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Gravity As a Thermodynamic Phenomenon

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Gravity As a Thermodynamic Phenomenon University of Patras Faculty of Science Department of Physics Division of Theoretical, Computational Physics and Astrophysics M.Sc. Thesis Gravity as a thermodynamic phenomenon Demetrios Moustos Supervisor Charis Anastopoulos Patra, February 2014 In memory of my grandmother Abstract The analogy between the laws of black hole mechanics and the laws of ther- modynamics led Bekenstein and Hawking to argue that black holes should be considered as real thermodynamic systems that are characterised by entropy and temperature. In particular, Bekenstein argued that the entropy of a black hole 3 is S = (kBAc )=(4G~), where A is the area of its horizon. In addition, Hawking showed that the temperature of a black hole is T = (~κ)=(2πckB), where κ is its surface gravity. Black hole thermodynamics shows a deeper connection between thermody- namics and gravity. This perspective motivated several ideas that suggest an interpretation of gravity as a thermodynamic phenomenon. The original idea is due to Jacobson. He first inverted the reasoning and showed that the Einstein's equation can be viewed as an equation of state. Later, Padmanabhan showed that the gravitational equations can be interpreted in terms of thermodynamics. He also showed that one can derive the gravitational equations from the thermo- dynamics of spacetime. More recently, Verlinde argued that gravity is an entropic force. The above arguments are components of the broader view, first formulated by Sakharov, that gravity is not a fundamental force, but an emergent one. It arises in the limit of some underlying| yet unknown| microscopic theory, in the same sense that hydrodynamics or elasticity emerge from molecular physics. In this thesis, we examine the arguments of Jacobson, Padmanabhan and Verlinde that suggest the interpretation of gravity as a thermodynamic theory. Acknowledgements I would like to thank the examination committee composed of C. Anastopoulos, D. Sourlas and A. Terzis. Especially, I am deeply grateful to my supervisor Dr. Charis Anastopoulos for his support, his confidence, and his valuable guidance during the years of my graduate studies. Allow me also to thank my family and my friends. The present thesis would not have been completed without their support. Contents Abstract ii Acknowledgements iii 1 Introduction 1 1.1 Black holes . 3 1.1.1 The laws of black hole mechanics . 4 1.1.2 Black hole entropy and the generalized second law . 6 1.2 Hawking effect . 8 1.2.1 Bogolyubov transformations . 8 1.2.2 Hawking radiation . 11 1.3 Unruh effect . 14 1.3.1 Rindler wedge . 14 1.3.2 Thermal Green functions . 15 1.3.3 Unruh temperature . 17 1.4 Entropy bounds . 18 1.4.1 Holographic principle . 19 1.5 Gravity as a thermodynamic phenomenon . 22 1.5.1 Plan of the thesis . 22 2 Thermodynamics of spacetime 24 2.1 The Einstein Equation of state . 24 2.2 Non-equilibrium thermodynamics of spacetime . 27 2.3 Summary and remarks . 32 3 Thermodynamic aspects of gravity 36 3.1 Einstein's equation as a thermodynamic identity on the horizon . 37 3.2 Structure of Einstein-Hilbert action . 38 3.2.1 Holographic structure of gravitational action . 39 3.2.2 Surface term and horizon entropy . 40 3.2.3 Free energy of spacetime . 41 3.2.4 Gravity from spacetime thermodynamics . 42 iv Contents v 3.3 Equipartition of energy in the horizon degrees of freedom . 46 3.4 Gravitational equations as entropy balance condition . 49 3.5 Extremisation of spacetime's entropy functional . 52 3.6 Summary and remarks . 56 4 Gravity as an entropic force 60 4.1 Entropic force . 60 4.2 Emergence of the Newton's laws . 61 4.2.1 Inertia and Newton's second law . 62 4.2.2 Newton's law of gravity . 65 4.2.3 General matter distributions . 65 4.2.4 Newtonian gravity as an entropic force . 67 4.3 Relativistic generalisation . 69 4.4 Summary and remarks . 71 5 Conclusions 73 Appendix A Classical Irreversible Thermodynamics 76 Appendix B Static spherically symmetric spacetime 81 B.1 Imaginary time, periodicity and horizon temperature . 82 Appendix C Elements of the 3+1 formalism of general relativity 84 C.1 Hypersurfaces . 84 C.2 Intrinsic and extrinsic curvature . 85 C.3 The orthogonal projector . 86 C.4 Foliation of globally hyperbolic spacetimes . 88 C.5 3+1 decomposition of Riemann tensor . 89 C.6 The shift vector . 90 C.7 3+1 decomposition of Einstein equation . 92 Bibliography 94 Chapter 1 Introduction In 1687, Sir Isaac Newton's work Philosophiæ Naturalis Principia Mathematica was published for the first time. In this three-volume book, Newton formulated the three laws of classical mechanics and the law of universal gravitation. The latter states that every point mass M attracts any other point mass m with a gravitational force F that is proportional to the product of their masses, and inversely proportional to the square of the distance r = rer between them Mm F = −G e ; (1.1) r2 r where G is the Newton's gravitational constant. Acting on a particle of mass m the gravitational force accelerates it according to Newton's second law F = ma. Newton, with the aid of his laws, explained Kepler's laws of planetary motion and formulated the principles of kinematics of terrestrial bodies|with the study of which Galileo had dealt several years earlier. In a way analogous to electrostatics, one introduces a gravitational potential Φ. The Newtonian theory of gravity is then described by Poisson's equation r2Φ = 4πGρ, (1.2) where ρ is the mass density of an arbitrary continuous distribution of matter that generates the potential. The acceleration, now, of a body in the potential Φ is given by the latter's gradient a = −∇Φ. Albert Einstein published in 1905 [1] his theory of Special Relativity. In this theory, he unified the notions of space and time, introducing the notion of spacetime. He also modified properly the Newtonian laws of mechanics so that they would be invariant under a Lorentz transformation. Maxwell's equations of electromagnetism, formulated in 1865, are Lorentz invariant as well. Hence, the laws of mechanincs are consistent with the principles of special relativity, i.e., (i) the speed of light c is the same in any inertial frame and (ii) the laws of physics are 2 Introduction invariant in any inertial frame. However, inconsistency of Newton's gravitational law with the Special Relativity led Einstein to develop, in 1915 [2], the theory of General Relativity. In General Relativity, gravity is no longer regarded as a force, but as a manifestation of the curvature of the spacetime. Spacetime's curvature is generated by the presence of matter. It is worth pointing out that already in 1959 Le Verrier had reported a discrepancy between the observed rate of precession of the perihelion of Mercury's orbit and the theoretically|calculated within the framework of Newtonian theory|expected. Einstein, in order to formulate his theory, based on two principles: the Equiv- alence Principle and the Principle of General Covariance. The equivelence prin- ciple states that at every spacetime point in an arbitrary gravitational field, a locally inertial coordinate system can be chosen, such that, within a sufficiently small region of this point, all physics laws take the form of those of Special Rela- tivity. The principle of general covariance states that the equations that express the laws of physics should be generally covariant, i.e., they should preserve their form under general coordinate transformations [3, 4]. The content of General Relativity is summarized as follows [5, 6]. Spacetime is a four-dimensional manifold M endowed with a pseudo-Riemannian metric gµν. The curvature of spacetime is related to the matter distribution existed in it by the Einstein's equation1 1 8πG G ≡ R − Rg = T ; (1.3) µν µν 2 µν c4 µν where Gµν is the Einstein tensor, Rµν is the Ricci tensor, R is the Ricci scalar and Tµν is the stress-energy tensor. In vacuum, the Einstein's equation is reduced to Rµν = 0. Newtonian gravity's equation (1.2) is obtained by (1.3) in the limit of a weak gravitational field and slowly moving matter. General relativity gives the correct value of the precession of Mercury's peri- helion. This was probably the first success of the theory. The presence of gravi- tational fields causes also the bending of light. The correct value|calculated in the framework of Einstein's theory| of the bending of light was confirmed by Sir Arthur Eddington during the solar eclipse of 1919. Other significant results of Einstein's theory are the prediction of the existence of gravitational waves and the existence of black holes as solutions to Einstein's equation. A black hole is a region of spacetime where the gravitational field is so strong that even light cannot escape from its horizon, i.e., the boundary of the black hole. Schwarzschild found, in 1916, the first black hole solution to the Einstein's equation. In the framework of Newtonian gravity, John Michell in 1784 and P. 1We use the metric signature (− + ++). Greek indices take the values f0; 1; 2; 3g, whereas Latin indices denotes spatial coordinates and take the values f1; 2; 3g. From now on, we use units G = ~ = c = kB = 1 unless otherwise specified. 1.1 Black holes 3 S. Laplace in 1796 had suggested the existence of massive stars whose escape velocity exceeds the speed of light. However, in the 1970s, it was argued that black holes should be considered as real thermodynamic systems. Such systems are described by four laws, in correspondence with the standard laws of thermodynamics.
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