"If you can't join them, beat them !"

Combining Quantum theory and Gravity

Particle physicists noticed that there is something called gravity around 1983, when supergravity models came up.

The trouble with gravity is that it is not perturbatively renormalizable. What is more, every interesting question to which we want an answer from quantum gravity, is likely to be non-perturbative in character. Since we do not know how to handle even quantum electrodynamics non-perturbatively, there is little hope that similar techniques will bear fruit in quantizing gravity.

This fact gained reluctant acceptance rather slowly (circa the latter half of the 80s) among the high energy physicists. The last three decades witnessed significantly different and more imaginative approaches towards quantum gravity, but unfortunately -- often after a considerable amount of hope and hype -- none of them have led us to anywhere near answering the really important issues of quantum gravity.

Nevertheless, given the emotional investment of a generation of very talented physicists, the hope and the hype will continue for another decade or so before the die-hard supporters take out the life-support machine from the current models and write an obituary!

Meanwhile, I have been developing ideas which will survive this eventuality!

Gravity as an emergent phenomenon: Publications, Talks etc.

During the last decade (2002-), I have come up with a new perspective in which the field equations of gravity have the same status as those in emergent phenomena like fluid mechanics or elasticity. What is more, standard classical gravity seems to contain sufficient "internal evidence" to support this point of view. Here are some annotated references:

A popular article describing this approach from Scientific American (India), Jan 2011; click here for the pdf.

An article in a German popular magazine (with English translation at the end) describing my work is here.

These ideas allowed me to provide a completely novel way of approaching cosmology and providing a fundamental solution to the cosmological constant problem.

Two invited lectures on this are:

  • Cosmological Constant(s) I

  • Cosmological Constant(s) II

  • given at the mini-series "Cosmology and the Constants of Nature", June 2014 from the "Philosophy of Cosmology" project, Universities of Oxford and Cambridge.

    An invited review describing these ideas (co-authored with Hamsa Padmanabhan) is:

    Cosmological constant from the emergent gravity perspective, [arXiv:1404.2284]

    Here are a few, somewhat more technical reviews:

  • Invited Review for the MPLA Special Issue on "New trends on theory of gravity":

    Emergent Gravity Paradigm: Recent Progress, [arXiv:1410.6285]

  • A more conceptual description,

    A Dialogue on the Nature of Gravity, [arXiv:0910.0839]

    [Based on my lectures given at several conferences which appeared in the proceedings of the meeting 'The Foundations of Space & Time: Reflections on Quantum Gravity', Cape Town, Aug 2009.]

  • A comprehensive invited review:

    Thermodynamical Aspects of Gravity: New insights, [arXiv:0911.5004]

    My work has received prizes from the Gravity Research Foundation six times during 2002-2014:
      Second Prize essay; 2002: The Holography of gravity encoded in a relation between Entropy, Horizon Area and the Action for gravity, Gen.Rel.Grav., 34, 2029-2035 (2002)
      Fifth Prize essay; 2003: Why gravity has no choice: Bulk spacetime dynamics is dictated by information entanglement across horizons, Gen.Rel.Grav., 35, 2097-2103 (2003)
      Third Prize essay; 2006: Gravity's immunity from the vacuum: the holographic structure of semiclassical action, Gen.Rel.Grav., 38, 1547-1552 (2006)
      First Prize essay; 2008: Gravity - The Inside Story, Gen. Rel. Grav., 40, 2031-2036 (2008)
      Fifth Prize essay; 2012: The secret life of the spacetime, Int.Jour.Mod.Phys. D, 21, 1241005 (2012)
      Third Prize essay; 2014: What drives the time evolution of the spacetime geometry?, Int.Jour.Mod.Phys., D 23 1441003 (2014)

    Gravity as an emergent phenomenon: Semi-technical Description

    Atoms of spacetime

    A simple example of an emergent phenomenon in physics is the dynamics of a fluid or the elastic properties of a solid. Many of the phenomena in fluid mechanics can be understood in terms of dynamical equations involving the density, velocity, etc. which are well-defined when the fluid is treated as a continuum. We, however, know that the real fluid is made of discrete molecules with empty space in between. Macroscopic variables like density, velocity, etc. have no microscopic significance and certainly cannot be used to understand the microscopic physics of molecules making up the fluid. In this description, one thinks of fluid mechanics as an emergent phenomenon, with its own dynamical variables and corresponding equations of motion that are valid at scales much bigger than the typical intermolecular separations.

    Analogously, one can imagine the description of spacetime in terms of the metric, curvature etc. as an emergent phenomenon valid at scales large compared to some critical length scale which, could possibly be the Planck length. It is then conceivable that the general theory of relativity is similar to the description of, say, fluid mechanics. Variables like the metric, etc. (being analogous to the density, velocity etc. in fluid mechanics) have no significance in the microscopic description of spacetime. Just as the proper description of molecules of a fluid requires the introduction of new degrees of freedom and a theoretical formalism (based on quantum mechanics), the microscopic description of spacetime will require the introduction of new degrees of freedom ('atoms of spacetime') and a theoretical formalism. These new degrees of freedom, and the theoretical framework, could be widely different from the description based on the metric, etc., just as the quantum description of molecules is quite different from that of a fluid based on density, velocity etc. In that case, quantization of the metric itself, will be similar to quantizing the elastic vibrations of a solid; it will, at best, get you the phonons but not the atoms! Hence, this approach is not useful for unravelling the microscopic structure of spacetime, any more than quantizing the density and velocity of a fluid will help us to understand molecular dynamics.

    Boltzmann paradigm: "If you can heat it, it must have microstructure"

    Incredibly enough, the microscopic degrees of freedom (atoms of matter or spacetime) are noticeable even at large, macroscopic scales and, in fact, were noticed even by the cavemen! Prehistoric men knew the difference between a hot body and a cold one and the heat content is a direct signature of the underlying microscopic degrees of freedom.

    So, we do have one key link between the microscopic and emergent phenomena which we can exploit. A fluid or a gas exhibits thermal phenomena which involves the concepts of temperature and transfer of heat energy. If the fluid is treated as a continuum and is described by density, velocity etc., all the way down to microscopic scales, then it is not possible for it to exhibit thermal phenomena.

    As first stressed by Boltzmann, the heat content of a fluid arises due to random motion of discrete microscopic structures which must exist in the fluid. These new degrees of freedom - which we now know are related to the actual molecules - make the fluid capable of storing energy internally and exchanging it with the surroundings. Given an apparently continuum phenomenon which exhibits temperature, Boltzmann could infer the existence of underlying discrete degrees of freedom.

    The Boltzmann interpretation of thermal behaviour has two other attractive features. First, while the existence of the discrete degrees of freedom is vital in such an approach, the exact nature of the degrees of freedom is largely irrelevant. For example, whether we are dealing with argon molecules or helium molecules is largely irrelevant in the formulation of gas laws, and such differences can be taken care of in terms of a few well-chosen numbers (like, e.g., the specific heat). This suggests that such a description will have a certain amount of robustness and independence as regards the precise nature of the microscopic degrees of freedom.

    Second, the entropy of the system arises due to our ignoring the microscopic degrees of freedom. Turning this around, one can expect the form of the entropy functional to encode the nature of the microscopic degrees of freedom. If we can arrive at the appropriate form of entropy functional, in terms of some effective degrees of freedom, then we can expect it to provide the correct description. (Incidentally, this is why thermodynamics needed no modification due to either relativity or quantum theory. An equation like, for example, TdS = dE + PdV will have universal applicability as long as the effects of relativity or quantum theory are incorporated in the definition of S(E,V) appropriately.)

    Move on from matter to spacetime. We know that the spacetime horizons, and more generally null surfaces, that arise in general relativity are endowed with temperature and entropy. This suggests that one could use a thermodynamic description to link the standard description of gravity with the statistical mechanics of - as yet unknown - microscopic degrees of freedom using a suitably defined entropy functional.

    A 'top-down' approach to microstructure of spacetime

    My work, starting from 2002, has developed this paradigm into a powerful theoretical structure and has discovered several new results. [In the description below, the numbers refer to the papers in my list of publications; when the link is clickable, you can obtain the arXiv version of the paper.]

    Historically, the idea the gravity could be an emergent phenomenon was first emphasised by Sakharov in the sixties, and one specific implementation of this paradigm was suggested by Ted Jacobson in 1995. (Curiously enough, nobody followed it up seriously until 2002 when I published the first paper on my approach.) When I started out, I first wanted to prove that the conventional approach to gravity itself contained a sufficient number of hints indicating that it is an emergent, thermodynamic force. I achieved this in two ways.

    Gravitational Field equations reduce to a thermodynamic identity

    First, I showed that the field equations of gravity reduce to the identity TdS = dE + PdV when evaluated on any horizon. Starting from the elementary case of a spherically symmetric horizon in Einstein gravity in 2002, this has now been shown (by the work done by me and collaborators, as well as groups elsewhere in the world) to be valid for an impressively large class of spacetimes and models of gravity like the Lovelock theory in D-dimensions (see the first paragraph of Section 5.2 of the review arXiv:0911.5004, for a list!). The PdV term is crucial and makes it technically different from the first law of BH dynamics, the Clausius relation, etc. [and, of course, PdV = Fdx]. See Refs. 147, 181, 188, 199.

    Gravitational action functionals are holographic

    Second, I realized that, if gravity has thermodynamical origin, then the action functional describing gravity must encode this information. It was known for a long time that the action principle in Einstein gravity has a bulk term and surface term and the latter is ignored/removed to obtain the field equations. I and my collaborators could show that there is a "holographic" relationship between the surface and bulk terms of the action functionals, not only in the case of Einstein's theory but in a much wider class of gravitational theories. In other words, the bulk and the surface terms are related by a holographic identity, which has a simple physical origin. It was known previously that the surface term in the action is related to the entropy of the horizon, which is a bit of a mystery because the surface term was ignored while obtaining the field equations. The holographic relation shows how this works. Taking this idea further, one can interpret the action functional in a wide class of gravitational theories as the free energy of the spacetime. (It was possible to obtain these results only because I worked with action principles, while others have concentrated on the field equations.) See Refs. 143, 150, 165, 171, 184, 211.

    The thermodynamic connection goes far beyond Einstein's theory

    These results show that the connection between thermodynamics and gravity goes far beyond Einstein's theory of gravity. In other words it is deeply connected with the fact that gravity is described by the spacetime structure and is not specific to Einstein's theory. Since this generality is telling us something deep and beautiful, concentrating just on Einstein's theory is a wrong approach. Other people working in this area have been focusing mostly on Einstein's theory, which misses some key elements. For a detailed review of the results in theories more general than Einstein's gravity, see Ref. 235.

    Determining the Avogadro number of spacetime

    I could also show that the microscopic degrees of freedom obey a principle of equipartition in static geometries. This was done for Einstein gravity in 2004, and generalised to a wide class of gravitational theories in 2009-10. It should be stressed that Boltzmann's equipartition law, E=(1/2)NkT is a direct link between the microscopic degrees of freedom and macroscopic physics. The E and T in this relation can be defined in the continuum limit of thermodynamics, but N has no meaning in thermodynamics, in which it is infinite! The finite value of N contains information about the statistical mechanics. Remarkably enough, one can prove an identical relation in the case of a general class of gravitational theories and read off N - which is equivalent to determining the Avogadro number of spacetime! In a paper appearing in the Physical Review in June 2010, I have shown how this can be achieved. See Refs. 162, 204, 205, 208.

    Gravitational field equations from entropy density of spacetime

    Given this backdrop, it is obvious that one should be able to derive the field equations for a wide class of gravitational theories, from purely thermodynamic arguments. This can done by introducing an expression for the entropy density of spacetime in terms of the horizons perceived by local Rindler observers and maximising it for all such observers. The expression for entropy density immediately links macroscopic physics to the underlying microscopic degrees of freedom in a very simple and transparent manner. This work, done in 2007-08, goes far beyond the earlier attempts because it addresses a class of models far more general than Einstein's theory and uses extremum principles rather than equations of motion. See Refs. 186, 191, 198, 204, 205.

    What really drives the evolution of the spacetime?

    The field equations can be expressed in a completely thermodynamic language! In dynamic spacetime, the rate of change of gravitational momentum is related to the difference between the number of bulk and surface degrees of freedom. All static spacetimes maintain holographic equipartition; i.e., in these spacetimes, the number of degrees of freedom in the surface is equal to the number of degrees of freedom in the bulk. It is the departure from holographic equipartition that drives the time evolution of the spacetime. This result, which is equivalent to Einstein’s equations, provides an elegant, holographic description of spacetime dynamics. See Refs. 239, 245.

    Implications for the cosmological constant problem

    We need three ingredients to solve the cosmological constant problem: (a) The gravitational field equations must be made invariant under the addition of a constant to the Lagrangian so that gravity is "protected" from the shift in the zero level of the energy densities. (b) At the same time, the solutions to the field equations must allow the cosmological constant to influence the geometry of the universe, because without it, we cannot possibly explain the observed accelerated expansion of the universe. (c) We need a new, fundamental physical principle to determine the numerical value of the cosmological constant since it cannot be introduced as a low energy parameter in the Lagrangian if the theory satisfies (a) above.

    One can further show that the cosmological constant problem cannot be solved in any theory of gravity interacting with matter which satisfies the following three conditions: (1) The theory is generally covariant. (2) The matter equations of motion are invariant under the addition of a constant to the matter Lagrangian. (3) The gravitational field equations are obtained by an unrestricted variation of the metric tensor in the total action. While all the three criteria stated above seem very reasonable, they together will prevent us from solving the cosmological constant problem; so we need to give up at least one of them. Assuming we do not want to give up general covariance of the theory or the freedom to add a constant to the matter Lagrangian, we can only tinker with the third requirement.

    The emergent paradigm does exactly this and thus provides a natural solution to cosmological constant problem. Since the field equations now arise from varying a thermodynamical potential rather than the metric, we bypass (3) above and obtain field equations which are immune to bulk (vacuum) energy! [See Ref. 191.] The cosmological constant appears only as an integration constant to the solutions of the field equations! It is then possible to provide a new physical principle which determines its numerical value.

    Expansion of the universe as a quest for holographic equipartition

    Emergent perspective of gravity provides a novel way of studying cosmology in which the expansion of the universe can be interpreted as equivalent to the emergence of space itself. In such an approach, the dynamics evolves towards a state of holographic equipartition, characterized by the equality of number of bulk and surface degrees of freedom in a region bounded by the Hubble radius. This principle correctly reproduces the standard evolution of a Friedmann universe! . Further, (a) it demands the existence of an early inflationary phase as well as late time acceleration for its successful implementation and (b) highlights a new conserved quantity for our universe (called CosMIn) and fixes its value to be (4 pi). (c) This, in turn relates the cosmological constant to other parameters in high energy physics and correctly predicts its value. Alternatively, the approach predicts the value of the inflationary energy scale (with a factor of 5), if we use observed value of the cosmological constant. This is a unique, parameter-free prediction which is falsifiable, which makes this approach specially attractive. See Refs. 236, 242.

    Can it be just a coincidence?

    Could it be that the links between gravitation and thermodynamical laws are just a coincidence?

    No! I think they are quite deep!

    When I started out in 2002, I thought that this was probably an accidental analogy, but three facts have convinced me that this is fairly deep. First, the results extend far beyond Einstein's gravity and have led to a deeper understanding of the structure of these theories; I consider this feature to be important. Second, it has passed every test I could throw at it in the last eight years. Third, there are several peculiar mathematical features in these theories which have no explanation in the conventional approach but fit very nicely in the thermodynamic perspective.

    In fact, I am now sufficiently convinced of the importance of this connection to include a chapter titled "Gravity as a Emergent Phenomenon", in my recently published textbook, "Gravitation: Foundations and Frontiers", by Cambridge University Press (2010)!

    What are the implications for quantum gravity?

    Where do attempts to quantize gravity (which are "bottom-up") fit into this body of work (which is "top-down")? The picture I have in mind for spacetime is similar to that of a solid or gas made of atoms and described by the theory of elasticity or gas dynamics. Gravitational field equations are analogues of the laws of elasticity. The fact that the laws of elasticity or gravity are emergent phenomena does not mean we do not need to develop a quantum theory for the true degrees of freedom of the system - which are the atoms making up the solid or gas and some unknown entities in the case of spacetime. It only means that the variables used in describing the elasticity [like density, elastic constants, etc.] or gravity [metric, curvature, etc.] will not be of relevance in the microscopic theory and one needs to discover the correct degrees of freedom and then quantise them.

    I interpret my entire program as a "Top-down" approach, from classical theories of gravity to discover the nature of the underlying true degrees of freedom. Once I have them, I need to develop a quantum theory for these degrees of freedom.

    Earlier Work

    Here is a set of highlights from my earlier work: