UNIVERSITY OF SOUTHAMPTON Faculty of Social & Human Sciences School of Mathematics Covariant Thermodynamics & Relativity by C´esarSim´onL´opez-Monsalvo arXiv:1107.1005v1 [gr-qc] 6 Jul 2011 Thesis for the degree of Doctor of Philosophy March 2011 UNIVERSITY OF SOUTHAMPTON ABSTRACT FACULTY OF SOCIAL AND HUMAN SCIENCES SCHOOL OF MATHEMATICS Doctor of Philosophy COVARIANT THERMODYNAMICS AND RELATIVITY by C´esarSim´onL´opez-Monsalvo This thesis deals with the dynamics of irreversible processes within the context of the general theory of relativity. In particular, we address the problem of the ‘infinite’ speed of propagation of thermal disturbances in a dissipative fluid. Although this problem is not new, its best known solution - the Israel and Stewart second order expansion - has an effective, rather than fundamental, character. The present work builds on the multi-fluid variational approach to relativistic dissipation, pioneered by Carter, and provides a dynamical theory of heat conduction. The novel property of such approach is the thermodynamic interpretation associated with a two-fluid system whose constituents are matter and entropy. The dynamics of this model leads to a relativistic generalisation of the Cattaneo equation; the constitutive relation for causal heat transport. A comparison with the Israel and Stewart model is presented and its equivalence is shown. This discussion provides new insights into the not-well understood definition of a non-equilibrium temperature. A crucial feature of the multi-fluid approach is the interaction between its constituents. It is a well known fact that when two, or more, fluids interact, instabilities may occur. Within this work, the two-stream instability analysis is extended to the relativistic domain. As far as the author is aware, such extension has not been discussed in the literature. The analysis allows to assess the stability and causality of relativistic models of matter and their linear deviations from thermodynamic equilibrium directly from their equations of state or, equivalently, their Lagrangian densities. For completeness, a brief digression on a consistent (stable and causal) ‘first-order’ model is also included. Finally, the road to follow is laid by posing some physical applications together with some future perspectives and closing remarks. To sum up, the variational approach to heat conduction presented in this thesis consti- tutes a mathematically promising formalism to explore the relativistic evolution towards equilibrium of dissipative fluids in a dynamical manner and to get a deeper conceptual understanding of non-equilibrium thermodynamic quantities. Moreover, it might also be useful to explore the more fundamental issues of the irreversible dynamics of relativity and its connections with the time asymmetry of nature. i Contents Abstract i Contents ii List of figures v Declaration of authorship vi Acknowledgements vii 1 Introduction 1 1.1 The empirical nature of Thermodynamics . 2 1.2 Historical problems of relativistic heat conduction . 5 1.3 The multi-fluid approach to thermal dynamics . 7 1.4 The state of the art . 10 2 Relativity and Thermodynamics 12 2.1 Spacetime and matter . 12 2.1.1 The principles of relativity . 13 2.1.2 The symmetry of general relativity . 15 2.1.3 Equations of motion . 17 2.2 Irreversible processes . 22 2.2.1 Equilibrium thermodynamics . 23 2.2.2 Linear Irreversible Thermodynamics . 26 2.2.3 Extended Irreversible Thermodynamics . 32 2.3 Irreversible processes in general relativity . 34 ii Contents 3 Relativistic kinetic theory 36 3.1 Symplectic structure of phase space . 36 3.1.1 Forms and motion . 39 3.1.2 Volume elements . 44 3.1.3 Individual mass-shell volume measure . 45 3.2 Boltzmann's equation . 46 3.3 Entropy . 47 3.3.1 H-theorem . 51 3.4 Macroscopic fluid equations . 54 3.4.1 Thermal equilibrium and the Tolman-Ehrenfest effect . 56 3.4.2 Deviations from thermal equilibrium . 58 4 Relativistic theories of dissipation 60 4.1 First-order theories . 61 4.1.1 Eckart's model . 62 4.1.2 Landau & Lifshitz theory . 67 4.2 Second-order theories . 67 4.2.1 The Israel & Stewart second-order theory . 68 4.2.2 Carter's theory of dissipation . 72 5 Variational thermal dynamics 76 5.1 Convective construction . 77 5.1.1 Lagrangian and Eulerian variations . 80 5.1.2 Eulerian variation of conserved currents . 83 5.2 The two-fluids action principle . 84 5.2.1 The unconstrained variation . 85 5.2.2 The constrained variation . 88 5.3 Thermodynamic interpretation . 91 5.3.1 The temperature problem . 92 5.4 Rheometric covariant thermodynamics . 94 5.4.1 Internal dynamics: the relativistic Cattaneo equation . 96 5.4.2 Further remarks on the internal thermal dynamics . 100 5.4.3 Equivalence with Israel & Stewart . 102 5.4.4 Covariance and entropy production: an electrodynamic analogue . 104 6 Adiabatic perturbations 106 6.1 Linear perturbations of thermal equilibrium . 107 6.2 Sound waves in a single fluid . 110 iii Contents 6.3 Sound waves in the matter and entropy system . 114 6.3.1 The completely decoupled case . 118 6.3.2 Cross-constituent coupling . 119 6.3.3 The role of entrainment . 122 7 Non-adiabatic perturbations 129 7.1 Transverse modes: first stability criterion . 130 7.2 Longitudinal modes: causality and the onset of instability . 132 7.3 What have we learned? . 137 8 Closing remarks 139 8.1 Further developments . 140 8.1.1 Microscopic foundations of relativistic thermal dynamics . 140 8.1.2 The `near' and `far' from equilibrium entropy conditions . 143 8.1.3 Classical origins of the arrow of time . 144 8.2 Conclusion . 145 A Regular and Anomalous Heat Conduction 147 Glossary 150 Bibliography 150 iv List of figures 2.1 This figure shows a comparison of the evolution of a thermal disturbance modelled with the heat (left) and telegrapher (right) equations. The vertical axis represents time, while the horizontal corresponds to one-dimensional space. The initial condition is a delta function at the origin of the spatial direction. The evolution occurs towards the top of the page. We can see that, for the heat equation, the pulse spreads over the entire region without any lag. On the contrary, the evolution provided by the telegraph equation propagates a thermal signal which dissipates away to merge the long term solution of the heat equation (the transition between `fast' and `slow' heat flux). The bottom graph shows a how the two distinct evolutions become indistinguishable in the `long' run. Here, the `z'-axis corresponds to tem- perature, while the spatial and temporal directions should be obvious from the description of the top images. 33 6.1 Plots of the real (top) and imaginary (bottom) parts of the mode frequencies 2 σn as functions of w, for cs = 0:5, b = 1:0, and βns = 0. The merger of two frequencies, and subsequent non-zero imaginary parts, signal the presence of a two-stream instability. 123 6.2 Plots of the real and imaginary parts of the mode frequencies σn as functions \2 2 of y and C , for cs = 0:5, and b = 1:0. The lines of intersection and subsequent merger of two frequencies signal the presence of a two-stream instability. 128 8.1 Gravity and entropy production: What can our intuition tell us about the future of this initial configuration? [This image is property of Steve Strawn, splutphoto.com] ..................................146 v Declaration of authorship I, C´esarSim´onL´opez-Monsalvo, declare that the thesis entitled: Covariant Thermodynamics and Relativity, and the work presented in the thesis are both my own, and have been generated by me as the result of my own original research. I confirm that: • this work was done wholly or mainly while in candidature for a research degree at this University; • where any part of this thesis has previously been submitted for a degree or any other qualification at this University or any other institution, this has been clearly stated; • where I have consulted the published work of others, this is always clearly attributed; • where I have quoted from the work of others, the source is always given. With the exception of such quotations, this thesis is entirely my own work; • I have acknowledged all main sources of help; • where the thesis is based on work done by myself jointly with others, I have made clear exactly what was done by others and what I have contributed myself; • parts of this work have been published as: 1. C. S. Lopez-Monsalvo and N. Andersson, Thermal Dynamics in General Relativity, Proc. Roy. Soc. A 467:738759, March 2011. arXiv:1006.2978v1 [gr-qc]. 2. L. Samuelsson, C. S. Lopez-Monsalvo, N. Andersson and G. Comer, Relativistic two- stream instability, Gen. Rel. Grav. 42, 413-433, 2010. arXiv:0906.4002v1 [gr-qc]. 3. C. S. Lopez-Monsalvo, Heat conduction in relativistic systems: alternatives and per- spectives, In IoP Gravitational Physics Group newsletter December 2010. arXiv:1011.6628 [gr-qc]. Date: Signed: vi Acknowledgements Although immensely rewarding, the path of science - I believe - is hardly ever smooth. In this work, I had the fortune to be assisted by two truly inspirational guides. I am greatly indebted to Nils Andersson, whose helpful advice always arrived with incisive clarity. He has contributed more than anyone to direct my brief explorations on thermodynamics and relativity. I am also thankful to James Vickers for sharing with me a little bit of his mathematical sophistication. Even his tiniest contribution always showed me a clearer way. To both of them, all my admiration. However, should the reader find errors and imprecisions in this work, those are the sole consequence of my own misconceptions. I would like to thank in advance my examiners: Ian Hawke and David Matravers, for the time and effort spent reading this thesis.