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On the Connections Between Thermodynamics and General Relativity On the Connections between Thermodynamics and General Relativity Jessica Santiago VICTORIAUNIVERSITYOFWELLINGTON Te Whare Wananga¯ o te UpokooteIkaaM¯ aui¯ School of Mathematics and Statistics Te Kura Matai¯ Tatauranga A thesis submitted to the Victoria University of Wellington arXiv:1912.04470v1 [gr-qc] 10 Dec 2019 in fulfilment of the requirements for the degree of Doctor of Philosophy in Mathematics. Victoria University of Wellington 2019 \H´ametaf´ısica bastante em n~ao pensar em nada. (...) O mist´erio das cousas? Sei l´ao que ´emist´erio! O ´unico mist´erio ´ehaver quem pense no mist´erio. Quem est´aao sol e fecha os olhos, Come¸caa n~ao saber o que ´eo sol E a pensar muitas cousas cheias de calor. Mas abre os olhos e v^eo sol, E j´an~ao pode pensar em nada, Porque a luz do sol vale mais que os pensamentos De todos os fil´osofos e de todos os poetas. A luz do sol n~ao sabe o que faz E por isso n~ao erra e ´ecomum e boa. (...)" Alberto Caeiro O Guardador de Rebanhos | V \There's enough metaphysics in not thinking about anything. (...) The mystery of things? I have no idea what mystery is! The only mystery is there being someone who thinks about mystery. Who is in the sun and shut their eyes, Starts not knowing what the sun is And to think a lot of things full of heat. But they open their eyes and see the sun And can't think about anything anymore, Because the sunlight is worth more than the thoughts Of all philosophers and all poets. The sunlight does not know what it's doing So it's never wrong and it's common and good. (...)" Alberto Caeiro O Guardador de Rebanhos | V i ii Abstract In this thesis, the connections between thermodynamics and general relativity are explored. We introduce some of the history of the interaction between these two the- ories and take some time to individually study important concepts of both of them. Then, we move on to explore the concept of gravitationally induced temperature gradients in equilibrium states, first introduced by Richard Tolman. We explore these Tolman-like temperature gradients, understanding their physical origin and whether they can be generated by other forces or not. We then generalize this con- cept for fluids following generic four-velocities, which are not necessarily generated by Killing vectors, in general stationary space-times. Some examples are given. Driven by the interest of understanding and possibly extending the concept of equilibrium for fluids following trajectories which are not generated by Killing vec- tors, we dedicate ourselves to a more fundamental question: can we still define thermal equilibrium for non-Killing flows? To answer this question we review two of the main theories of relativistic non-perfect fluids: Classical Irreversible Thermo- dynamics and Extended Irreversible Thermodynamics. We also take a tour through the interesting concept of Born-rigid motion, showing some explicit examples of non-Killing rigid flows for Bianchi Type I space-times. These results are impor- tant since they show that the Herglotz{Noether theorem cannot be extended for general curved space-times. We then connect the Born-rigid concept with the re- sults obtained by the relativistic fluid's equilibrium conditions and show that the exact thermodynamic equilibrium can only be achieved along a Killing flow. We do, however, introduce some interesting possibilities which are allowed for non-Killing flows. We then launch into black hole thermodynamics, specifically studying the trans- iii Planckian problem for Hawking radiation. We construct a kinematical model con- sisting of matching two Vaidya spacetimes along a thin shell and show that, as long as the Hawking radiation is emitted only a few Planck lengths (in proper distance) away from the horizon, the trans-Plackian problem can be avoided. We conclude with a brief discussion about what was presented and what can be done in the future. iv Acknowledgments I would firstly like to thank myself, for opening my heart to changes, for facing the challenges with courage and determination (sometimes too much), for committing to learn how to accept what I cannot change, for understanding that the amount I can change around me is not much, and that the first and only possible place to start is inside me. I would also like to thank my supervisor, Matt, for giving me this opportunity, for opening doors in my life and for being present. Thanks for dedicating so much of your time to others and sorry if I asked more than you could give at times. Thanks for having such a big heart with your students. To my office mates and friends Sebastian, Del and Alex. Thanks for all the discussions about physics, life, books, yoga and capitalism. You guys are great. To Finn, who helped me during the moments I most needed. Thanks for helping me to grow, for your company, support, complex food (haha) and, of course, for proofreading my thesis. To Jasmine, Meenu, Susan, Emma and Ellis my beautiful lovely gals. Each one of you is so special and your friendship has nurtured my heart and soul in different and complementary ways. Thank you thank you thank you. You gals are amazing and I really hope to see you around this big world a lot of times. Special shot to Jasmine for proofreading. To my brazilian family in Wellington: Maduro, Gisele, Marina and Andrew. Thanks for the music, the friendship, the parties, and the music again. Parab´ens! To Rosie, Brendan, Hamish, Toni, Teressa and Will. I love living with you guys. Thanks for all the chats, company, introduction to K-pop and delicious food. To Andr´e,who put a lot of effort brainwashing me to meditate and for being a v great friend and a great presence in my life. Thank you. To Cesar and Andr´eLandulfo, who helped with my research. To Ana, Jessica and Natalia. Our friendship is stronger than any distance! You are forever flawless for me. To Matheus, Lucas, Everton, Zanoli, Patricia, Pedro, Otavio and Xin. And to so many others that have inspired joy, peace and love into my heart. Por ultimo, as pessoas que me deram a base para ser quem eu sou e que, mesmo longe, se mantiveram presentes em minha vida e no meu cora¸c~ao. Minha m~ae, diva linda maravilhosa. Voc^e´euma mulher incr´ıvel. Obrigada por me dar colo mesmo pelo whatsapp. Meu pai e meus irm~aos, Adonias, Alan, Alex e Sheila. S´ode pensar em voc^esmeu cora¸c~ao fica mais feliz. Obrigada por estarem presentes. E `aAmanda, minha pequena. Eu amo te ver crescer essa pessoa t~ao sorridente, engra¸cada, cheia de energia e carinhosa. Obrigada por existir na minha vida. I would also like to thank Victoria University of Wellington for the scholarship. vi Contents 1. Introduction1 1.1. Outline...................................6 1.2. Notation and conventions . .7 2. Thermodynamics and General Relativity9 2.1. Thermodynamics . .9 2.1.1. Thermodynamic Equilibrium . 10 2.1.2. The Laws of Thermodynamics . 14 2.2. General Relativity . 19 3. Gravity-induced temperature gradients 31 3.1. The weight of heat . 32 3.2. The physics behind Tolman temperature gradients . 35 3.2.1. The static weak field approximation . 35 3.2.2. Planck's blackbody spectrum . 36 3.2.3. How to measure temperatures . 38 3.3. Electrically induced temperature gradients? . 41 3.3.1. Maxwell's argument . 41 3.3.2. The impossibility of electrically induced temperature gradients 43 3.4. The general case extension . 46 3.5. The rotating universe example . 54 3.6. Black Hole examples . 61 vii Contents 3.7. Covariant Thermodynamics . 67 4. Can we still define thermal equilibrium for non-Killing flows? 73 4.1. RelativisticFluids............................. 75 4.2. Kinematics of fluids and spacetime optics . 76 4.2.1. Shear, expansion and vorticity . 79 4.2.2. The rate of deformation tensor . 82 4.3. Rigidbodymotion ............................ 84 4.3.1. Herglotz-Noether theorem . 85 4.3.2. Killing, conformal Killing and geodesic congruences . 86 4.3.3. Non-geodesic non-Killing congruences: . 91 4.4. Non-Perfect Fluids . 99 4.4.1. Classical Irreversible Thermodynamics . 102 4.4.2. Extended Irreversible Thermodynamics . 107 4.5. The possibility of equilibrium along non Killing flows . 113 5. The trans-Planckian Problem 125 5.1. Introduction . 126 5.2. Static approximation case . 129 5.3. Piecewise Vaidya spacetime . 132 5.3.1. Vaidya spacetime in null coordinates . 133 5.3.2. Matching null coordinates outside/inside . 134 5.3.3. Thin-shell tangent and normal . 135 5.4. Exterior region | outgoing Hawking radiation . 136 5.4.1. Blueshift/redshift . 136 5.4.2. Evading trans-Planckian physics . 138 5.4.3. From Unruh temperature to Hawking temperature . 143 5.5. Interior metric and the final fate of the Vaidya model black hole . 144 5.6. Models for evaporation scenarios . 148 5.6.1. Non equal masses case . 149 5.6.2. Mass matching case . 151 5.6.3. The \time matching" case . 152 viii Contents 5.6.4. The empty-interior massive shell (a consistency check) . 153 5.7. Remarks.................................. 154 6. Famous Last Words 155 A. Some technical results on the Vaidya model 159 Publications related to the PhD 165 Bibliography 167 ix x List of Figures 2.1. Junction conditions . 27 3.1. Temperature gradient caused by a gravitational field . 39 3.2. External observer looking at blackbody radiation from a box . 40 3.3. Gedankenexperiment: Electric heat engine . 45 3.4. Rotating cylinder . 55 3.5.
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