In Search of Lost Thermodynamics. Erik Curiel

Transcription

1 In Search of Lost Thermodynamics Erik Curiel Munich Center For Mathematical Philosophy Ludwig-Maximilians-Universität and Black Hole Initiative Harvard University Erik Curiel (MCMP; BHI) In Search of Lost Thermodynamics 1 / 35

2 How Thermodynamical Are Black Holes? Erik Curiel (MCMP; BHI) In Search of Lost Thermodynamics 2 / 35

3 A LOT Erik Curiel (MCMP; BHI) In Search of Lost Thermodynamics 3 / 35

4 Outline 1 The Problem 2 Wald s Way 3 In the Shadow of Young QFT-CST in Flower 4 Planck s Way 5 Sodom and Gomorrah 6 Thermodynamics Regained Erik Curiel (MCMP; BHI) In Search of Lost Thermodynamics 4 / 35

5 The Problem 1 The Problem 2 Wald s Way 3 In the Shadow of Young QFT-CST in Flower 4 Planck s Way 5 Sodom and Gomorrah 6 Thermodynamics Regained Erik Curiel (MCMP; BHI) In Search of Lost Thermodynamics 5 / 35

6 The Problem PROBLEMS: 1 zeroth law 2 Hawking radiation not from internal degrees of freedom, but external scattering field 3 only reason for entropy is to save Second Law, but we don t think that s fundamental anyway I ll consider only first problem Erik Curiel (MCMP; BHI) In Search of Lost Thermodynamics 6 / 35

7 Wald s Way 1 The Problem 2 Wald s Way 3 In the Shadow of Young QFT-CST in Flower 4 Planck s Way 5 Sodom and Gomorrah 6 Thermodynamics Regained Erik Curiel (MCMP; BHI) In Search of Lost Thermodynamics 7 / 35

8 Wald s Way Kerr Black Holes region of no escape (event horizon) asymptotically flat: isolated system stationary (asymptotic time-translation symmetry): in equilibrium rotational symmetry: has angular momentum zero electric charge no hair : characterized by mass, angular momentum, electric charge like a thermodynamical system (and a fundamental particle!) (Hayward s dynamical trapping horizons are fascinating, but beyond scope of this talk) Erik Curiel (MCMP; BHI) In Search of Lost Thermodynamics 8 / 35

9 Wald s Way Thermodynamical Objects? late 1960s Wheeler poses fundamental puzzles about black holes and the Second Law of Thermodynamics 1970 Penrose shows how to extract energy; Hawking proves Area Theorem; Christodoulou characterizes irreducible mass, defines reversible and irreversible processes; Geroch s infamous Gedankenexperiment Bekenstein proposes entropy proportional to surface area, formulates Generalized Second Law 1973 Bardeen, Carter and Hawking prove four Laws of Black-Hole Mechanics; most think analogy with thermodynamics purely formal using quantum effects, Hawking shows black holes radiate like perfect black bodies: after brief resistance, most now think black holes are true thermodynamical objects Erik Curiel (MCMP; BHI) In Search of Lost Thermodynamics 9 / 35

10 Wald s Way The Status of the Analogy? A purely formal analogy? Or are black holes truly thermodynamical objects? Are the black-hole laws the laws of thermodynamics extended to cover black holes? Erik Curiel (MCMP; BHI) In Search of Lost Thermodynamics 10 / 35

11 Wald s Way Orthodoxy: Yes! Hawking radiation justifies the claim that κ is a physical temperature the postulated validity of the Generalized Second Law justifies the claim that A is a physical entropy (N.b.: black-hole entropy is NOT entropy of Hawking radiation, and is not based on it!) Erik Curiel (MCMP; BHI) In Search of Lost Thermodynamics 11 / 35

12 In the Shadow of Young QFT-CST in Flower 1 The Problem 2 Wald s Way 3 In the Shadow of Young QFT-CST in Flower 4 Planck s Way 5 Sodom and Gomorrah 6 Thermodynamics Regained Erik Curiel (MCMP; BHI) In Search of Lost Thermodynamics 12 / 35

13 In the Shadow of Young QFT-CST in Flower Philosophical Memory and the Conceptual Identity of Thermodynamics concepts, principles, and relations among them get modified in philosophically rich ways in extension to new framework new avenues to attack old problems Erik Curiel (MCMP; BHI) In Search of Lost Thermodynamics 13 / 35

14 In the Shadow of Young QFT-CST in Flower Scientific Memory and the Physical Identity of Thermodynamics delineate what of standard thermodynamics black holes do and do not satisfy constraints on form of underlying statistical theory (quantum gravity?) Erik Curiel (MCMP; BHI) In Search of Lost Thermodynamics 14 / 35

15 Planck s Way 1 The Problem 2 Wald s Way 3 In the Shadow of Young QFT-CST in Flower 4 Planck s Way 5 Sodom and Gomorrah 6 Thermodynamics Regained Erik Curiel (MCMP; BHI) In Search of Lost Thermodynamics 15 / 35

16 Planck s Way Fundamental Principles and Features of Standard Thermodynamics not just four laws, but also: state space of equilibrium states temperature as mediator/measure of physical couplings classification of processes as adiabatic, reversible, and quasi-static important conjugacy relations between intensive and extensive variables additivity of entropy equilibrium states maximize entropy and minimize free energy the Clausius and Kelvin Postulates... Erik Curiel (MCMP; BHI) In Search of Lost Thermodynamics 16 / 35

17 Planck s Way Extending Thermodynamics into New Fields How to Decide Whether Blackbody Radiation Is Thermodynamical? need Gibbsian heat term to save First Law? analogue of temperature physically couples in right way to ordinary thermodynamical systems? violations of Second Law without entropy of blackbody radiation (e.g., perpetuum mobile of 2nd kind: Clausius and Kelvin Postulates)? analogues of as much of the rest as possible... Erik Curiel (MCMP; BHI) In Search of Lost Thermodynamics 17 / 35

18 Planck s Way THEN... blackbody radiation is real thermodynamical system BUT... differences point to need to modify statistical grounding, and place constraints on how to do it Erik Curiel (MCMP; BHI) In Search of Lost Thermodynamics 18 / 35

19 Planck s Way can we do same for black holes? Erik Curiel (MCMP; BHI) In Search of Lost Thermodynamics 19 / 35

20 Sodom and Gomorrah 1 The Problem 2 Wald s Way 3 In the Shadow of Young QFT-CST in Flower 4 Planck s Way 5 Sodom and Gomorrah 6 Thermodynamics Regained Erik Curiel (MCMP; BHI) In Search of Lost Thermodynamics 20 / 35

21 Sodom and Gomorrah Zeroth Law Thermodynamics The temperature T is constant throughout a body in thermal equilibrium. Black Holes The surface gravity κ is constant over the event horizon of a stationary black hole. Erik Curiel (MCMP; BHI) In Search of Lost Thermodynamics 21 / 35

22 Sodom and Gomorrah REAL Zeroth Law Two bodies in equilibrium with a third are in equilibrium with each other. Erik Curiel (MCMP; BHI) In Search of Lost Thermodynamics 22 / 35

23 Sodom and Gomorrah Fundamental Role of Transitivity construction of state space of equilibrium states state-function of intensive quantity physically identified as temperature equilibrium implies constancy of temperature temperature as mediator and measure of thermal coupling definition of entropy, its additivity characterization of processes as adiabatic, reversible, and quasi-static conjugacy of intensive and extensive quantities entropy maximization, free-energy minimization Erik Curiel (MCMP; BHI) In Search of Lost Thermodynamics 23 / 35

24 Sodom and Gomorrah CONSTANCY OF TEMPERATURE ALONE DOES NOT SUFFICE! Erik Curiel (MCMP; BHI) In Search of Lost Thermodynamics 24 / 35

25 Thermodynamics Regained 1 The Problem 2 Wald s Way 3 In the Shadow of Young QFT-CST in Flower 4 Planck s Way 5 Sodom and Gomorrah 6 Thermodynamics Regained Erik Curiel (MCMP; BHI) In Search of Lost Thermodynamics 25 / 35

26 Thermodynamics Regained A Strengthened Zeroth Law for Black Holes? Yes! Claim: Constancy + Kelvin Postulate Transitivity Recall: The Kelvin Postulate A transformation whose only final result is to transform into work heat extracted from a source that is at the same temperature throughout is impossible. Erik Curiel (MCMP; BHI) In Search of Lost Thermodynamics 26 / 35

27 Thermodynamics Regained What Is Mutual Equilibrium? claim: all we need is a sufficient condition Condition there will be no spontaneous transfer of heat from one system to another other, and neither will spontaneously perform work on the other, when two bodies in mutual equilibrium are brought into contact Erik Curiel (MCMP; BHI) In Search of Lost Thermodynamics 27 / 35

28 Thermodynamics Regained Deriving the Zeroth Law (Crude Sketch) three ordinary thermodynamical systems A, B and C 1 assume A and C in mutual equilibrium, and B and C, but not A and B 2 bring A and C into contact: no heat spontaneously transferred, so they are same temperature (similarly for B and C) A and B same temperature 3 bring A and B in contact 4 ex hypothesi, one must spontaneously perform work on the other 5 violation of the Kelvin Postulate 6 they are in mutual equilibrium Erik Curiel (MCMP; BHI) In Search of Lost Thermodynamics 28 / 35

29 Thermodynamics Regained Mutual Equilibrium for Black Holes so: what are work and heat for black holes? Erik Curiel (MCMP; BHI) In Search of Lost Thermodynamics 29 / 35

30 Thermodynamics Regained Irreducible Mass M irr a convex, positive-definite function of total mass and angular momentum; total mass cannot be reduced below initial value of M irr by any physical process M 2 irr := 1 2 [M 2 + (M 4 J 2 ) 1 2 ] so M 2 = M 2 irr + J 2 4M 2 irr M 2 irr Erik Curiel (MCMP; BHI) In Search of Lost Thermodynamics 30 / 35

31 Thermodynamics Regained Irreducibility of Irreducible Mass A = 16πM 2 irr thus Second Law implies Mirr 2 through any physical process cannot decrease Erik Curiel (MCMP; BHI) In Search of Lost Thermodynamics 31 / 35

32 Thermodynamics Regained Free Energy, Heat, and Work for Black Holes since M cannot be reduced below M irr : free energy := M M irr so quantity of heat transferred in any process: change in total energy minus change in free energy: Q BH := M irr (> 0) so work : change in total energy not due to change in free energy (rotational and radiative processes) Erik Curiel (MCMP; BHI) In Search of Lost Thermodynamics 32 / 35

33 Thermodynamics Regained The Kelvin Postulate for Black Holes A transformation whose only final result is that a quantity of heat is extracted from a stationary black hole and transformed entirely into work is impossible. Erik Curiel (MCMP; BHI) In Search of Lost Thermodynamics 33 / 35

34 Thermodynamics Regained Proof (Crude Sketch): 1 assume such a transformation possible 2 if heat extracted is purely gravitational, then change in irreducible mass must be strictly greater than change in total mass (i.e., irreducible mass must increase) 3 so area (entropy) must also increase, violating the Postulate (rigorous proof à la Wald s physical process version of the GSL; straightforward emendation to accommodate heat transfer by Hawking radiation) Erik Curiel (MCMP; BHI) In Search of Lost Thermodynamics 34 / 35

35 Thermodynamics Regained Promissory Note construction of equilibrium state-space, etc. Erik Curiel (MCMP; BHI) In Search of Lost Thermodynamics 35 / 35