Cosmologist Sergei Odintsov on Newton’s law, modifications of general relativity, and the essence of gravity

faq | November 15, 2016

Gravity is described as gravitational force, the gravitational interaction, one of four interactions which we so far know in the universe, the others being strong, weak and electromagnetic interactions. The gravitational force somehow is different from the other three which act at very small scales, at very small distances. In turn, gravity, as far as we understand it now, is important at very large distances which are comparable with the distance of our universe, with the large-scale structure of the universe. In this sense the solar system size is still small to see all the expected effects of the gravity.

History of Research

The history of the topic of gravity as it is understood now probably starts from Newton’s law. Of course, not going into what people were thinking before. Eventually one of the first persons who tried to understand earth gravity in a more or less consistent mathematical description was Newton. His law is very simple. Two bodies with masses attract each other with the force which is proportional to their masses and inversely proportional to the square of distance between them. Of course, we can immediately apply this law to the solar system. But we have to be quite careful. We should consider the body in a sense of a center of mass, the body should be considered as kind of a particle, very massive particle located in the center of mass of the body. Even in this description there appeared some problems related for instance with the distribution of density in the body. Even though we can approximately consider planet as some model where all mass is located in the center of mass, it still occupies some space in the solar system. So it is already not so trivial on the level of Newton’s law.


Image: Sir Isaac Newton. Sir Godfrey Kneller. 1689 / Wikimedia Commons

When we started to study the motion of the bodies, spherical bodies like planets, already in the solar system we found that there are deviations from Newton’s law. And that is one of indications on which the generalization of Newton’s gravitation was proposed. Actually, at the beginning of XX century there were proposed several different gravitations. But among them the best in mathematical sense as more concise theory, in physical sense as a theory which satisfies the Solar system tests and turns out to be independent of the coordinates system choice is general relativity or Einstein gravity. Everything works perfectly within the solar system with general relativity. So when we investigate the universe evolution and its regions which are located extremely far from our Solar system we apply general relativity. However, it is not apriori known that GR is correct at large scales. Hence, in principle we may think about other theories which coincide with general relativity predictions in Solar system but may deviate of it at very large scales or at very early universe.

General Relativity

General relativity is a covariant theory to describe gravity. Of course, mathematically it is very beautiful, but in the same time very complicated theory. The reason is that when we apply it to the evolution of the large-scale universe, basically speaking we arrive to a system of differential equations. Eventually, very few systems of differential equations have a general solution in mathematics. This particular one is a very complicated mathematical system. Of course, physical considerations help to simplify this system and to make it reasonable so that it may be really applied for the mathematical description of the universe history. Space-time of the universe is characterized by the so-called metric tensor, naively speaking, it is kind of invariant variable. Metric tensor consists of sixteen components but due to the fact that it is symmetric not all components are independent of each other.


Image: Schematic spacetime curvature /

Actually only ten components are independent ones. However, our astronomical and astrophysical observations strongly indicate towards isotropic and homogeneous universe. Furthermore, this universe is nearly spatially flat. If this is true, then the degrees of freedom or better say independent components of the metric tensor are reduced. Effectively it is reduced to only one component which is called the universe scale factor and which represents kind of the main cosmological variable. Hence, scale factor of the universe is a function which describes the time-evolution of the universe. Thus, instead of the system of differential equations in theory of GR we get only single differential equation of the second order for this scale factor.

Of course, it is much simpler to solve single equation than system of equations. In addition, we have conservation law of matter. Such the universe is often called Friedmann-Robertson-Walker universe. Basically speaking, Friedman-Robertson-Walker universe is a specific and eventually most realistic universe which is isotropic and homogeneous at large scales and depends only on the scale factor. Of course, one can consider other universes which may also appear as solutions of the equations of GR. This system became playground for our understanding of the universe.

Quantum Gravity

First, gravitational interaction is extremely weak if compared to the other three fundamental interactions. For example, we know that stable configurations like atoms are not created thanks to gravitational force, it is impossible to confine them by gravitational force. Gravity is extremely weak at microscopic scales. Eventually we know that four fundamental interactions appear and act in a different way but it is expected that they are consequences of some unknown fundamental unification theory. For instance, there exists the well-established theory which unifies electroweak and strong interactions (the corresponding construction is called Grand Unified Theory).

However, the corresponding fundamental theory was realized at the very early universe when physical laws looked different due to the extreme external conditions. At the very early universe, the gravitational theory was essentially quantum theory. Quantum gravity theory acts at the energies which are typical of Planck energy i.e. at the very early universe. Current elementary particle physics is studied at the universe where typical energies are very small, typical curvature is very small but typical sizes are very large (of billions of light years).

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We do not know much about quantum gravity but number of attempts to formulate and study it were made. There is so-called perturbative quantum gravity approach where quantum gravity is described in close analogy with quantum chromodynamics (theory which explains weak interactions) and quantum electrodynamics (theory which explains electromagnetic interactions). These attempts go back to 50s, 60s and still continue. One more direction is called loop quantum gravity which is kind of mathematical realization related with different constraints from so-called hamiltonian formulation. There is very popular direction which was developed during last years which is called string theory which represents the theory of extended objects (not particles). However, despite the many promises and despite to the fact that this theory has been developed up to a very complex and very rich mathematical structure, this theory did not bring us to the final quantum gravity: still many questions remain. Then in such situation when no consistent quantum gravity exists we may search the manifestations of quantum gravity at the early universe around inflationary epoch. In the same line it lies the study of gravitational waves because with the recent observation of gravitational waves we understood that gravitational theory basically has the same structure as other fundamental interactions. So there is particle which is called graviton to describe this theory.

Modifications of general relativity

Actually, we need to know the behavior of single variable (scale factor). If we know the behavior of this variable through the whole history of the universe, eventually we understand the evolution of the universe. However, it is an extremely hard task. For example, different works show that the universe passed very fast inflationary early time acceleration at early universe. It means that at the early universe right after the epoch when quantum gravity was important by some reasons the universe started to expand more and more quickly. This accelerated expansion is called an inflationary era. After the accelerated inflation we believe the inflation stopped and the universe transited to the era which is called deceleration era. During this very interesting epoch when universe expanded with decceleration, it passed the radiation dominance era and then matter-dominance era. Effectively, at this stage the universe started to look as it is now: with usual matter (stars, planets, etc). Over 6 billion years ago due to some unknown reason the universe started to expand again with acceleration. Such current universe is called dark energy universe. How to explain the transition to dark energy era?

If we consider only gravity then we should do something in order to reach this acceleration because usual GR cannot explain it (forgetting about famous cosmological constant problem). So the simplest task is to modify gravity in some way. If one adds the cosmological constant to the action of gravitational theory one can consider the Einstein gravity with cosmological constant. Such theory may describe the inflationary era, the accelerated evolution of the universe. However, immediately there appears a number of problems. First, it is difficult to get to the exist from such era. In other words, we need to introduce somehow dynamical scenario for the cosmological constant. The simplest generalization in this case is to consider not GR but its slight generalization which is scalar-tensor theory, i.e. gravity plus scalar field.

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Let us consider general relativity coupled with scalar field with some potential. In this case there appears the scalar degree of freedom (scalar particle) and this extra scalar degree of freedom may play the role of inflaton field, i.e. the fuel of universe to expand with acceleration. At the same time at some point the specific form of such theory may provide the exit from inflation and then the universe may go to standard universe evolution. This line was very popular in 80th and it was developed by number of researchers. There were proposed hundreds of models where inflation is described by some specific scalar field with some specific potential. So far this is eventually the the richest combination of inflationary scenarios in theoretical cosmology.

If we add to the theory extra fields like scalar field, then these extra fields should appear in our theory of elementary particles. So if we use inflaton we should identify with some scalar particle of high-energy physics. For instance, there were many speculations that it could be the Higgs particle which seems to be discovered recently. However, if inflation was caused by Higgs boson the question is why it stopped to act after inflation? Furthermore, why it acted in one way at the early universe, then effectively disappeared at radiation-matter-dominance era and started to act in another way later during dark energy era? In other words, the consideration of extra fields which cause the inflationary expansion leads to number of fundamental problems.

Modified Gravity

Due to these problems the natural question is: could it be that there are no extra dark components of the universe and everything is caused by only the gravity itself. Imagine that we have some gravitational theory with unknown geometrical structure (in other words, with unknown gravitational Lagrangian which characterizes this structure). This Lagrangian may look different at different universe epochs. So in a sense with the expansion of the universe we have a change of gravitational Lagrangian or, let us say, we have a change of effective gravitational force. For instance, some terms of the gravitational theory are dominant at the very early universe. Then in this case this gravitational Lagrangian would cause the inflationary expansion. During inflationary expansion the curvature of the universe was decreasing, so typical energies, typical characteristic scales, energy scales were decreasing. Then the original Lagrangian was also changed. The change of this Lagrangian caused the exit from the inflation to the deceleration era.

During this deceleration era, this radiation and matter-dominance eras our gravitational theory is approximately general relativity. So general relativity perfectly describes intermediate universe. However, during this deceleration era typical curvature is decreasing. Somehow 6 billion years ago some other gravitational terms which were irrelevant before started to play an important role due to the this decrease of physical curvature, the role of dark energy. So now we live in the universe which expands with acceleration. In this way we have in a sense quite unique and fundamental picture of universe evolution history. We have gravitational theory which describes unified evolution of the universe. We do not need any extra components to describe one era then other dark component to describe next era. In fact, in our works with Prof. S.Nojiri of Nagoya university we proposed several such modified gravity theories which describe the universe evolution quite well.

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Nevertheless, a very important and very interesting property of extended gravity theories is the following. When I was a student I was told at that time by my professor that the only theory which can pass the solar system tests is GR. Now 30 years after we have a number of theories, I would say, classes of theories which are realistic within the solar system. They are realistic in the same sense as GR but they behave in a different way at the very early universe and at the very late universe if compare with GR. The only problem is that we do not know yet how to derive such modified gravity theory from some fundamental theory due to lack of quantum gravity. Eventually it should be some effective theory which comes from quantum gravity. However, due to lack of consistent quantum gravity this approach still remains to be phenomenological. However, recently some spectacular development of such theories is observed. Let me just tell several words. There are modified gravities which depend only on the curvature invariant. These are theories which are called f(R) gravities, there are theories which are called mimetic gravities. There are unimodular modified gravities, string-inspired gravities, etc. etc. These theories appeared during the last 10-20 years as alternatives to GR to describe the universe in unified way from the very beginning till the current stage.

Current Methods to Study Gravity

Many researchers continue their activities in the scalar-tensor theory. For instance, the study where inflation is caused by scalar like Higgs particle, or by some fluids is very active now. However, much attention is also paid to modified gravity. Each week we have 10-15 papers related in one way or another to modified gravities of different types. Black holes are studied in such theories too. Furthermore, we get better astrophysical data and in some observational data there appear indications that GR is not good enough to understand some phenomena occurring at the universe. From general paradigm in theoretical cosmology 20 years ago, each research work should have proven that GR is correct. Now there are papers where there appear claims that GR eventually is not enough to describe whole astrophysics and in general cosmology. It is quite promising.

We need to get a number of observational data with which we can compare our theoretical cosmology predictions. That is all. So far we have few observational cosmological effects. We need to extend this number and we need to get the cosmological parameters with higher and higher accuracy. That is the purpose of all current research in cosmology. There are observations like gravitational waves experiment or so-called Planck experiment to get quantitative predictions about inflation. These experiments which are run by teams of several hundred, sometimes up to one thousand people should clarify observational features of the universe. But not only observations, but also some extra experiments to check the accuracy on the Earth or in the space, in solar system. Not only to observe but to measure the deviations from predicted values. For instance, to estimate the possible corrections to Newton gravitational constant which seems to be still in very good connection with result predicted by general relativity.

Open questions

I do not rule out the possibility that even current paradigm of the universe evolution history will change completely. For example, there are attempts to change the inflationary universe paradigm by so-called bounce universe or cyclic universe. It may happen that in future we will see some proposal of such or other sort. Second, I do not rule out the possibility that eventually we may get in near future observational indications which will change some details of evolution of some era of the universe. However, for me the main open question is why the universe looks so. Why the gravitational theory appears in such form? Why we do not see anti-gravitation? Indeed, we have some rules which are considered to be correct up to some point and we apply them to describe the universe evolution. However, we do not have the answer why precisely such theory is correct theory of the universe evolution? Why precisely this type of the evolution of the universe is the consequence of one of the solutions of our gravitational theory. In this sense I believe that further study of gravity may help to answer to this question, at least somehow. We continue the study of gravitational theories so that such theories would be consistent with the current universe evolution and with current observational data. That is one of the main lines I continue to work on.

Professor of Cosmology, Institut de Ciències de l'Espai
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