 50–32 Ma time interval, but both models include the five major asteroids. From the above analyses of various astronomical solutions, it follows that the critical parameters that lead to discrepancies among solutions beyond 40 Ma are the initial conditions (e.g., La2004 vs La2010a–d), followed by the step size of the numerical integration (e.g., La2010a vs La2010b). We use phase and coherence analysis to explore to what extent the hiatus dataset and these different eccentricity model predictions are in phase. We find a strong coherence between the two signals of 0.83–0.93 for cycles at 2.4 Myr, confirming the common 2.4 Myr cyclicity among geological and astronomical signals (Fig.  5 ).

a Detrended hiatus-frequency data (black), and bandpass filtered data of 2.4 Myr eccentricity-related cycle (red). b Bandpass filters (0.44 ± 0.1 cycles/Myr) of the 2.4 Myr eccentricity cycle from astronomical solutions La2004, La2010a–d and La2011.

2π-MTM (multi-taper spectral method) cross coherence and phase spectra of the hiatus and La2004, La2010a–d, and La2011 eccentricity data for the interval 42–32 Ma. Note the strong coherence at the 2.4 Myr cycle band (gray bar) between the eccentricity and hiatus signals.

The vast majority of Cenozoic hiatuses were formed by erosion and redistribution of sediments by bottom currents 19 , 24 , 25 . An assessment of the paleo-water depth at which each hiatus in our dataset formed versus regional reconstructions of the carbonate compensation depth (CCD) — the depth at which the rate of supply of carbonate is balanced by its dissolution 26 , shows that only a small number of hiatuses from a dozen holes may either be the result of carbonate dissolution or non-deposition of carbonate 19 . Dissolution events are difficult to detect stratigraphically because they can be extremely brief and require a quantitative assessment of fragmentation of foraminifer tests 27 . These events can be linked to geologically rapid (<10 kyr) injections of p CO 2 into the atmosphere 28 resulting in a coupled decline of carbonate saturation and pH of the oceans, but they are too short to be detected as hiatuses in our dataset. On longer timescales, carbonate saturation of the ocean is regulated primarily by continental weathering, not by atmospheric p CO 2 , resulting in a decoupling of carbonate saturation and pH 29 . This means that longer ‘carbonate crash’ events, such as the middle to late Miocene crash in the Pacific lasting ~2–3 Myr, are linked to processes unrelated to atmospheric p CO 2 and include deep-water exchange causing a reduction in carbonate accumulation rates 30 . Therefore, our dataset, which includes only a small number of hiatuses below the CCD, reflects processes driven by deep-water modification of seafloor sediment.

We propose that maxima in the Eocene hiatus cycles correspond to orbitally-forced intensification of deep-water circulation and erosive bottom current activity, linked to eccentricity maxima which are associated with peaks in insolation and seasonality 8 . The mechanism that could link insolation cycles of shorter timescales to insolation cycles of longer timescales is amplitude modulation (AM), which is a common feature in the theoretical astronomical variations. The fundamental Milankovitch cycles, 19–23 kyr precession, 41 kyr obliquity, and 100 and 400 kyr eccentricity, are modulated by longer orbital cyclicities (e.g., 1.2 Myr, 2.4 Myr, etc.). The energy transfer from shorter to longer cycles as a result of an AM process is detected in climate and carbon-cycle sedimentary records with highly resolved proxies, where both high and low astronomical frequency bands were recognized (e.g., refs. 9 , 31 ). A recent general circulation model parameter sensitivity study 32 concluded that global annually-averaged surface-air temperature rises by a significant 1.75 °C when orbital eccentricity increases from 0 to Earth’s maximum of 0.07 33 , with more pronounced warming at the poles. Ocean modeling studies have suggested a connection between eccentricity forcing and changes in deep-ocean circulation in greenhouse climates (e.g., ref. 34 ). In addition, high seasonal climate variability related to eccentricity maxima has been linked to the warming of intermediate seawater 12 and increases in intensity in deep-water circulation 35 . Decadal observational data and high-resolution modeling indicate that mesoscale eddies also intensify in a warming climate system 36 , 37 . Eddy kinetic energy often reaches the seafloor at >5 km water depth 38 , and is recognized as a driver of seafloor erosion and contourite drifts 39 where it reaches values three times higher than over the average global ocean 40 . The Drake Passage sedimentary record over the last 140 kyr suggests that strengthened westerly winds during warm climates resulted in a stronger Antarctic Circumpolar Current (ACC) and higher bottom current velocities 41 . Recent high-resolution simulations of paleo-ocean currents following the opening of circum-Antarctic gateways point to a long-term weakening of gyres associated with progressive global cooling 42 , suggesting that deep-reaching eddy-induced hiatuses should also be expected to decline as a consequence. The long-term decline in deep-sea hiatuses since the middle Miocene (Fig.  2a ) has been linked to late Cenozoic cooling and reduced vigour of abyssal circulation 19 , lending further support to enhanced seafloor erosion during warmer climates.

We argue that the 2.34 Myr hiatus cycles over the 70–57 Ma greenhouse interval are also paced by the 2.4 Myr eccentricity forcing via the same combination of climate-dependent oceanographic mechanisms. The regularity of the cycles suggests the absence of a chaotic transition over this time interval and may provide geological constraints on orbital solutions that are highly uncertain beyond ~55 Ma 6 . The longer 3.15 Myr and 2.56 Myr hiatus cycles since ~34 Ma (Fig.  2d ) have likely been augmented by non-orbital forcings amplifying the removal of seafloor sediments. In particular, the relatively short (~10 Myr) Oligocene interval recording the 3.15 Myr cycles is bounded by two bifurcation phases (Fig.  2d ) signaling instability in the climate-ocean system. Westerhold et al. 43 showed that astronomically paced climate cycles are less predictable during the Oligocene to present icehouse climate compared to warmer climates, suggesting increasingly complex, stochastic climate dynamics. Even though this interpretation is based on shorter Milankovitch cycles, their results are consistent with both the Oligocene and Miocene hiatus cycle frequency bifurcations we observe, as well as the divergence between orbital forcing and hiatus cycle frequencies.

Hiatus-frequency bifurcations

Cycle bifurcations are transitions from a unimodal to a bimodal signal caused by a change in the dynamic state of a system 44 . Cycle bifurcations can equally arise from the modulation of close high-frequency cycles by low-frequency cycles, which is a common feature in Milankovitch astronomical cyclicities 9 , 31 . We suggest that bifurcations in the hiatus signal represent disturbance of the background 2.4 Myr eccentricity cycles caused by major changes in ocean circulation. These changes are chiefly driven by the opening and closing of ocean gateways modifying ocean circulation and regional patterns of seafloor erosion and hiatus formation. The bifurcation interval at ~57–52 Ma (Fig.  2d ) coincides with a severe narrowing of the Norwegian-Greenland Seaway between ~56 and 53 Ma that may have altered the Atlantic deep-ocean circulation 45 . The Earth also experienced a series of short-lived but frequent extreme global warming events 46 exemplified by the Palaeocene–Eocene Thermal Maximum (PETM) 47 during which Earth’s hothouse climate was more stochastic compared to a warmhouse climate due to its nonlinear response to astronomical forcing 43 .

Astronomical models of the orbital motion of the inner planets predict frequent chaotic resonance transitions from libration to circulation ~65 to ~45 Ma 5 , 15 , 20 , 21 . These models show various possibilities in terms of their predicted resonance passages over the 40–70 Ma interval 6 , with the nominal La2010a astronomical solution showing frequent resonances over this time interval, in contrast with other models 6 . As astronomical model predictions are uncertain, they require geological confirmation 4 , 14 . We test possible observable resonance passages from our hiatus-frequency data through local perturbation in the g4–g3 eccentricity term, and compare them with those theoretically predicted from six astronomical models.

Our evolutive FFT spectrogram of the hiatus-frequency data shows evidence for the record of g4–g3 eccentricity term (Fig.  6a ), along with a potential frequency shift at ~56 Ma. The evolutive FFT spectrogram of the nominal La2010a eccentricity data highlights several frequency shifts, in particular at the g4–g3 eccentricity term (Fig.  6b, c and Supplementary Fig.  S1 ). The most important shift is at nearly 56 Ma detected in the eccentricity. This concomitant shift in g4–g3 eccentricity frequency in the hiatus and astronomical data hints at a very likely 56 Ma chaotic transition in the inner Solar System, expressed in the critical resonant argument θ = (s4–s3)–2(g4–g3) 48 .

a Taner bandpass of hiatus-frequency data (cutoff frequencies: 0.28 and 1.8 cycles/Myr, Taner filter roll-off = 10 20 ), along with evolutive FFT spectrogram (window = 6 Myr, step = 0.1 Myr). b – e Taner low-pass La2004 and La2010a,b,c eccentricity data (cutoff frequencies: 0 and 1.8 cycles/Myr, Taner filter roll-off = 10 20 ), along with evolutive FFT spectrogram (window = 6 Myr, step = 0.1 Myr). c Taner low-pass La2004 eccentricity data (cutoff frequencies: 0 and 1.8 cycles/Myr, Taner filter roll-off = 10 20 ), along with evolutive FFT spectrogram (window = 6 Myr, step = 0.1 Myr). Horizontal, dashed-white lines indicate potential chaotic excursions, while pink arrows indicate the return to a new stable state for the longer excursions (this does not concern the shorter excursions because their time occurrence is brief). Note the common chaotic excursion at nearly 56 Ma among the La2004 and La2010a,b,c models, matching a perturbation in the record of g4–g3 (Mars–Earth) eccentricity-related cycle in the hiatus-frequency data. In the hiatus data there is a potential shift in g4–g3 orbital term from a dominant period around 2 Myr into a dominant period around 1 Myr (see Panel a ). For a detailed interpretation of all potential chaotic transitions within the studied 70–40 Ma time interval and using more astronomical solutions see Supplementary Figs.  S1 , S2 and S3 .

We also show that the La2004 astronomical model 5 exhibits a significant deviation in the g4–g3 eccentricity term at 56 Ma (Fig.  5c ), potentially reflecting a chaotic transition at the timing of the PETM. Although this possible chaotic transition at the PETM is better expressed in the La2010a than in the La2004 model 6 , our geological hiatus record shows a better correlation, at the g4–g3 eccentricity cycle band, with the La2004 model. This reflects that the La2004 model is likely more reliable at longer timescales than the La2010 models 9 , 14 , 22 , 23 . Additionally, the La2010b and La2010c solutions, which were acquired differently (see “Methods” section), show a concomitant shift in g4–g3 eccentricity term at nearly 56 Ma.

The initiation of the Oligocene bifurcation, lasting from 34 to 30 Ma (Fig.  2d ), coincides with the Eocene-Oligocene Transition, which marks a tipping-point in Earth’s climate system from a greenhouse to an icehouse state 43 . Combined opening and deepening of the Drake Passage, and the deepening of the Tasman Gateway, connected all major oceans, profoundly affecting global ocean circulation and instigating the ACC 49 . Enhanced erosion of the seafloor is recorded as an increase in hiatus frequencies in the South Atlantic, Indian, and South Pacific oceans 19 , and widespread contourite drifts 40 . Increased vigour of deep-sea currents and surface eddies during this time likely explains the lengthening of the hiatus cycles through excess removal of seafloor sediment in addition to orbital forcing. The following phase of relative stability lasted only 8 Myr before the onset of the following Miocene bifurcation starting at ~22 Ma (Fig.  2d ), which we also attribute to major oceanographic changes superimposed on the eccentricity forcing. These include the initiation and gradual strengthening of the proto–Atlantic Meridional Overturning Circulation (AMOC) associated with the complete opening of the Drake Passage at ca. 23–17 Ma 50 , the early Miocene deepening of the Fram Strait and the Greenland-Scotland Ridge 49 , and the closure of the Tethys seaway at ca. 20–13.8 Ma 51 . Sensitivity experiments using a fully coupled ocean-atmosphere general circulation model have shown that progressive closure of the Tethys seaway strengthens the AMOC as well as the ACC 52 . The renewed uplift of the Greenland-Scotland Ridge between ~18 Ma and 15 Ma 49 may be at least partly responsible for extending the duration of contemporaneous bifurcation (Fig.  2d ) by a temporary weakening of the AMOC.

Chaotic orbital transition at ~56 Ma

The orbits of planets in the Solar System are not stable (chaotic) at timescales of tens to hundreds of Myr 48 , 53 . The chaotic diffusion in the Solar System hampers the establishment of precise astronomical solutions in deep geological time, given that uncertainties in model computation increase substantially over time 48 , 54 . One of the features of chaos is the emergence of frequent secular resonances in astronomical models arising especially in the inner Solar System 48 , 53 . The secular resonances are expressed as linear combinations of long-period orbital cycles that are detectable in the sedimentary record. Geological detection of long-period orbital cycles associated with resonant arguments allows the mapping of the timing of chaos and represents valuable constraints for astronomical modeling 15 . One of the main resonant arguments relating long orbital eccentricity period (g4–g3) and long orbital inclination period (s4–s3) is θ = (s4–s3)−2(g4–g3), which is theoretically well recognized (e.g., ref. 5 ), and its associated periodicities can be captured in Cenozoic sedimentary records 9 , 18 . The present mean periods of s4–s3 and g4–g3 are 1.2 and 2.4 Myr, respectively. However, due to chaos, these periods may deviate in the geological past, beyond 40 Ma 5 , 6 , 20 . The study of the evolution of the g4–g3 cycle in early Cenozoic strata can provide constraints on the timing of chaos and hence on astronomical modeling 5 , 15 . Astronomical modeling shows that the critical resonant argument θ has a present-day ratio of 2:1 between g4–g3 and s4–s3 (i.e., s4–s3 = 2(g4–g3)); however, when shifting from one resonance to another it has been shown that the time span of a transition which characterizes chaos is not easily predictable. The geological record offers a good opportunity to potentially capture the chaotic transition and possibly determine its duration. Geological studies of chaos in the inner Solar System are rare, but the few existing studies have shown promising results 13 , 14 , 15 , 22 . Westerhold et al. 13 used a range of ocean drilling geological data to locate such a (transient) transition between ~55 Ma and ~52 Ma at Ocean Drilling Program (ODP) sites 1258, 1262, 1265, and 1267 from the equatorial and South Atlantic Ocean that are also included in our large global hiatus dataset (Fig.  1 ). Their early Eocene time interval coincides almost exactly with the bifurcation in the hiatus frequencies in our study. The hiatus data show a shift at ~56 Ma of the g4–g3 eccentricity term from a periodicity of ~2 Myr to a periodicity of ~1 Myr, followed by a return to the previous periodicity of ~2 Myr (Fig.  6 ). This points to a potential transient chaotic transition in the orbital motion of the inner Solar System lasting ~2.6 Myr. The timing of the transition evident in our hiatus dataset is consistent with the results of Westerhold et al. 13 who found a ~1.2 Myr periodicity between 55 Ma and 52 Ma bounded by a periodicity of ~2.4 Myr at times older than 55 Ma and younger than 52 Ma in their study. Our analysis shows evidence for a chaotic transition in the g4–g3 eccentricity term at ~56 Ma occurring in four astronomical solutions (La2004, La2010a, La2010b, and La2010c), correlating with a contemporaneous bifurcation in the hiatus data (Fig.  6 ).

We note that the detection of potential chaotic transitions in the geological record appear to be sensitive to the onsets of transitions, rather than their mid-ages, terminations or durations. The end of a transition is not always evident (Fig.  6 and Supplementary Figs.  S1 , S2, and S3 ), while the onset is easier to capture 22 . This observation is helpful for the comparison and correlation of different records documenting this phenomenon. It is likely that the onset of the transition found in Westerhold et al.’s study 13 corresponds to the same transition we detect in our hiatus data and in La2010b and La2010c models (Fig.  6 ). Although the assessment of the duration of a chaotic transition is somewhat ambiguous, our hiatus data point to a duration of ~2.6 Myr versus only ~1.5 Myr in La2010a, ~4.2 Myr in La2004, and potentially ~4.8 Myr in the La2010b and La2010c models. Despite significant advances in astronomical modeling, integrating high-precision initial conditions and possible effects from major asteroids, the time interval 50–40 Ma is still subject to uncertainties 6 , 20 . The validity of any astronomical model within this time interval needs to be tested by correlation with the geological record 5 , 6 , 15 , 21 . The chaotic transition at ~56 Ma that we infer from the hiatus-frequency data broadly matches the La2004, La2010a, La2010b, and La2010c models (Fig.  6 ). The detection of the chaotic transition in both our hiatus data and in Westerhold et al.’s 13 study, may provide a valuable constraint on astronomical modeling 5 . The onset of the transition coincides with the timing of PETM; however, any potential causal relationship between these phenomena requires further investigation of geological and proxy records.

Deep-sea hiatuses provide a remarkable 70 Myr-long record of ~2.4 Myr eccentricity cycles disrupted by episodes of profound tectonic, oceanographic, and climatic changes that have affected the vigor of ocean bottom currents driving deep-sea erosion. Our analysis reveals a potential chaotic transition at ~56 Ma in Earth-March orbital motions, which coincides with the timing of the PETM. The discovered link between deep-sea hiatus-frequency maxima and grand cycle eccentricity peaks, corresponding to increased insolation and seasonality, suggests that warmer climates are associated with more vigorous deep-ocean circulation. Elucidating the role of astronomical forcing in driving deep-ocean circulation is important for better understanding the future response of the ocean to global warming.

Hiatus dataset

Our investigation of deep-sea hiatus cyclicity through time uses the global compilation of Dutkiewicz and Müller 19 , which is based on carefully selected age-depth models for 293 deep-sea drill holes from the NSB (Neptune Sandbox Berlin) database 55 . The hiatus data used in this study are available via Zenodo at https://doi.org/10.5281/zenodo.8295655 . The age-depth models are based on a combination of biostratigraphic, magnetostratigraphic, and other event data, with outlier detection 55 . Sites for which the quality of the age-depth model was ranked good to excellent by NSB with a median error ranging from about ±0.5 Myr to as low as ±0.2 Myr for orbitally-tuned sections 55 were used. The overall uncertainty of the age models based on combined Deep-Sea Drilling Project (DSDP) and Ocean Drilling Program (ODP) data in the NSB database is estimated to be ±0.37 Ma for the Cenozoic 56 . This error is sufficient for the analysis of the 2.4 Myr (g4 –g3) eccentricity cycle and the 1.2 Myr (s4–s3) obliquity cycle, and is expected to be even smaller for the more accurately dated stratigraphic sections sampled by the subsequent Integrated Ocean Drilling Program and the International Ocean Discovery Program (IODP), and further revisions of existing age-depth models 55 . Our dataset comprises 370 hiatuses defined as stratigraphic discontinuities longer than 0.18 m.y. following Spencer‐Cervato 56 from all major ocean basins spanning the last 70 Myr. This conservative hiatus duration limit applies to our analysis because our dataset includes many of the holes (now revised but unchanged in terms of resolution) that Spencer‐Cervato (1998) used plus additional IODP sites with higher resolution. The dataset is spread across all major ocean basins and includes sites on submerged continental crust, normal oceanic crust, LIPs, proximal and distal to ocean gateways, and at variable water depths, thus minimizing sampling bias (Fig.  1 ). The number of holes increases steadily from 70 Ma to the present-day. Most regions are well represented from ~60 Ma with the exception of the South Pacific, the Southern Ocean and the Mediterranean (Fig.  7a ), which remain relatively unexplored compared to other regions of the global ocean. The North Pacific shows an increase in the number of holes penetrating stratigraphy younger than ~23 Ma relative to other regions, but these holes cover a disproportionately large area of the global ocean. The slight increase in the number of holes at ~25 Ma in the North Atlantic reflects seafloor erosion and contourite drift deposition during the Oligocene and Miocene 40 . The mode of hiatus duration is relatively uniform for the entire time series indicating the absence of hiatus duration bias even for the oldest sections (Fig.  7b ). This is consistent with empirical data suggesting that the effect of detecting more longer hiatuses with increasing time span is weakest in deep-water facies compared to facies on continental shelves 57 . The hiatus dataset spans a wide range of reconstructed paleo-water depths (Fig.  7c ) with roughly 15% of holes located on elevated ocean floor associated with a major LIP (Fig.  1 ). All age-depth models were converted to the Geological Timescale 2020 58 . Hiatuses were recognized as zero-slope intervals in each age-depth model 59 . The occurrence of hiatuses or their absence (i.e., the presence of conformities) was sampled at relatively dense intervals of 100 kyr to avoid bias in the detection of the onset or cessation of hiatuses. We note that the age of hiatus onset is a maximum constrained by the chronology of the underlying sediment considering that the volume of sediment removed is unknown 60 , while the age of hiatus termination is based on the chronology of overlying sediment and is more precise. A priori it is unknown what the combined effect of uncertainties in the age models and the volumes of missing sediment for individual sites is. Our approach of a big data analysis, combining a large number of sites widely distributed geographically and in terms of water depth, is designed to average out biases which may be present at individual sites.

a Number of holes drilled versus age of section penetrated. b Hiatus duration versus mean hiatus age. Black line represents the mode of hiatus duration based on a 5 Myr moving window and a robust maximum likelihood probability filter with outlier detection. c Backtracked hiatus paleo-water depth 19 versus hiatus duration showing that the majority of hiatuses are shorter than 5 Myr and occur across a wide range of paleo-water depths. Legend as in ( b ).

Spectral analysis

Long-term irregular trends in the hiatus data were measured and removed using the loess weighted average 61 . For spectral analysis, we used the multi-taper spectral method (MTM) associated with the harmonic F -test, with three 2π prolate tapers 62 . The F -test was used to seek evidence for individual lines within frequency bands with elevated amplitude that could be related to specific significant cyclicities. The MTM and F -test results were then compared to those of the basic Fourier transform squared Periodogram (unsmoothed periodogram, Matlab’s periodogram.m) along with the classical autoregressive AR (1) red-noise model. We used the Gaussian bandpass filter to extract the targeted cycles 63 . Additionally, we used the Fast Fourier Transform (FFT) amplitude spectrogram to study the evolution of cyclicities through time in the hiatus-frequency and Earth’s orbital eccentricity data 5 , 6 , 20 . The FFT spectrogram is based on a function computing a running periodogram of a uniformly sampled time series using FFTs of zero-padded segments, and normalized to the highest amplitudes (e.g., ref. 64 ). Finally, correlation of cyclicities between the hiatus and eccentricity signals was quantified using the 2π-MTM cross-spectral analysis provided in the Matlab cmtm.m routine 65 . We focused on the interval 42–32 Ma where the hiatus and eccentricity data share close 2.3 Myr cycles, and the reliability of the astronomical signal within the 42–32 Ma interval is guaranteed 5 , 6 . Six astronomical solutions, which were derived differently, have been analyzed: La2004 5 , La2010a, La2010b, La2010c, La2010d 6 , and La2011 20 (Table  S1 ). The main difference between the La2004 model and the La2010a–d and La2011 models is that the La2004 initial conditions are adjusted to the JPL (Jet Propulsion Laboratory) numerical ephemeris DE406 66 over −5000 yr to +1000 yr from the present date, while La2010a–d, La2011 are fitted to the high-precision planetary ephemeris INPOP 67 , 68 , 69 (INPOP: Intégration Numérique Planétaire de l’Observatoire de Paris) over longer time intervals. La2010a and La2010b are fitted to INPOP08a 68 over 0.58 Myr time interval with a smaller step size of the numerical integration (10 −3 ) for La2010a and a larger step size of the numerical integration (5 × 10 −3 ) for La2010b. The La2010a,b solutions include the effects from five major asteroids (Ceres, Vesta, Pallas, Iris, and Bamberga) 20 . La2010c is fitted to INPOP08a over 1 Myr and with a larger step size of the numerical integration (5 × 10 −3 ), but does not include the five major asteroids. La2010d is fitted to INPOP06 69 over 1 Myr with a larger step size of the numerical integration (5 × 10 −3 ), and including the five major asteroids. La2011 is fitted to INPOP10a (Fienga et al., 2011) over 1 Myr, which in addition includes the five major asteroids.

Data availability

The hiatus data used in this study are available via Zenodo at https://doi.org/10.5281/zenodo.8295655 .

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Acknowledgements

This research was supported by the Australian Research Council Future Fellowship grant FT190100829 to A.D. S.B. was supported by the French Agence Nationale de la Recherche (Grant ANR-19-CE31-0002-01) and the European Research Council under the European Union’s Horizon 2020 research and innovation program (Advanced Grant AstroGeo-885250).

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Adriana Dutkiewicz & R. Dietmar Müller

Sorbonne Université, CNRS, Institut des Sciences de la Terre Paris, ISTeP, 75005, Paris, France

Slah Boulila

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A.D. and R.D.M. conceived the initial idea for the study and prepared the datasets. S.B. performed the spectral analysis. A.D., S.B., and R.D.M. contributed to the interpretation of the results. A.D. wrote the manuscript with input from S.B. and R.D.M.

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Correspondence to Adriana Dutkiewicz .

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Dutkiewicz, A., Boulila, S. & Dietmar Müller, R. Deep-sea hiatus record reveals orbital pacing by 2.4 Myr eccentricity grand cycles. Nat Commun 15 , 1998 (2024). https://doi.org/10.1038/s41467-024-46171-5

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Received : 05 September 2023

Accepted : 15 February 2024

Published : 12 March 2024

DOI : https://doi.org/10.1038/s41467-024-46171-5

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