Gravity : An Emergent Perspective

Author:
T. Padmanabhan

Affiliation: Inter University Centre for Astronomy and Astrophysics (IUCAA), Pune, India

Historically, we thought of electrons as particles and photons as waves, time as absolute and gravity as a force. Padmanabhan, in his recent work "Gravity-the Inside Story" which won the First Award in Gravity Research Foundation Essay Contest 2008, suggests that we have similarly misunderstood the true nature of gravity because of the way our ideas evolved historically. When seen with the `right side up', the description of gravity becomes remarkably simple, beautiful, and explains features which we never thought needed explanation!

To understand what is involved in this appraoch one could compare the standard, historical, development of gravity with the approach developed by me over the last few years [1]. Historically, Einstein started with the Principle of Equivalence and --- with a few thought experiments --- motivated why gravity should be described by a metric of spacetime. This approach gives the correct backdrop for the equality of inertial and gravitational masses and describes the kinematics of gravity. Unfortunately there is no equally good guiding principle which Einstein could use that leads in a natural fashion to field equations G_ab= κ T_ab which govern the evolution of g_ab (or to the corresponding action principle). So the dynamics of gravity is not backed by a strong guiding principle.

Strange things happen as soon as: (a) we let the metric to be dynamical and (b) allow for arbitrary coordinate transformations or, equivalently, observers on any timelike curve examining physics. Horizons are inevitable in such a theory and they are always observer dependent. This conclusion arises very simply: (i) Principle of equivalence implies that trajectories of light will be acted by gravity. So in any theory which links gravity to spacetime dynamics, we can have nontrivial null surfaces which block information from certain class of observers. (ii) Similarly, one can construct timelike congruences (e.g., uniformly accelerated trajectories) such that all the curves in such a congruence have a horizon. What is more, the horizon is always an observer dependent concept, even when it can be given a purely geometrical definition. For example, the r = 2M surface in Schwarzschild geometry acts operationally as a horizon only for the class of observers who choose to stay at r > 2M and not for the observers falling into the black hole.

Once we have horizons which are inevitable, we get into more trouble. It is an accepted dictum that all observers have a right to describe physics using an effective theory based only on the variables (s)he can access. (This was, of course, the lesson from renormalization group theory. To describe physics at 10 GeV one shouldn't need to know what happens at 10¹⁴ GeV in "good" theories.) This raises the famous question first posed by Wheeler to Bekenstein [2]: What happens if you mix cold and hot tea and pour it down a horizon, erasing all traces of "crime" in increasing the entropy of the world? The answer to such thought experiments demands that horizons should have an entropy which should increase when energy flows across it.

With hindsight, this is obvious. The Schwarschild horizon -- or for that matter any metric which behaves locally like Rindler metric -- has a temperature which can be identified by the Euclidean continuation. If energy dE flows across a hot horizon of temperature T then the ratio dE/T = dS gives the entropy of the horizon. Again, historically, nobody including Wheeler and Bekenstein looked at the Euclidean periodicity in the Euclidean time (in Rindler or Schwarzschild metrics) before Hawking's result came! And the idea of Rindler temperature came after that of black hole temperature! So in summary, the history proceeded as follows:
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Principle of equivalence ( ~ 1908)

=> Gravity is described by the metric g_ab ( ~ 1908)

? Postulate Einstein's equations without a real guiding principle! (1915)

=> Black hole solutions with horizons (1916) allowing the entropy of hot tea to be hidden ( ~1971)

=> Entropy of black hole horizon (1972)

=> Temperature of black hole horizon (1975)

=> Temperature of the Rindler horizon (1975 -- 1976)
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There are several peculiar features in the theory for which there is no satisfactory answer in the conventional approach described above and they have to thought of as algebraic accidents. But there is an alternative way of approaching the dynamics of gravity, in which these features emerge as naturally as the equality of inertial and gravitational masses emerges in the geometric description of the kinematics of gravity. These new results also show that the thermodynamic description is far more general than just Einstein's theory and occurs in a wide class of theories in which the metric determines the structure of the light cones and null surfaces exist blocking the information. So instead of the historical path, we can proceed as follows reversing most of the arrows:
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Principle of equivalence

=> Gravity is described by the metric g_ab

=> Existence of local Rindler frames (LRFs) with a horizon around any event

=>Temperature of the local Rindler horizon H from the Euclidean continuation

=> Virtual displacements of H allow for flow of energy across a hot horizon hiding an entropy dS = dE=T as perceived by a given observer

=> The local horizon must have an entropy, S_grav

=> The dynamics should arise from maximizing the total entropy of horizon (S_grav ) plus matter (S_m) for all LRF's leading to field equations!
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The procedure uses the local Rindler frame (LRF) around any event P with a local Rindler horizon H. When matter crosses a hot horizon in the LRF -- or, equivalently -- a virtual displacement of the H normal to itself engulfs the matter, some entropy will be lost to the outside observers unless displacing a piece of local Rindler horizon itself costs some entropy S_grav, say. Given the correct expression for S_grav, one can demand that (S_matter + S_grav) should be maximized with respect to all the null vectors which are normals to local patches of null surfaces that can act locally as horizons for a suitable class of observers -- in the spacetime. This puts a constraint on the background spacetime leading to the field equations. To the lowest order, this gives Einstein's equations with calculable corrections [3]. More generally, the resulting field equations are identical to those for Lanczos-Lovelock gravity with a cosmological constant arising as an undetermined integration constant. One can also show, in the general case of Lanczos-Lovelock theory, the on shell value of S_tot gives the correct gravitational entropy, further justifying the original choice. Several peculiar features involving the connection between gravity and thermodynamics is embedded in this approach in a natural fashion. In particular:

♦ There are microscopic degrees of freedom ("atoms of spacetime") which we know nothing about. But just as thermodynamics worked even before we understood atomic structure, we can understand long wavelength gravity arising possibly from a corpuscular spacetime by a thermodynamic approach.

♦ Einstein's equations are essentially thermodynamic identities valid for each and every local Rindler observer [4]. In spacetimes with horizons and high level of symmetry, one can also consider virtual displacements of these horizons (like r_H → r_H + ε) and obviously we will again get TdS = dE + PdV .

♦ If the flow of matter across a horizon costs entropy, the resulting gravitational entropy has to be related to the microscopic degree of freedom associated with the horizon surface. It follows that any dynamical description will require a holographic action with both surface and bulk encoding the same information [5]. For the same reason, the surface term in the action will give the gravitational entropy. Both these features have been investigated in detail by me and my collaborators in the previous years.

Most importantly, this is not just a reformulation of Einstein's theory. Shifting the emphasis from Einstein's field equations to a broader picture of spacetime thermodynamics of horizons leads to a general class of field equations which includes Lanczos-Lovelock gravity. It is now no surprise that Lanczos-Lovelock action is also holographic, is related to entropy and has a thermodynamic interpretation.

References
[1] "Classical and Quantum Thermodynamics of horizons in spherically symmetric spacetimes", T Padmanabhan, Class. Quan. Grav., 19, 5387, (2002), Abstract [gr-qc/0204019];
"Gravity and the Thermodynamics of Horizons", T. Padmanabhan, Phys. Rept., 406, 49, (2005) [gr-qc/0311036];
"Dark Energy and its Implications for Gravity", T. Padmanabhan, (2008) [arXiv:0807.2356].
[2] This is based on what Wheeler told me in 1985, from his recollection of events; it is also mentioned in his book 'A Journey into Gravity and Space-time', [Scientific American Library, NY, 1990] page 221. I have heard somewhat different versions from other sources.
[3] "Dark Energy and Gravity", T. Padmanabhan, Gen.Rel.Grav., 40, 529-564 (2008). Full Text. [arXiv:0705.2533];
"Entropy of Null Surfaces and Dynamics of Spacetime", T. Padmanabhan, Aseem Paranjape, Phys.Rev. D75 064004 (2007). Abstract. [gr- qc/0701003];
[4] "Einstein's equations as a thermodynamic identity: The cases of stationary axisymmetric horizons and evolving spherically symmetric horizons", D. Kothawala, S. Sarkar, T. Padmanabhan, Phys. Letts, B 652, 338-342 (2007) [gr-qc/0701002];
"Thermodynamic route to field equations in Lanczos-Lovelock gravity", A Paranjape, S Sarkar and T Padmanabhan, Phys. Rev. D 74, 104015 (2006). Abstract.
[5] "Holography of gravitational action functionals", A Mukhopadhyay, T Padmanabhan, Phys. Rev. D74,124023, (2006). Abstract. [arXiv:hep-th/0608120].