Classical Black Holes Are Hot†‡
Erik Curiel§
11. Sept 2023

Contents
1 Introduction

2

2 The Laws of Black-Hole Mechanics and the Laws of Thermodynamics

6

3 The Standard Argument Does Not Work

9

4 Temperature and Entropy in Classical Thermodynamics

12† I thank The Young Guns of the Spacetime Church of the Angle Brackets for helpful discussion on an ear-lier draft of this paper, and Ted Jacobson and Carlo Rovelli for insightful and critical discussion of my proposed
Carnot-Geroch cycle for classical black holes. I also thank Harvey Brown, Craig Callender, Juliusz Doboszewski,
Robert Geroch, Sean Gryb, David Malament, John Manchak, Ramesh Narayan, Carina Prunkl, Katie Robertson,
Chris Smeenk, Karim Thébault, Jim Weatherall and Chris Wüthrich for many fruitful and enjoyable discussionsof all the issues I treat here, and in particular I thank Harvey for his relentless and, most of all, extraordinarily
helpful asking of me, “So what?”. Finally, I thank Robert Wald for mercilessly pushing me over the course of several enjoyable days of conversation on the problem of possible violations of the Generalized Second Law (which
I discuss in some detail in §6), with appreciation of Bill Unruh for serving as an amused Greek Chorus for theshow.
‡ This paper started life as several vestigial manuscripts that I began in the early 2000s, which, in an act ofSchumpeterian creative destruction, I merged around 2008. I began giving talks based on the material at physics
and philosophy departments and conferences around 2011. A manuscript more or less in this form, albeit withmany and variegated modifications, clarifications, emendations, deletions and just plain old changes over theyears, has been floating around the community since 2014. Even the title has oscillated over the years, between
the present bold declaration and the more diffident “Are Classical Black Holes Hot or Cold?”, reflecting my waxing and waning skepticism and enthusiasm with regard to my own arguments and conclusions at the time. Since
2018, both the manuscript and my attitude (enthusiasm), and so the title, have remained remarkably stableagainst peturbations, with only small evolutions in each. It is, therefore, time to publish the damn thing and get
it behind me. Also, because of its history, the paper does not engage in any serious way with recent philosophical literature on black hole thermodynamics, of which, I am happy to report there is a growing body. There remains, yet, the problem of existing references in other work to earlier versions of this paper—according to Google
Scholar currently there are 14 of them making their way in the world (who are those intrepid souls?), each having
its own peculiar target, depending on the date the author looked at the manuscript, and which version she or helooked at. If you have come to this paper from one of those references and not a reference to this published version, please take the time to make sure that what the author says about it reflects what appears in this version,and has not been changed since the earlier, cited version. I apologize for the bother.
§ Author’s affiliations: Munich Center for Mathematical Philosophy, LMU Munich; Black Hole Initiative,
Harvard University; email: erik@strangebeautiful.com

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5 Taking Classical Black Holes Seriously as Thermodynamical Objects

15

5.1

Irreducible Mass, Free Energy, “Heat” and Thermal Coupling of Black Holes . . . .

16

5.2

Carnot-Geroch Cycles for Schwarzschild Black Holes . . . . . . . . . . . . . . . . .

18

5.3

The Generalized Clausius and Kelvin Principles for Black Holes . . . . . . . . . . .

24

5.4

The Statistical and Quantum Connection . . . . . . . . . . . . . . . . . . . . . . .

24

6 Problems, Possible Resolutions, Possible Insights, and Questions

25

References

31

ABSTRACTIn the early 1970s it was realized that there is a striking formal analogy between theLaws of black-hole mechanics and the Laws of classical thermodynamics. Before thediscovery of Hawking radiation, however, it was generally thought that the analogy was
only formal, and did not reflect a deep connection between gravitational and thermodynamical phenomena. It is still commonly held that the surface gravity of a stationary
black hole can be construed as a true physical temperature and its area as a true entropy
only when quantum effects are taken into account; in the context of classical generalrelativity alone, one cannot cogently construe them so. Does the use of quantum field
theory in curved spacetime offer the only hope for taking the analogy seriously? I think
the answer is ‘no’. To attempt to justify that answer, I shall begin by arguing thatthe standard argument to the contrary is not physically well founded, and in any event
begs the question. Looking at the various ways that the ideas of “temperature” and“entropy” enter classical thermodynamics then will suggest arguments that, I claim,show the analogy between classical black-hole mechanics and classical thermodynamics should be taken more seriously, without the need to rely on or invoke quantum
mechanics. In particular, I construct an analogue of a Carnot cycle in which a blackhole “couples” with an ordinary thermodynamical system in such a way that its surface
gravity plays the role of temperature and its area that of entropy. Thus, the connection
between classical general relativity and classical thermodynamics on their own seems
to be already deep and physically significant, independent of quantum mechanics. I
conclude with a discussion of how my arguments, if successful, could enrich our conceptual understanding of such traditional, vexed issues as the nature of entropy as aphysical quantity and the character and status of the Second Law of thermodynamics.

1

Introduction

I aim in this paper to clarify the status of the analogy between black-hole mechanics restricted
to general relativity on the one hand (i.e., with no input from quantum field theory on curved
spacetime or from any other type of semi-classical calculation) and classical thermodynamics on
the other (“classical” in the sense that no quantum and no statistical considerations come into play).
Based on the striking formal similarities of the respective mathematical formulæ of the Zeroth,First, Second and Third Laws of classical thermodynamics and of the mechanics of black holes
in stationary, axisymmetric, asymptotically flat spacetimes, as I discuss in §2, the best particular2Classical Black Holes Are Hot

11. Sept 2023

analogies seem to be: (1) that between the surface gravity of a black hole as measured on its
event horizon and the temperature of a classical system; and (2) that between surface area ofthe horizon and entropy.1 When it is also noted that black holes, like ordinary thermodynamicalsystems, are characterized by a small number of gross parameters independent of any details about
underlying microstructure, and that each version of the First Law states a conservation principle
for the same quantity as the other, viz., mass-energy, it becomes tempting to surmise that somedeep or fundamental connection between black holes and thermodynamics is being uncovered. But
is it of real physical significance?
The conventional answer to this question is ‘no’. Because classical black holes seem to be perfectabsorbers, they would seem to have a temperature of absolute zero, even when they have non-zerosurface gravity (Bardeen et al. 1973). It is only with the introduction of quantum considerations,the standard account runs, in particular the derivation of Hawking radiation, that one finds grounds
for taking the analogy seriously (Wald 1999). And yet the startling and suggestive fact remainsthat one can derive laws for black holes formally identical to those of classical thermodynamicalsystems from the fundamental principles of general relativity itself with no aid from quantum field
theory in curved spacetime.
It is therefore important to recognize that we have no derivation of black-hole thermodynamics
from an underlying statistical framework that is widely acknowledged as anything more thantheoretical speculation based on possible—but not otherwise justified—“principles” of quantum
gravity. In the 19th Century, when we had no more or less direct experimental access to theunderlying atomic structure of matter, the best evidence for the idea that thermodynamics could
be “reduced” to statistical mechanics was the gross thermodynamical behavior of matter: the First
Law, with its validity hinging on the assumption of a kind of energy known as “heat”, about whichnothing else was known except that it was not a substance, implied that there had to be hidden
material degrees of freedom in the dynamics of which this form of energy was expressed, stored,
and evolved. Without such evidence, the development of a statistical mechanics of the underlying
micro-structure of matter would have been a work of pure theoretical speculation, albeit a beautiful
and potentially compelling one.
We are in the same situation now with black holes, except that we can have no direct experimental access to provide the strongest possible evidence for their thermodynamical behavior.
The strongest evidence possible for us in our current state of knowledge is the extension of well
supported theory into those regimes that we have no direct access to, based on nothing else but
untested principles and intuitions that seem, for various reasons, most plausible to us. In thissituation, stronger evidence will come from the extension of theory that is itself better supported
by empirical tests. Classical general relativity scores infinitely better in this regard than quantum
field theory on curved spacetime. If we can demonstrate that models of black holes in generalrelativity evince manifestly thermodynamical behavior based on theoretical arguments that extend only classical general relativity into those empirically inaccessible regimes, we will therefore
have provided significantly stronger evidence than could come from analyzing quantum effects incurved spacetimes for the idea that there is an underlying micro-structure possibly amenable to a
1. Both the surface gravity and the surface area in question are defined with respect to the orbits of the Killingfields in virtue of which the spacetime is qualified as ‘stationary’ and ‘axisymmetric’. See Wald (1984, ch. 12) for
details.

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statistical treatment that will explain that thermodynamical behavior.
Does the use of quantum field theory in curved spacetime, however, offer the only hope fortaking the analogy between the laws of black holes and those of thermodynamics seriously? I
think the answer is ‘no’. To attempt to justify that answer, I shall begin by arguing in §3 thatthe standard argument to the contrary is not physically well founded, and in any event begs thequestion. I will, therefore, in §4, look at the various ways that the ideas of “temperature” and“entropy” enter classical thermodynamics, which will suggest arguments that show the analogy
between classical black-hole mechanics and classical thermodynamics should be taken seriously
indeed, without the need to rely on or invoke quantum mechanics. If this is correct, then there
may already be a deep connection between general relativity and classical thermodynamics ontheir own, independent of quantum mechanics. If this is not correct, then one most ask what it isabout purely gravitational systems that, in contradistinction to ordinary matter, blocks them fromadmitting a classical thermodynamical description. Ordinary gases, liquids and solids, after all,have a well defined thermodynamics without accounting for quantum effects. Why should large
black holes be different?
My arguments in this paper, however, are not only negative. I do think that the connection
between gravitational and thermodynamical phenomena suggested by the formal equivalence of
their respective Laws is of real physical significance. My strongest argument in favor of this claim is
the construction, in §5, of the analogue of a Carnot cycle with the heat sink provided by a stationary
black hole. In the process, the black hole’s surface gravity and area play, respectively, the physicalroles of temperature and entropy of an ordinary heat sink in an ordinary Carnot cycle. The process
also grounds the construction of an absolute temperature scale that applies both to black holes
and to ordinary classical thermodynamical systems. Finally, there follows from the construction
the existence of a universal constant with the physical dimension needed to give surface gravity thephysical dimension of temperature and area the physical dimension of entropy. The icing on the
cake would be a proof of appropriate analogues of the Clausius and Kelvin Postulates—the bases
for the introduction of temperature and entropy in classical thermodynamics—in the context ofclassical black hole thermodynamics. I formulate their analogues in §5.3, and discuss them, but I
do not attempt to prove them.
If surface gravity and area couple to ordinary thermodynamical systems in the same way as
temperature and entropy, respectively, do, and if they are introduced into the theory using the same
constructions and arguments, then there can be no grounds for denying that they physically area real temperature and entropy. To put it more provocatively, if my claim is correct, then gravity
on its own, independent of its relation to the other three known fundamental forces so successfully
treated by quantum field theory, already is a fundamentally thermodynamical phenomenon.2 I
want to stress, nonetheless, that I do not consider quantum effects to be irrelevant when considering
possible relations between gravitational physics and thermodynamics. Indeed, in §5.4 I argue, inlight of the result of Kiefer (2004), that the way that the classical “vibratory” modes of a perturbed
classical black hole give rise to the Hawking temperature in the quantum regime validates the ideathat the thermodynamical nature of black holes already manifests itself in the classical regime. I
want only to argue here for the idea that classical black holes are thermodynamical objects in a
2. If one could show that the sorts of arguments I give here could be translated into the framework of Newtonian gravitational theory, that would provide even stronger support for this last claim.

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physically significant way, independent of quantum considerations. I emphasize that my arguments
are not intended to show that there are no conceptual or foundational problems at all with theattribution of thermodynamical properties and behavior to classical black holes, only that the
attribution suffers no problems worse than those already faced by classical thermodynamics when
one ignores statistical and quantum considerations.
I conclude the paper, in §6, with a discussion of possible problems with my arguments andconstructions, some remarks on possible lessons my conclusions, if correct, may yield, and someopen questions.
Before diving in, I should perhaps say, by way of background, that I am curious about thisquestion in the first place in part because of my curiousity about the larger question of the relation
between thermodynamical characteristics of a physical system and the possibility of always being
able to or indeed always being required to find an underlying statistical interpretation of thosethermodynamical characteristics. That the laws of black hole mechanics follow from the fundamental theory itself (in this case, general relativity), and are not as with classical thermodynamics
an independent adjunct connected to the underlying fundamental (Newtonian) theory through theuse of statistical devices, could suggest that thermodynamics has itself more of the nature of afundamental theory than has been thought since the advent of statistical mechanics—or at leastthat thermodynamical characteristics and quantities of physical systems may be fundamental tothem in some way analogous to that of other fundamental characteristics and dynamical quantities,
such as the possession of a stress-energy tensor, for example, and its satisfaction of some form ofcovariant conservation principle.
In a similar vein, these sorts of results may also perhaps lend support to the idea that generalrelativity is an effective field theory, and the Einstein field equation only an equation of state, à la
Jacobson (1995) and Padmanabhan (2005). If that is true, then the entire program of “quantizing
gravity” may be misguided from the start, as it would be equally mistaken to try to quantize theclassical thermodynamics of an ordinary gas rather than its statistically modeled molecular dynamics.3 Yet another possibility, independent of but perhaps complementary to that just mentioned, isthat one may take my arguments as showing that the signature of quantum gravity, in particular
the traces of whatever statistical quantities it may give us for making traditional sense of the thermodynamical phenomena I discuss here, show up already in purely classical, non-statistical theory,
thus giving us clues to what kind of statistical theory we may need underlying it, by looking at differences between black-hole thermodynamics and ordinary thermodynamics. Finally, and I thinkmost importantly, my arguments lend prima facie support to projects (especially in cosmology)
that want to attribute entropy generically to purely classical “gravitational degrees of freedom”, asin the work of Clifton et al. (2013), and as required by Penrose’s Conformal Curvature Hypothesis(Penrose 1979), and several programs in quantum gravity (Rovelli 2008; Amelino-Camelia 2013;
Ashtekar et al. 2015b; Elvang and Horowitz 2015).
I do not intend to investigate these larger issues here, however. I intend to investigate only the
3. Carlo Rovelli remarked to me in conversation that in fact there is no reason not to quantize a classical theory of gross matter such as, e.g., Navier-Stokes theory: we would get a theory that describes quantum corrections
to the classical theory, such as phonons. While this is an important observation, it does not touch the heart of thematter, which is that such a quantization could not produce a fundamental theory of the matter, as it is hopedthat a quantization of general relativity would produce a fundamental quantum theory of gravity.

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status of the analogy between the laws of classical thermodynamics on the one hand and those of
black-hole mechanics in classical general relativity on the other. I mention these larger issues only
to give some of my motivation for this work, and to place it in the context of important work being
done in many branches of theoretical physics today.
There are other motivations behind this project as well. Although philosophers of physics haverecently begun to work on issues arising from proposals for theories of quantum gravity, many ofwhich take as one of their starting points the seemingly thermodynamical character of gravitational
phenomena as exemplified by the laws of black-hole mechanics, almost no philosophical work has
been done investigating the nature of this seemingly thermodynamical character as revealed by
the structures of general relativity and of quantum field theory formulated on curved, relativistic
spacetimes.4 Because general relativity and quantum field theory are well entrenched, clearly andrigorously articulated physical theories, I believe it behooves philosophers to study them, if not
before, at least in conjunction with work done on quantum gravity.
Finally, I believe that to study the thermodynamics of black holes, whether in the purely
classical regime or when quantum effects are also taken into account, may have a rich conceptual
payoff in another direction: as suggesting new avenues of attack on old, well known, and deeply
recalcitrant problems such as the nature of entropy as a physical quantity, and the characterand status of the Second Law of thermodynamics. When physical quantities and principles aretaken from their original, familiar context and extended into utterly new regimes—as is the casewith entropy, temperature and the Second Law for black holes—one may fairly hope that themodifications and emendations required both conceptually and formally for the understanding of
those quantities and principles in the new regime will lead to a deeper and more finely nuanced andmulti-faceted understanding of them in the original context as well. This certainly happened whenentropy, temperature and the Second Law were extended at the end of the 19th Century from their
original home among gases, fluids and some solid states of matter to encompass electromagnetic
radiation. I think we can with confidence expect the same to happen here.

2

The Laws of Black-Hole Mechanics and the Laws of Thermodynamics
The four laws of black hole mechanics, which are analogous to those of thermodynamics,
were originally derived from the classical Einstein equation. With the discovery of thequantum Hawking radiation, it became clear that the analogy is, in fact, an identity.
How did classical general relativity know that the horizon area would turn out to be aform of entropy, and that surface gravity is a temperature?
Ted Jacobson (1995)“Thermodynamics of Spacetime: The Einstein Equation of State”
Within the context of general relativity, one can derive laws describing the behavior of black

holes in stationary, asymptotically flat spacetimes bearing a remarkable resemblance to the classical4. While this was true when I wrote the first draft of this paper in 2014, I am happy to report that the situation is rapidly changing. See, e.g., Wallace (2018; 2019), Dougherty and Callender (2016), Prunkl and Timpson(2019), and Wüthrich (2019).

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laws of equilibrium thermodynamics. I restrict attention to the asymptotically flat case, becausethat is the simplest natural analogue of an isolated system for black holes in general relativity.5 I
restrict attention to stationary black holes because those are the simplest natural analogue of an
equilibrated system for black holes in general relativity.
Now, for the laws themselves:6
Zeroth Law
Thermodynamics
The temperature T is constant throughout a body in thermal equilibrium.7
Black HolesThe surface gravity κ is constant over the event horizon of a stationary black hole.
First Law
Thermodynamics
dE = T dS + p dV + Ω dJ
where E is the total energy of the system, T the temperature, S the entropy, p thepressure, V the volume, Ω the rotational velocity and J the angular momentum.8
Black HolesδM =1κδA + Ωbh δJbh
8π

5. By restricting attention to the asymptotically flat case, I exclude black holes in de Sitter and anti-de Sitter spacetimes (Hawking and Page 1983; Bousso and Hawking 1998; Gomberoff and Teitelboim May 2003). The
former is important because it is believed that the actual universe will asymptotically approach de Sitter spacetime; the latter is important because, with reflecting boundary conditions, anti-de Sitter acts as an effective “box”
for its constituents, and much of ordinary thermodynamics takes place in such bounded systems. The generalization of the idea of a black hole and of the Four Laws to the non-asymptotically flat case by Hayward (June 1994),
by the use of so-called dynamical trapping horizons, is of great interest, but to treat them would take us beyond
the scope of this paper. Also, I will not discuss the so-called Minus-First Law of Brown and Uffink (2001); muchwork has been done to prove, or at least argue for, its correlate in black-hole mechanics (though not referred toas such in that literature), that perturbed black holes tend to settle down to a unique equilibrium state, and, inparticular, that the sorts of perturbations I consider here do not destroy the event horizon. There are now strongplausibility arguments in favor of it (Hollands and Wald 2013), but its status in black-hole mechanics is still, tomy mind, very much up for grabs, though, as a betting man, my money is on there being arguments for it at least
as strong as for the Third Law (which, perhaps, is not to say very much).
6. For proofs of the laws for black holes, see Wald (1984, ch. 12), Israel (1986), Wald (1994), and Gao andWald (2001).
7. This is not the standard formulation of the thermodynamical Zeroth Law, which is rather transitivity ofequilibrium, i.e., “if two systems are in thermal equilibrium with a third, then each is in thermal equilibrium withthe other”. (See, e.g., Fowler and Guggenheim (1939, ch. ii*, §222, p. 56).) This is not a problem for my purposes,
because the formulation I use (which is the standard one in the context of black hole thermodynamics) and thestandard formulation for ordinary thermodynamics are effectively equivalent when the systems at issue are assumed to be thermally homogeneous, in the sense that each system contain no boundary with a permeability toheat flow different than that of some other part of the system, as is the case for all the types of system my constructions rely on.
8. Strictly speaking, this is not the First Law, but rather the Gibbs Relation, which is equivalent to the FirstLaw for thermodynamical systems in equilibrium and for systems that deviate from equilibrium only “quasistatically”. Since all my arguments involve only systems in equilibrium, and, as is standard in thermodynamicalarguments, systems that deviate from equilibrium only by quasi-stationary effects, this is not a problem.

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where M is the total black hole mass, A the surface area of its horizon, Ωbh the “rotational velocity” of its horizon (Wald 1984, ch. 12, §3, pp. 319–320), Jbh its total
angular momentum, and ‘δ’ denotes the result of a first-order, linear perturbation of thespacetime.9
Second Law
Thermodynamics
δS ≥ 0 for any process in an isolated system.10
Black HolesδA ≥ 0 in any process.11
Third Law
Thermodynamics
T = 0 is not achievable by any process.12
Black Holesκ = 0 is not achievable by any process.
The most striking architectonic similarity between the characterization of ordinary thermodynamical systems in equilibrium by the Laws of thermodynamics and the characterization of
stationary black holes is that in each case the behavior of the system, irrespective of any idiosyncracies in the system’s constitution or dynamical history, is entirely captured by the values of a
9. For an exact definition and thorough discussion of the perturbations used, see Gao and Wald (2001). Thereis an oddity about this formulation of the law, however, that I have not seen addressed in the literature but issurely worth puzzling over. While the δ acting on M is the same as that acting on Jbh , it is not the same as that
acting on A. The δ acting on M and Jbh represents a perturbation of a quantity taken asymptotically at spatial
infinity; the other represents perturbations taken “at the event horizon”. I know of no other physically significant
equation where different differential operators act on different mathematical spaces in such a way that, as in thiscase, there’s no natural mapping between them. What’s going on here?
10. Again, this is not the usual formulation of the Second Law in classical thermodynamics, which is standardly
given as the Clausius or Kelvin Postulate (e.g., Fermi 1937, §7). Because the principle of entropy increase follows
from either Postulate (§§11–13), and because the appropriate analogues for those Postulates hold for black holes(§5.3 below), this again is not a problem for my arguments.
11. Note that, because we are considering by fiat only asymptotically flat black holes, the appropriate analogueof an isolated classical thermodynamical system, it would be redundant to stipulate that the process take placein an isolated system. Indeed, the Area Theorem (as the Second Law for black holes is often called) is a result inpure differential geometry, the only input of the Einstein field equation being to provide a physical interpretationof the so-called null energy condition. That condition, more or less, rules out only macroscopic fluxes of negative
energy, so the scope of the quantifier in “any process” in the statement of the Law should be taken very broadly
indeed. In particular, one need not even assume the process is quasi-static, nor even that the processes are restricted to the sorts of first-order, linear perturbations used in the formulation and proof of the First Law. (SeeCuriel 2017 for a discussion of the physical content of the null energy condition and its role in the proofs of thelaws for black holes.)12. I think this is a defective statement of the Third Law of thermodynamics. (See, e.g., Schrödinger 1960,
Aizenman and Lieb 1981, and Wald 1997 for a discussion of some of its problems.) Schrödinger (1960) provides
a far more satisfactory statement of the Third Law, which I think carries over into black-hole thermodynamics. I
do not have room to go into the matter here, though—the standard version of the Law, as given here, suffices formy purposes.

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small number of physical quantities, 6 for ordinary thermodynamical systems, 4 for black holes:in the former case, they are temperature, entropy, pressure, volume, angular velocity and angularmomentum;13 in the latter, they are surface gravity, area, angular velocity and angular momentum. The Zeroth and Third Laws suggest that we take the surface gravity of a black hole as theanalogue of temperature. The Second Laws suggest that we take area as the analogue of entropy.
1
This is consistent with the First Law, if we treat 8π
κδA as the Gibbsian “heat” term for a systemin thermal equilibrium. Indeed, if we do so then the analogy for the First Law becomes exact:

relativistically, energy just is mass, so the lefthand side terms of the First Law for ordinary systems
and for black holes are not just analogous, they are physically identical; moreover, Ωbh δJbh as awork term is physically identical to the corresponding term in the law for ordinary systems. Finally, the Penrose process (Penrose and Floyd 1971) allows one to energy extraction from rotating
black holes with an upper limit on efficiency, the maximal efficiency being achieved when and only
when the area of the black hole—the analogue of its entropy—does not change. This is identicalto the case of ordinary heat engines.
Now the force of the question motivating this paper should be clear: the mathematical analogy
is perfect, and there are already some indications that the analogy may reach down to the levelof physics, not just mathematics. But how far should we take the analogy? What can it mean totake seriously the idea that the surface gravity of a black hole is a physical temperature, and its
area a physical entropy?

3

The Standard Argument Does Not Work

There are well-known difficulties with taking the surface gravity of a classical black hole to represent
a physical temperature. One important method for defining the thermodynamic temperature ofan object derives from the theory of thermal radiation from black bodies. If a normal black body
immersed in a bath of thermal radiation settles down to thermal equilibrium, it will itself emit
thermal radiation with a power spectrum characteristic of its equilibrium temperature as measured
using a gas thermometer. This power spectrum can then be used to define a temperature scale. It is
this definition of thermodynamic temperature that is almost always (at times implicitly) invokedwhen the claim is made that if one considers classical general relativity alone then black holes,
being perfect absorbers and perfect non-emitters, have an effective temperature of absolute zero.14
To try to be a little more precise, I will offer a reconstruction of the standard argument. It is
not given in exactly this form by anyone else in the literature, but I think it captures both the spirit
and the letter of the orthodox view; in any event, it does not set up a straw man. Put a Kerr blackhole in a box with perfectly reflective sides, which are far from the event horizon (in the sense thatthey are many times farther away from the event horizon “in natural spacelike directions” than its
own “natural” diameter). Pervade the box with thermal radiation. According to classical generalrelativity, the black hole will absorb all incident thermal radiation, and emit none, until eventually
all thermal radiation in the box (outside the event horizon) has vanished, so the black hole must
13. Of course, the First Law guarantees that not all these quantities will be independent, and, if one is considering a particular species of thermodynamical system, then one may have available an equation of state that willfurther reduce the number of independent quantities, but all that is beside the point for my purposes.
14. See for example the remarks in Bardeen et al. (1973), Carter (1973) and Wald (1999). There is another argument for attributing the temperature absolute zero to all classical black holes, that it seems to be possible touse them to convert heat into work with 100% efficiency. I address this in §6.

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have a temperature of absolute zero. Thus, the surface gravity κ, which is never zero for a nonextremal Kerr black hole, cannot represent a physical temperature of the black hole in classicalgeneral relativity. Conventional wisdom holds, as a result, that if the formal similarities mentioned
above were all there were to the matter then they would most likely represent a merely accidental
resemblance or perhaps would indicate at best a superficial relationship between thermodynamics
and black holes, but in any event would not represent the laws of classical thermodynamics asextended into the realm of black holes.15
In 1974, using semi-classical approximation techniques Hawking (1974; 1975) discovered that,when quantum particle-creation effects near the black hole horizon are taken into account, stationary, axisymmetric black holes appear to radiate as though they were perfect black-body emittersℏ
in thermal equilibrium with temperature
κ. It is this result that is generally taken to justify
2π
the view that the resemblances between the laws of black hole mechanics and the laws of classicalthermodynamics point to a fundamental and deep connection among general relativity, quantum
field theory and thermodynamics, and in particular that κ does in fact represent the physicaltemperature of a black hole, and therefore A its entropy.16
I have several problems with this orthodoxy. First, I find the physical content of the standard
argument not to stand up to scrutiny. While it is true that the Kerr black hole in the box, according
to classical general relativity, will emit no blackbody radiation while it absorbs any incident on
it, that is not the end of the story. Classical general relativity does tell us that the Kerr blackhole will emit some radiation, viz., gravitational radiation, while it is perturbed by the infalling
thermal radiation, and that gravitational radiation will in fact couple with the thermal radiation
still outside the black hole. If we are trying to figure out whether purely gravitational objects, suchas black holes, have thermodynamical properties, we should surely allow for the possibility thatgravitational radiation, or, indeed, the exchange of “gravitational energy” in any form, may countas a mediator of thermodynamical coupling.17 Indeed, just as electromagnetic radiation turned
out to be a medium capable of supporting a physically significant coupling of electromagnetic
systems with classical thermodynamical systems, it seems prima facie plausible that gravitationalradiation may play the same role for gravitational systems. Just as “heat” for an electromagnetic
system may be measured by electromagnetic radiation, at least when transfer processes are at issue,
so it may be that “heat” for a gravitational system may be measured by gravitational radiation,
15. The remarks of Wald (1984, p. 337), for example, are exemplary in this regard.
16. See again, for example, the remarks of Wald (1984, p. 337) and the more forceful claims of Wald (1999).
Indeed, some of the most important researchers in the field make even stronger claims. Unruh and Wald (1982,
p. 944), for example, claim that “the existence of acceleration radiation [outside the event horizon, a fundamentally quantum phenomenon,] is vital for the self-consistency of black-hole thermodynamics.” In fact, that A isa true physical entropy is logically and physically independent of the fact that κ is a true physical temperature,but many physicists tend to erroneously conflate confirmation of the former claim with confirmation of the latter,even if only implicitly. (I should emphasize that neither Wald 1984, nor Unruh and Wald 1982, nor Wald 1999 isguilty of this; see, e.g., Preskill 1994 and Aghapour and Hajian 2016 for explicit statements to that effect.) For
the purposes of this paper, in order to make the standard case as strong as possible, I will assume the implicationfrom attribution of temperature to attribution of entropy works. See Visser (1998, 2003) for illuminating discussion of the issue.
17. I use scare-quotes for ‘gravitational energy’ because that is an infamously vexed notion in classical generalrelativity, with no cogent way known to localize it, and indeed strong reasons to think there can be no localization of it in general. (See, e.g., Curiel 2019a.) I discuss this issue, and the potential problems it may raise for myarguments, in §6.

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or any form of exchange of gravitational energy, again at least when transfer processes are atissue.18 Electromagnetic energy is just not the relevant quantity to track when analyzing thethermodynamic character of purely gravitational systems.
Second, I do not think this definition of temperature is the appropriate one to use in the context
of a purely classical description of black holes, for the electromagnetically radiative thermal equilibrium of systems immersed in a bath of thermal radiation is essentially a quantum and statisticalphenomenon, by which I mean one that can be correctly modeled only by using the hypothesisthat radiative thermal energy is exchanged in discrete quanta and then computed correctly onlywith the use of statistical methods. To use that characterization of temperature to argue that wemust use quantum mechanics in order to take surface gravity seriously as a physical temperature,
therefore, is to beg the question. If my qualm is well founded, it follows that the standard argument does not bear on the strength of the analogy as indicating a real physical connection between
classical general relativity and thermodynamics. After all, if one is trying to determine the statusof the analogy between classical gravitational theory and classical thermodynamics independentlyof any quantum considerations, then the most appropriate characterizations of temperature to use
are those grounded strictly in classical thermodynamics itself. (I make the idea behind this qualm
precise in §6, in discussing possible problems with my arguments.)
There is yet another prima facie problem, however, with trying to interpret surface gravity as
a true temperature and area as a true entropy, which my arguments so far do not address: neither
has the proper physical dimension. In geometrized units, the physical dimension of temperature ismass (or energy), and entropy is a pure scalar. The physical dimension of surface gravity, however,is mass−1 , and that of area mass2 . There are, moreover, no purely classical universal constants
available to fix the dimensions by multiplication or division.19 The only available universal constantto do the job seems to be ℏ, which has the dimension mass2 , but ℏ of course is quantum to thecore, and so not appropriate for use in my arguments.20 I cannot address this problem at this stageof my arguments. Remarkably, however, a natural sequela to my construction of the appropriate
analogue of a Carnot cycle for black holes, in §5.2, shows that the existence of a universal constantin the classical regime with the proper dimension is guaranteed.
There are other good reasons for being skeptical of the orthodox argument as well. First,
though it is not to the best of my knowledge acknowledged in the literature, Hawking radiation
is not blackbody radiation in any standard sense of the term. Standard blackbody radiation is
generated solely by internal degrees of freedom of the system itself (vibrational modes of atoms andelectrons of the black body, e.g.), and the type of radiation emitted (electromagnetic) is governedby the nature of the micro-constituents of the body. Hawking radiation, however, comes fromthe “interaction” of the black hole with an external quantum field, and the radiation scatteredis always and only of that type, independent of whatever “stuff” one thinks may constitute the18. It is folklore in the field that the energy in gravitational radiation cannot play this role, as it does not seem
to provide the proper spectrum for the thermal nature of Hawking radiation that grounds the definition of the
Hawking temperature. I address this issue in §5.4.
19. All the classical universal constants, such as the speed of light and Newton’s gravitational constant, are dimensionless, when expressed in geometrized units (Curiel 2019a). That itself is a puzzling state of affairs, whichsurely deserves investigation.
20. I am grateful to Ted Jacobson and Carlo Rovelli for pushing me on the issue of the physical dimensions of
the quantities, and on the seeming need to introduce ℏ to make things work out properly.

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micro-structure of the spacetime geometry of the event horizon. Indeed, in standard derivationsof Hawking radiation, back-reaction of the quantum field on the metric is ignored (i.e., the field istreated as “test matter”, its stress-energy not contributing to spacetime curvature), so excitationsof micro-degrees of freedom of the black hole are necessarily excluded from playing a role in thegeneration of the Hawking radiation. Thus, Hawking radiation is not generated by internal degrees
of freedom of the black hole.21
The strongest arguments that attribution of a thermodynamical entropy to black holes is needed
to save Second Law, moreover, do not essentially rely on quantum phenomena. They rather
depend on showing that, if one does not attribute entropy to a black hole, then it is trivial toreduce the entropy of a causally isolated system (the region outside the event horizon) simply
by throwing entropic stuff into the black hole. The “classical” thermodynamics of black holes,
especially the Area Law, serve an essential evidentiary and epistemic role in such arguments.
Without an understanding of black-hole entropy as a truly thermodynamical entropy, in the senseof ordinary thermodynamics, therefore, we can have no real evidence in the first place that black
holes have a micro-structure appropriate for a statistical treatment of its dynamics that would
yield a physically significant accounting of its entropy. This is in perfect analogy with the epistemic
situation regarding classical thermodynamics and the statistical mechanics of molecular kinetics ofordinary bodies.
Before continuing, I make one final point about what I see as a serious problem with theorthodoxy. The Hawking temperature of black holes that we have good reason to take seriously
as possible denizens of our world is staggeringly, mind-blowingly small. We have strong, secure
evidence (if one takes general relativity seriously, which I do) that there is a black hole at thecenter of the Milky Way of approximately four million Solar masses. Its Hawking temperature
is approximately 10−14 Kelvin. Even much, much, much smaller black holes, say a remnant of
a supernova of “only” 10 Solar masses, will have a Hawking temperature of approximately 10−9
Kelvin. That is a billion times colder than the ambient cosmic microwave background radiation.
Radiation having temperature of that magnitude is utterly negligible with regard to essentiallyevery known type of physical processes that could take place in the black hole’s vicinity. It isdifficult, at best, to see how such utterly negligible effects can guarantee the self-consistency ofa thermodynamical account of black holes (footnote 16), if by “thermodynamical” one means at
a minimum: has a physically substantive and conceptually coherent account of how appropriate
kinds of interactions can occur with other types of physical systems we manifestly understand asthermodynamical. I expand on this point in §§4–5.

4

Temperature and Entropy in Classical Thermodynamics

I think there are grounds for taking the analogy very seriously even when one restricts oneself tothe classical theories, without input from or reliance on quantum theories. To make the case morepoignant, imagine that we are physicists who know only classical general relativity and classicalthermodynamics, but have no knowledge of quantum theory. How could we determine whether or
not to take black holes as thermodynamical objects in a substantive, physical sense, given that
we know the deep formal analogy between the two sets of laws? In such a case, we ought to
21. I discuss this in more detail in Curiel (2019b).

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look to the way that temperature and entropy are introduced in classical thermodynamics andthe various physical roles they play there. If the surface gravity and area of black holes can be
introduced in the analogous ways and play the analogous physical roles, I contend that the global
analogy is already on strong ground. In other words, the surface gravity and area must play thesame role in the new theory with regard to to modeling interactions and processes (such as mutual
equilibration and heat flow) as, respectively, temperature and entropy do in the original theory
with regard to the the analogous interactions and processes there. If, moreover, it can be shownthat surface gravity couples to ordinary classical thermodynamical systems in the same formal
way as ordinary temperature does, then there are no grounds for denying that it is a true physicaltemperature.22 And if area for black holes is related to surface gravity and to the proper analogueof heat in the same way as entropy is to ordinary temperature and heat, and if it is required forformulating an appropriately generalized Second Law, then there are no grounds for denying that
it is a true physical entropy. Indeed, it was exactly on grounds such as these that physicists in the
19th century concluded that the power spectrum of blackbody radiation itself encoded a physicaltemperature and entropy, not merely that there was an analogy between thermodynamics and thetheory of blackbody radiation.
There are three fundamental, related ways that temperature is introduced in classical thermodynamics, which themselves ground the various physical roles temperature can play in the theory(how it serves as the mediator of particular forms of coupling between different types of physicalsystems, e.g.). The first derives from perhaps the most basic of the thermodynamic characteristics
of temperature and is perhaps most definitive of the cluster of ideas surrounding the concepts of“temperature” and “heat”: it is that when two bodies are brought into contact, heat will spontaneously flow from the one of higher temperature to the one of lower temperature.23 The second
arises from the fact that increase in temperature is positively correlated with increases in the capacity of a system to do work.24 This fact allows one to define an empirical scale of temperature
through, e.g., the use of a gas thermometer: the temperature reading of the thermometer is madedirectly proportional to the volume of the thermometric gas used, which is itself directly proportional to the work the gas does on its surrounding container as it expands or contracts in response
to its coupling with the temperature of the system being measured. The utility of such a scaleis underwritten by the empirical verification that such empirical scales defined using a multitudeof different gases under a multitude of different conditions are consistent among one another.25
22. Since entropy directly mediates no coupling between thermodynamical systems, the same argument is notavailable for it. This is one of the properties of entropy that makes it a truly puzzling physical quantity: there isno such thing, not even in principle, as an entropometer. To ascertain the entropy of a system, one must calculate
it from the values of other quantities that one can directly measure. This perhaps is intimately related to the factthat entropy is the only fundamental quantity that has a fundamentally modal character: it is a measure of how
many microstates a system could consistently occupy given its macrostate; but that measure is not a property ofthe microstate alone it actually does occupy, as energy, temperature, pressure, volume, et al., are.
23. It is important for some of my later arguments to note that this characterization of comparative temperaturedoes not preclude processes in which heat at the same time flows from the colder body to the hotter. It says onlythat it is always the case that heat flows from hotter to colder, irrespective of what may or may not happen in
the reverse direction.
24. See, e.g., the exemplary remarks of Sommerfeld (1964, p. 36): “Thermodynamics investigates the conditions
that govern the transformation of heat into work. It teaches us to recognize temperature as the measure of thework-value of heat. Heat of higher temperature is richer, is capable of doing more work.”
25. Planck (1926, §1, p. 1) remarks that quantitative exactness is introduced into thermodynamics through this13

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The third arises from an investigation of the efficiency of reversible, cyclic engines, viz., Carnot
cycles, which yields a definition of the so-called absolute temperature scale associated with thename of Kelvin (Fermi 1937, §§8–10). It is the possibility of physically identifying the formallyderived absolute scale with the empirically derived scale based on capacity to do work (increase involumes, e.g.) that warrants the assertion that they both measure the same physical quantity.26
Likewise, there are (at least) three ways that entropy enters classical thermodynamics. The
first historically, and perhaps the most physically basic and intuitive, is as a measure of how muchenergy it takes to transform the heat of a thermal system into work: generally speaking, the freeenergy of a thermodynamical system is inversely proportional to its entropy.27 The second is asthat perfect differential dS into which temperature, as integrating factor, transforms exchanges ofheat dQ (ch. iv): the integral of dQ along a quasi-stationary path between two equilibrium statesin the space of states of a thermodynamical system is not independent of the path chosen, whereasdQ
the integral ofis. (Indeed, Sommerfeld 1964 uses this fact to conclude that entropy is a trueTphysical property of a thermodynamical system, whereas heat content is not.) The third also arises
from the analysis of the Carnot cycles, as a determinant of its efficiency (Fermi 1937, ch. iv).
Now, the following fundamental theorem of classical thermodynamics provides the basis bothfor the definition of the absolute temperature scale and for the introduction of entropy as theperfect differential derived from exchanges of heat when that temperature is used as an integrating
factor.
Theorem 4.1 (Carnot 1824). Any two reversible, cyclic engines operating between temperatures
T2 and T1 (as measured using gas thermometry) have the same efficiency. The efficiency of any
non-reversible engine operating between T2 and T1 is always less than this.
This theorem is a direct consequence of either the Clausius or the Kelvin postulate, which
can be argued on physical grounds both to be equivalent to each other and to directly imply theprinciple of entropy increase (for the proofs of which see, e.g., Fermi 1937):
Postulate 4.2 (Lord Kelvin). 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.
Postulate 4.3 (Clausius). A transformation whose only final result is to transfer heat from a bodyat a given temperature to a body of a higher temperature is impossible.
I claim that these last two postulates, and the fact that they provide grounds for the proof ofthe efficiency theorem, for the introduction of temperature and entropy as physical quantities, andfor proof of the principle of entropy increase, encode essentially all that is of physical significancein the ways I sketched that both temperature and entropy enter into classical thermodynamics.
The Clausius Postulate captures the idea that when two bodies are brought into thermal contact, heat flows from the body of higher temperature to the other. The Kelvin Postulate captures
observation, for changes of volume admit of exact measurements, whereas sensations of heat and cold do not, noreven comparative judgments of hotter and cooler on their own.
26. Maxwell (1891, chs. viii, xiii) gives a beautifully illuminating discussion of the physical basis of the equivalence of the absolute temperature scale with the one based on gas thermometry.
27. Again, the discussion of Maxwell (1891, ch. xii) about this idea is a masterpiece of physical clarification andinsight.

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the idea that the capacity of a body to do work on its environment tends to increase as its temperature increases. If one could show that appropriately formulated analogues to these two propositions
about classical black holes hold in general relativity, with surface gravity playing the role of temperature and area that of entropy, one would have gone a long way towards showing that surface
gravity is a true thermodynamical temperature and area a true entropy. If one could further showthat ordinary thermodynamical systems equilibrate with black holes in a way properly mediated bytheir ordinary temperature and by the black hole’s surface gravity, so as to allow for the construction of a Carnot-like cycle and the definition of an absolute temperature scale, the analogy would
have been shown to be far more than analogy: it would be physical equivalence in the strongest
possible sense. I sketch ways to prove all these propositions in §5 below.

5

Taking Classical Black Holes Seriously as Thermodynamical Objects

What is needed, first, is a way to characterize “thermal coupling” between black holes and ordinary
thermodynamical systems: granted that “heat” in the gravitational context is gravitational energyof a particular form, such as that carried in the form of gravitational radiation or that responsible
for red-shift effects in monopole solutions, then it follows that black holes are not perfect absorbers.
When there is an ambient electromagnetic field, the black hole will radiate gravitationally as it
absorbs energy and grows from the infalling electromagnetic radiation. So to conclude that surface
gravity is a physical temperature, one need show only that the gravitational energy exchanged
between a black hole and other thermodynamical systems in transfer processes depends in the appropriate way on the surface gravity of the event horizon.28 This approach has prima facie physicalplausibility: to take the energy in gravitational radiation, e.g., to be the gravitational equivalentof heat is the same as to take the energy in electromagnetic radiation to be the electromagnetic
equivalent of heat—it is what couples in the appropriate way to the average kinetic energy ofmolecules in ordinary thermodynamical systems, viz., what makes it increase and decrease, andthat with respect to which equilibrium is defined.
Just as the concept of “thermal coupling” had to be emended in the extension of classicalthermodynamics to include phenomena associated with radiating black bodies, so we should expect
it to be in this case. In classical thermodynamics before the inclusion of black-body phenomena,
thermal coupling meant immediate spatial contiguity: heat was known to flow among solids, liquids
and gases only when they had boundaries touching each other.29 In order to extend classicalthermodynamics to include black-body phenomena, the idea of thermal coupling had to be extended
as well: two black bodies thermally couple when and only when the ambient electromagnetic fieldeach is immersed in includes direct contributions from the electromagnetic radiation emitted by
the other. They do not need to have contiguous boundaries.
In order to characterize the correct notion of thermal coupling among systems including black
holes (or more generalized purely gravitational systems, such as cosmological horizons), we firstneed to characterize an appropriate notion of “heat” for black holes, and the concomitant notion
28. I will discuss in §6 below the fact that there is no well defined notion of localized gravitational energy in
general relativity, and how that may bear on my arguments.
29. This fact, perhaps, contributed to the historical idea that heat was a fluxional, perhaps even fluid, substance, such as phlogiston or caloric.

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of free energy. That will put us in a position to construct the appropriate generalization of Carnot
cycles for them, and so to formulate the appropriate generalizations of the Clausius and Kelvin
Postulates for such systems.

5.1

Irreducible Mass, Free Energy, “Heat” and Thermal Coupling ofBlack HolesIn analyzing the ideas of reversibility and irreversibility for processes involving black holes,
Christodoulou (1970) introduced the irreducible mass Mirr of a black hole of mass M and angular momentum J:30
2
:=
Mirr

1
1 2
[M + (M 4 − J 2 ) 2 ]2(5.1.1)Inverting the definition yields
2
M 2 = Mirr
+

1 J2
2
4 Mirrand so, for a Kerr black hole,
M > Mirr(Clearly, Mirr = M for a Schwarzschild black hole, i.e., one with vanishing angular momentum.)
A simple calculation for a Kerr black hole, moreover, shows that,2
A = 16πMirr(5.1.2)Thus, it follows from the Second Law that Mirr itself cannot be reduced by any physical process,and so any process in which the irreducible mass increases is a physically irreversible process. The
total mass of a black hole, therefore, cannot be reduced below the initial value of Mirr by any
physical process. In principle, therefore, the free energy of a black hole is just M − Mirr , in so far
as its total mass M represents the sum total of all forms of its energies, and Mirr represents theminimum total energy the black hole can be reduced to.31
In classical thermodynamics, it makes no sense to inquire after the absolute value of the quantityof heat a given system possesses. In general, that is not a well defined property accruing to a system.
One rather can ask only about the amount of heat tranferred between bodies during a given process(Maxwell 1891, chs. i, iii, iv, viii, xii). Consider, then, a classical thermodynamical system withtotal energy E and free energy Ef . E − Ef is the amount of energy unavailable for extraction, what
Kelvin called its dissipated energy, Ed . Say that through some quasi-stationary process, we know
not what, both E and Ed change so that the system now has less free energy than it did before;
therefore, the entropy of the system must have increased, which can happen only when it absorbs
heat, which will in general be the difference between the total change in energy and the change
in free energy. If they both change so that the system has more free energy, the same reasoning
applies, and it must have given up a quantity of heat equal to that difference.
30. I will discuss only Kerr black holes, not Kerr-Newman black holes, which also have electric charge, as theensuing technical complications would not be compensated by any gain in physical comprehension.
31. Some—e.g., Wald (1984, ch. 12, §4)—interpret M − Mirr as the rotational energy of a Kerr black hole, inso far as extracting that much energy from a black hole would necessarily reduce its angular momentum to zero.
Based on the arguments I will give in this section, I prefer to think of it as a thermodynamical free energy, whichcannot necessarily be decomposed in a canonical way into different “forms”, e.g., that much heat and that muchrotational energy, etc.

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These remarks suggest defining the “quantity of heat transferred” to or from a black hole during
any quasi-stationary thermodynamical process to be the change in its free energy, which is to say
the change in total black hole mass minus the change in its irreducible mass,32∆M − ∆Mirr(5.1.3)If, for instance, the irreducible mass of a black hole does not change, while the total mass decreases,
then it would have given up a quantity of heat. As a consistency check, it is easy to see that,
according to this definition, when an ordinary thermodynamical system in equilibrium is dumped
into a Kerr black hole, the black hole absorbs the quantity of heat the ordinary matter contained
as characterized by the Gibbs relation, viz., its temperature times its entropy, as only that energy
contributes to its total mass without directly changing its angular momentum. In this regard, it ishighly suggestive that Schwarzschild black holes are exactly those with no free energy: a sphericallysymmetric vacuum spacetime, by Birhoff’s Theorem, must be in equilibrium (i.e., must be static),
and in particular must be either Minkowski or Schwarzschild spacetime, and so spherical symmetry
seems to be the purely gravitational equivalent of “heat death”.
Based on this characterization of “quantity of heat transferred”, I claim that the appropriate
notion of thermal coupling for systems involving black holes is any interaction where there is achange in the black hole’s free energy. For purely gravitational interactions, this includes emission
and absorption of that part of the energy of gravitational radiation not due to angular momentum,
energy exchange due to simple monopole- or multipole-moment couplings in the near-stationary
case, and so on.
Some care must be taken in applying this definition to Schwarzschild black holes, however.
Because M = Mirr for a Schwarzschild black hole, one can never give up heat while remaining
Schwarzschildian. Schwarzschild black holes, essentially, have achieved heat death—one cannot
extract energy from them without perturbing them in an appropriate way, in particular a way
involving transfer of angular momentum. Similarly, they cannot absorb heat in a straightforward
sense: if one absorbs ordinary heat from a classical thermodynamical system, say, being throwninto it, then after it settles down again to staticity it will once again have its total mass equal toits irreducible mass (unless, again, it acquires angular momentum in the process, and so becomes a
Kerr black hole). In this case, I think it still makes sense to say the black hole has absorbed heat,in so far as, between the time the system is thrown in and the time the black hole equilibrates
again, its irreducible mass will not be equal to its total mass. The maximum of this difference,
during the equilibration process, will presumably equal the energy of the system the black holeabsorbed; as the perturbations are radiated away and the horizon relaxes back to perfect sphericity(and so staticity), the irreducible mass will come once again to equal its total mass.33 Thus, for
32. I thank Harvey Brown for drawing to my attention the fact that Carathéodory (1909b), in his groundbreaking axiomatization of classical thermodynamics, introduced the notion of heat in a way very similar to this,not as a primitive quantity as is usually done, but as the difference between the internal and the free energies of asystem.
33. There are many challenges one could reasonably pose to the approximations involved in attempting to carryout such a calculation with anything approaching rigor (which I have not done), in particular for trying to define
and calculate the total mass and irreducible mass of a non-stationary black hole. They are, however, all the same
sort of challenge one could pose to the analogous problem in classical thermodynamics when trying to calculatechanges in entropy for systems not in equilibrium, so there is no problem here peculiar to black-hole thermodynamics.

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the thermodynamics of purely gravitational systems in general relativity, spherical symmetry is
the state corresponding to a maximally equilibrated state for a given amount of mass-energy.34

5.2

Carnot-Geroch Cycles for Schwarzschild Black HolesAs I remarked at the end of §4, the strongest evidence that the formal equivalence of the lawsof black holes and those of ordinary thermodynamical systems in fact constitutes a true physicalequivalence, and that surface gravity is a physical temperature and area a physical entropy, would
consist in a demonstration that black holes thermally couple with ordinary thermodynamical systems in such a way that κ plays the same role in that coupling as ordinary temperature would if thesystem at issue were coupling with another ordinary thermodynamical system and not with a blackhole, and the same for area. My proposed construction of the appropriate analogue for a Carnot
cycle including black holes, which I give in this subsection, will kill three birds with one stone:
not only will it show that κ can be characterized as the absolute temperature of the black holeusing the same arguments as classical thermodynamics uses to introduce the absolute temperaturescale; it will do so by showing that in the coupling of black holes with ordinary thermodynamicalsystems, κ does in fact play the physical role of temperature and area that of entropy; and it will
have as a natural corollary the existence of a universal constant that renders the proper physicaldimensions to surface gravity as a measure of temperature and area as a measure of entropy.35
I call the constructed process a “Carnot-Geroch cycle” both to mark its difference from standard
Carnot cycles, and because it relies essentially on the mechanism at the heart of the most infamous
example in this entire field of study, Geroch’s thought experiment of slowly lowering towards a blackhole a box filled with thermal matter, with the argued consequence being that classical black holes
must have temperature absolute zero.36 (I discuss Geroch’s original example and argue that it does
not in fact support the conclusions he wanted to draw from it in §6 below.) I will first sketch thesteps of the proposed cycle informally, then work through the calculations. (The sketch contains
two implicit assumptions, which I state and discuss after.)
Reversible Carnot-Geroch Cycle Using a Schwarzschild Black Hole as a Heat Sink
1. Start with a small, empty, essentially massless, perfectly insulating box “at infinity”, oneside of which is the outer face of a piston initially completely depressed so that the box
has no empty volume, and the other side a sliding door attached to a large reservoir ofequilibrated ideal gas at fixed temperature T0 (which one may think of as a heat bath);in particular, the box is “small” in the sense that it will experience negligible tidal forces
as it is lowered toward the black hole.
34. It would be of interest to compare these arguments to those in favor of holography.
35. I thank Ted Jacobson for bringing to my attention after I wrote this paper the insightful analysis of Sciama(1976), in some ways quite similar to mine. (See Jacobson 2005 for a prècis of Sciama’s analysis.) Sciama, however, in conformity with orthodoxy, uses quantum systems all the way through and assumes that the analogy
between black holes and ordinary thermodynamical systems is merely formal when one does not take quantum
effects into account.
36. According to Jakob Bekenstein (private correspondence) and Robert Wald (conversation), Geroch first
proposed the example during a colloquium he gave at Princeton in December, 1971. (Bekenstein tells me thathe considers it the first attempt to attribute a temperature to a black hole.) I cannot resist pointing out thatmy construction is essentially a jiu jitsu move against Geroch’s original intent, turning the force of the example
against itself, using Geroch’s proposed mechanism to show that surface gravity really is a temperature.

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2. Very quickly open the door so the face of the piston in contiguous with the reservoir
gas, and then very slowly (“quasi-statically”, so that the process is well approximated as
an isentropic process) draw the piston back through the inside of the box, so filling thebox with gas, so the gas does work against the piston as it moves; when the piston has
withdrawn part but not all of the way to the opposite side of the box, quickly seal thebox, leaving the space opened by the piston filled with a mass of the gas M0 in thermal
equilibrium at temperature T0 , and with entropy S0 ; assume the entire energy of thebox, including the gas, is negligible compared to the mass of the black hole, and alsothat the mass of the box itself is negligible compared to that of the gas in it.
3. Very slowly, lower the box towards the black hole using an essentially massless rope;
during this process, an observer inside the box would see nothing relevant change; inparticular, as measured by an observer co-moving with the box, the temperature, volume
and entropy of the gas remain constant.37
4. At a predetermined fixed proper radial distance from the black hole, stop lowering thebox and hold it stationary.
5. Very slowly, draw the piston back even further, so lowering the temperature of the gas
to a fixed, pre-determined value T1 while keeping its entropy the same; the value of thetemperature is to be fixed by the requirement that the change in total entropy vanishes
during the next step (i.e., entropy of black hole plus entropy of everything outside blackhole does not change after the gas is dumped into the black hole).
6. Open the box and eject the gas out of it by using the piston to push it out, so the gas
freely falls into the black hole delivering positive mass-energy and positive entropy toit, and the piston returns to its initial state; by the way the temperature of the gas was
fixed in the previous step, this is an isentropic process.
7. Pull the box back up to infinity (which takes no work, as the box now has zero massenergy, and so zero weight), so it returns to its initial state.
Because the total entropy remains constant during every step in the process, these cycles arereversible in the sense of classical thermodynamics. Because the irreducible mass of the black holeincreases, however, it is not an irreversible process in the sense of black-hole mechanics.38
The description of the process contains the following two implicit assumptions: first, that it
makes sense to attribute a physical temperature Tbh and entropy Sbh to a black hole (though we do
not yet know what they are); and second, that the entropy of ordinary thermodynamical systems
and the entropy of the black hole are jointly additive. (These are the same assumptions one needs
37. The mass-density distribution of the gas would change, increasing towards the side facing the black hole;this, however, does not affect the analysis, since this is what one expects for a system in thermal equilibrium in aquasi-static “gravitational field”. In any event, given our assumption about the size of the box, this effect is negligible.
38. I plan, in future work, to investigate the possibility of constructing another form of Carnot-Geroch cycle fora Kerr black hole, one that exploits its angular momentum in such a way as to make the process both reversible
in the sense of classical thermodynamics and physically reversible according to black-hole mechanics. This would
run contrary to the claims of Deng and Gao (2009) that Hawking radiation is necessary for such cycles to be reversible.

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to argue that blackbody electromagnetic radiation has a consistent thermodynamics; I discuss anddefend them in §6 below.) Finally, let us now make one final assumption: that the appropriate
temperature at which to eject the gas into the black hole for the entire cycle to be isentropic (T1 instep 6) is that one would expect for a thermally equilibrated body in thermal contact with anotherat temperature Tbh sitting the given distance away in a nearly-static gravitational field. It willAthen follow that the physical temperature must be 8πακ and the physical entropy , where κ isα
the black hole’s surface gravity, A its area, and α is a universal constant with physical dimension
m2 .
Let the static Killing field in the spacetime be ξ a (timelike outside the event horizon, null on
it).

39

1

1

Let χ = (ξ n ξn ) 2 , aa = (ξ n ∇n ξ a )/χ2 be the acceleration of an orbit of ξ a , and a = (an an ) 2 .

Then a standard calculation (e.g., Wald 1984) shows thatκ = lim(χa)
where the limit is taken as one approaches the event horizon in the radial direction, i.e., near theblack hole χa is essentially the force that needs to be exerted “at infinity” to hold an object sothat it follows an orbit of χa , which is to say, to hold it so that it is locally stationary. Thus χ isessentially the “redshift factor” in a Schwarzschild spacetime.
Let the total energy content of the box when it is initially filled at infinity be E0 (as measured
with respect to the static Killing field). In particular, E0 includes contributions from the rest massof the gas M0 , and from its temperature T0 and entropy Sb ; let W0 be the work done by the gas
as it pushes against the piston in filling the box. By the Gibbs relation and by the First Law ofthermodynamics, therefore, we can compute the quantity of heat Qb initially in the box:
Qb = T0 Sb = E0 + W0
As the box is quasi-statically lowered to a proper distance ℓ from the event horizon, its energy
as measured at infinity becomes χE0 , where χ is the value of the redshift factor at ℓ. Thus, theamount of work done at infinity in lowering the box isWℓ = (1 − χ)E0
(Recall that we assumed the box to be so small that χ does not differ appreciably from top tobottom.) This is not standard thermodynamical work, as the volume of the gas, as measured bya co-moving observer, has not changed. It is rather work done by “the gravity of the black hole”.
Because we assumed that the mass-energy of the box plus gas is negligible compared to that of theblack hole, we ignore the attendant change in gravitational energy associated with the black hole,just as in the construction of an ordinary Carnot Cycle the heat given up or absorbed by a heatbath is assumed not to change its total heat content (and so not to change its temperature).
Now, when the box is held at the proper distance ℓ from the black hole and the piston slowly
pushes or pulls so as to change the temperature of the gas from T0 to T1 (as measured locally),
the piston does work (as measured at infinity)
W1 = χ(E0 − E1 )39. I use the abstract-index notation for tensorial objects; see Wald 1984 for an exposition. Note that letters
subscripted to signs designating scalars, in particular ‘Q’, ‘S’, ‘T ’, ‘W ’ and ‘E’, are not indices, but labels to keep
track of the same quantity belonging to different physical systems (or different states of the same system).

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where E1 is the locally measured total energy of the gas after the gas’s (locally measured) volume
has been changed by the piston. When the gas has reached the desired temperature T1 , the box is
opened and the piston pushes the gas quasi-statically out of the box, so it will fall into the black
hole; in the process, the piston does work W2 (as measured at infinity).40
Now, by the First Law, the total amount of energy the gas has as it leaves the box isE1 −W2= T1 Sbχ(5.2.4)as measured locally. In order to compute the total amount of energy and the total amount of heat
dumped into the black hole as measured at infinity, we must compute the temperature of the box
as measured from there. It is a standard result (Tolman 1934, p. 318) that the condition for abody at locally measured temperature T to be in thermal equilibrium in a strong, nearly static
gravitational field is that the temperature measured “at infinity” be χT . Thus the temperature of
the box as measured from infinity will be χT1 . It follows from equation (5.2.4), therefore, that the
total amount of heat dumped into the black hole is
χT1 Sb = χE1 − W2
But χE1 = χE0 − W1 and χE0 = E0 − Wℓ , so
χT1 Sb = E0 − Wℓ − W1 − W2
The expression on the righthand side of the last equation, however, is just the total amount ofenergy in the box as measured at infinity, and so χT1 Sb is the total amount of energy the blackhole absorbs, as measured from infinity, which is entirely in the form of heat.
Now, because we have assumed that the entropy for the gas and for the black hole is additive,
the total change in entropy is
χT1 SbTbh
For the process to be isentropic (∆S = 0), it must be that
∆S = −Sb +χT1 Sb= Sb
Tbh(5.2.5)Tbh, precisely the temperature one would expect for a thermally equilibrated body inχthermal contact with another body at temperature Tbh a redshift distance χ away (Tolman 1934).

Thus, T1 =Write Qbh for the amount of heat the black hole absorbs (= χT1 Sb ), so equation (5.2.5) becomes
Qbh
= Sb
Tbh
Now, in the limit as the box, and so the heat and entropy it contains, becomes very small (while
the temperature remains constant), we may think of this as an equation of differentials,
δQbh
= dSb
Tbh

(5.2.6)

40. One may worry that this process cannot be quasi-static, not even in principle, in so far as the phase-spacevolume available to the gas as it is expelled from the box and before it is absorbed by the black hole is, in principle, unbounded, i.e., the entropy of the gas increases by an arbitrary amount. That conception of entropy, however, is appropriate only in a treatment explicitly grounded in statistical mechanics, and so is irrelevant to myarguments. Because the temperature and total energy of the gas do not change as it falls into the black hole, itsclassical entropy does not change.

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This expresses the well known fact that temperature plays the role of an integrating factor forheat. Since δQbh is the change in mass of the black hole, dMbh , due to its being the entirety ofthe energy absorbed, there follows from the First Law of black-hole mechanics41
8πδQbh= dA
κ(5.2.7)

Thus, κ is also an integrating factor for heat. It is a well known theorem that if two quantities areboth integrating factors of the same third quantity, the ratio of the two must be a function of thequantity in the total differential, and so in this caseTbh
= ψ(A)
κ(5.2.8)Tbh= ϕ(Sb ) for some ϕ, but we will not need to use that.) Itκfollows from equations (5.2.6) and (5.2.7) thatfor some ψ. (It is also the case that1
ψ(A)dA = dSb
8π(5.2.9)and so integrating this equation yields the change in the black hole’s area, ∆A as a function of Sb ,say ∆A = θ(Sb ). (From hereon, we fix some arbitrary standard value for A, and so drop the ‘∆’,as is done in analogous calculations in ordinary thermodynamics.)
In order to complete the argument, and make explicit the relation between A and Sb , and at
the same time fix the relation between κ and Tbh , consider two black holes very far apart, andotherwise isolated, so there is essentially no interaction between them. Perform the Geroch-Carnot
cycle on each separately. Let A1 and A2 be their respective areas, θ1 and θ2 the respective functions
for those areas expressed using Sb1 and Sb2 , the respective entropies dumped into the black holesby the cycles, and let θ12 (Sb12 ) be the function for the total area of the black holes considered asa single system, expressed using the total entropy Sb12 dumped into the system. Both the total
area of the black holes and the total entropy dumped in are additive (since the black holes, and so
the elements of the Carnot-Geroch cycles, have negligible interaction), i.e.,
θ1 (Sb1 ) + θ2 (Sb2 ) = θ12 (Sb12 ) = θ12 (Sb1 + Sb2 )(5.2.10)Differentiate each side, first with respect to Sb1 and then with respect to Sb2 ; because θ12 issymmetric in Sb1 and Sb2 ,
dθ1
dθ2
=
dSb1
dSb2
41. At least two conceptually distinct formulations of the First Law of black-hole mechanics appear in the literature, what (following Wald 1994, ch. 6, §2) I will call the physical-process version and the equilibrium version.
The former fixes the relations among the changes in an initially stationary black hole’s mass, surface gravity, area,
angular velocity, angular momentum, electric potential and electric charge when the black hole is perturbed bythrowing in an “infinitesimally small” bit of matter, after the black hole settles back down to stationarity. The latter considers the relation among all those quantities for two black holes in “infinitesimally close” stationary states,or, more precisely, for two “infinitesimally close” black-hole spacetimes. Clearly, I am relying on the physicalprocess version, for the most thorough and physically sound proof and discussion of which see Gao and Wald
(2001). Because the First Law is a theorem about black holes that makes no thermodynamical assumptions, its
use here is not circular, and in particular does not bear on the thermodynamical interpretation of the black hole’s
behavior I am arguing for.

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Classical Black Holes Are Hot

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Since the parameters of the two black holes and the two cycles are arbitrary, it follows that there
is a universal constant α such that

dθ
dA
=
=α
dSb
dSb
for all Schwarzschild black holes. It now follows directly from equations (5.2.8) and (5.2.9) thatTbh = 8πακ(5.2.11)and from equation (5.2.5) thatA
(5.2.12)
αup to an additive constant we may as well set equal to zero.42 α is guaranteed by construction toSbh =have the proper dimensions to give Tbh the physical dimension of temperature (mass, in geometrizedunits), and Sbh the physical dimension of entropy (dimensionless, in geometrized units). Of course,“behind the scenes” as it were, we know that α must ultimately be expressed as some multiple of ℏ,
since we believe the world to be at bottom quantum in nature. The salient—and remarkable—fact
here is that the existence of such a constant can be demonstrated using purely classical arguments.43
As a consistency check, it is easy to compute that the total work performed in the process,
WT = W0 + Wℓ + W1 + W2
equals the total change in heat of the box during the process, Qb − χT1 Sb , exactly as one should
expect for a Carnot cycle. One can use the total work, then, to define the efficiency of the process
in the standard way,η :=WT
χT1 Sb=1−
Qb
Qb

from which it follows that

8πακT0Thus, one can use the standard procedure for defining an absolute temperature scale based on theη =1−efficiency of Carnot cycles (Fermi 1937), and one concludes that the absolute temperature of theblack hole is indeed 8πακ.
I conjecture that essentially the same construction and calculations can be carried out for Kerr
black holes, with only a few slight modifications needed (e.g., to hold the box stationary outside theevent horizon now means to hold it co-rotating with the event horizon, so the angular work-term forthat must be accounted for in the calculation of total work done). The same results should hold; inparticular, one derives the same constant α, in this case by considering three Carnot-Geroch Cycles
performed separately on one Schwarzschild and two Kerr black holes, each at a great distance fromthe others, and then performing essentially the same calculation that starts at equation (5.2.10)
above. I plan to investigate this in future work.
Unfortunately, one cannot use similar arguments as in the classical case to prove the analogueof theorem 4.1, as the Carnot-Geroch Cycle for Schwarzschild black holes is not reversible in thephysical sense.
42. In contradistinction to classical thermodynamical systems, geometrized units for the entropy of black holes
can be naturally constructed: let a natural unit for mass be, say, that of a proton; then one unit of entropy is thatof a Schwarzschild black hole of unit mass. Why does classical black-hole thermodynamics allow for the construction of a natural unit for entropy when purely classical, non-gravitational thermodynamics does not?
43. One might well take this as a strong sign that the signature of quantum gravity is legible already in the deep
structure of the classical theory.

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5.3

11. Sept 2023

The Generalized Clausius and Kelvin Principles for Black Holes

Although I consider the construction of the Carnot-Geroch Cycle and the arguments based on it
to be the most decisive in favor of conceiving of classical black holes as truly thermodynamical
objects, I think it is still worthwhile to consider the appropriately translated analogues of theClausius and Kelvin Postulates hold for black holes as well. Because those Postulates provide theground for all ways of introducing temperature and entropy in classical thermodynamics, and forproving non-decrease of entropy, to show that they hold of black holes as well will show that thephysical behavior of black holes conforms as closely as possible to that of classical thermodynamics
in all fundamental respects. Because I conjecture that they admit of rigorous proof, I call them‘principles’, not ‘postulates’.
To construct the analogue of the Clausius Postulate for black holes, we need to translate
postulate 4.3 using the appropriate concepts. In particular, for temperature, we need to use
surface gravity, and for quantity of heat, we need to use the definition relevant to interactions
involving black holes stated in §5.1. This gives the following.
Principle 5.3.1 (Generalized Clausius Principle for Black Holes).
Given two stationary black holes with mass and irreducible mass M1 , M2 and M1,irr , M2,irr , respectively, with M1 > M2 ( i.e., the temperature of the former is less than that of the latter),then there is no physical transformation whose only final result is that ∆M1 − ∆M1,irr > 0 and∆M1 − ∆M1,irr = −(∆M2 − ∆M2,irr ).
The standard arguments in favor of the Clausius and Kelvin Postulates (as given, e.g., in Fermi
1937, ch. 3), which rely on the impossibility of constructing a perpetuum mobile of the second
kind, do not translate straightforwardly into the context of general relativity, where there is no
general principle of the conservation of energy. Such a transformation is trivially impossible for
Schwarzschild black holes, since it can never be the case that ∆M − ∆Mirr > 0, so the Principle isof possible interest only for Kerr, Reissner-Nordström, and Kerr-Newman black holes, which arebeyond the scope of this paper.
The same considerations yield the following translation for the Kelvin Postulate.
Principle 5.3.2 (Generalized Kelvin Principle for Black Holes).
There is no physical transformation whose only final result is that ∆M −∆Mirr > 0 for a stationary
black hole and the harvested energy is transformed entirely into work.
Again, the standard arguments from ordinary thermodynamics will not work here, and, again,
the principle is trivial for Schwarzschild black holes. That both principles hold in general relativity
would by no means be trivial. Nothing a priori would suggest that they do.

5.4

The Statistical and Quantum Connection

The fundamental constant whose existence I derive is of course just ℏ in disguise, but it is extraordinary that traces of it strong enough for its derivation make themselves known already inthe purely classical theory. This is in line with Ted’s plaintive query (Jacobson 1995), “How didclassical general relativity know that the horizon area would turn out to be a form of entropy, andthat surface gravity is a temperature?”24

Classical Black Holes Are Hot

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It is folklore in the field that the energy in gravitational radiation cannot play the role of heat,as it does not seem to provide the proper spectrum for the thermal nature of Hawking radiationthat grounds the definition of the Hawking temperature. There are two responses that I thinkshow the folklore to be mistaken. First, and most importantly, the definition of the Hawking
temperature from the existence of Hawking radiation is a fundamentally quantum and statisticalconstruction, not a classical one. The construction of §5.2 shows that the Hawking temperature
can be defined in a purely classical way, so one need not worry about an underlying statistical or
quantum structure to ground it.
For those who still worry how the quantum, statistical nature of the Hawking temperature
can be embodied in the classical regime, where there seems no possible way for an underlying
classical statistical theory to ground it, there is perhaps a better answer. Generic perturbations ofa black hole give rise to oscillations or vibrations in the event horizon, characterized by a spectrumof individual modes each with its own frequency, known as ‘quasi-normal modes’. (Kokkotas
and Schmidt 1999). Kiefer (2004) shows that quantization of these modes naturally gives rise to
the thermal spectrum characteristic of Hawking radiation, and so to the standard definition of the
Hawking temperature. In so far as quasi-normal modes encode the energy spectrum of gravitationalradiation emitted by the perturbed black hole, therefore, such energy in the classical regime canand does embody the thermodynamical properties of black holes as given by a statistical, quantum
analysis.44 Note that this story does not contradict the claim that our current understanding ofHawking radiation does not arise from the dynamics of the microstructure of the event horizon.
After all, the standard derivations of Hawking radiation ignores back-reaction, so the generated
thermal radiation of the ambient quantum field cannot be generated by any dynamics associatedwith the stationary event horizon.
Recall that such a quantization of the classical frequency spectrum of the electromagnetic field
is exactly how the quantum, statistical based definitions of temperature and entropy are given
for blackbody radiation. The classical thermodynamics of black holes, therefore, stands in every
important way in perfect analogy to the thermodynamics of electromagnetic radiation, of which
no one would question the thermodynamical nature.

6

Problems, Possible Resolutions, Possible Insights, andQuestions

I conclude the paper with a brief discussion of some prima facie problems with my arguments,suggestions for their resolutions, an examination of what insights my conclusions, if correct, may
offer, and some general questions in the field that I think still need to be addressed, possibly withthe help of my arguments and conclusions.
An obvious complaint against the argument based on the construction of the Carnot-Geroch
Cycle is that it is circular: why assume a classical black hole has an entropy in the first place? The
best answer to this is implicit in the questions Wheeler initially posed in the late 1960s that inspired
the entire field of black-hole thermodynamics in the first place: if we don’t assume black holes haveentropy, then we would, with effortless virtuosity, be able to achieve arbitrarily large violations
44. For this argument to be entirely satisfying, however, the appropriate calculations need to be carried out inthe classical regime, which I have not yet done. It is technically a difficult problem.

25

Classical Black Holes Are Hot

11. Sept 2023of the Second Law of thermodynamics. The world external to a black hole is isolated from theinterior of the black hole. So, take your favorite highly entropic system and throw it into a black
hole: the entropy of that system vanishes from the external world, so spontaneously lowering thetotal entropy of a causally isolated system in such a way that arbitrarily large decreases can be hadwith arbitrarily small costs of energy. The only escape from this possibility is to assign the blackhole itself an entropy in such a way that, when an ordinary entropic system passes into a blackhole, then the black hole’s entropy increases at least as much as the entropy of the system entering
it. This proposition is referred to as the Generalized Second Law: the total entropy of the world,
viz., the entropy of everything outside black holes plus the entropy of black holes, never decreases(Bekenstein 1973, 1974). To bang on a now-familiar drum in a now familiar rhythm, this was thesame form of reasoning, driven by the same kinds of worry, used to justify the first attempts toattribute entropy to blackbody radiation. I do not think there is a similar, independently motivatedreason for attributing a temperature to the black hole a priori. That justification can come only
from the fact that to do so yields a consistent theory of thermodynamics for classical black holes:
surface gravity in the Carnot-Geroch cycle ends up playing every important physical role thattemperature does in an ordinary one: it is the integrating factor for energy that yields entropy; it
determines the conditions for mutual equilibrium; and so on.
This attempt to answer the first problem leads naturally to the next, possibly the most serious
potential problem: the derivation of the relation between black-hole entropy and area based on
the Carnot-Geroch cycle does not by itself guarantee that there is no process that violates theGeneralized Second Law. In particular, though in footnote 36 I claimed to turn Geroch’s infamousthought-experiment on its head, nothing seems to preclude Geroch’s original use of it to argue
that, were classical black holes to have physical temperature, it would have to be absolute zero
independently of what value its surface gravity had. If one arranges matters just so, the weight
lifted by the lowering of the box will have extracted all the energy content of the box when it
reaches the event horizon; one can then dump into the black hole the stuff in the box, which still
has its original entropy but zero mass-energy; thus, one will have converted thermal energy into
work with 100% efficiency, implying the black hole must have temperature absolute zero. Because
the matter dumped into the black hole has no mass-energy, the area of the black hole does notincrease; because the matter still has its original entropy, however, the total entropy of the world
outside the event horizon has decreased, thus violating the Generalized Second Law.
There are (at least) two possible responses.45 First, one can note that the procedure requires
measurements of arbitrarily fine precision: the violation of the Generalized Second Law occurs
only if the matter has exactly zero stress-energy when it is released precisely when the box iscontiguous with the event horizon. Otherwise, the area of the black hole will increase, and will
always do so in way so as to preserve the Generalized Second Law. If one holds that classicalthermodynamics is only an effective theory in the first place, as seems reasonable, then the notionof arbitrarily precise measurements never gets off the ground.46 In particular, the exact precision
45. Perhaps the most influential response in the physics literature to this problem is given by Unruh and Wald(1982). I will not consider their response, as it essentially relies on quantum effects.
46. This is not the same issue as arises with arguments over the possibility of a Maxwell demon, though thedemon may have to make arbitrarily fine measurements in order to function. The Maxwell demon in classicalstatistical mechanics will eventually thermalize, and so one will have to continually produce a new demon from alow-entropy source in order to produce arbitrarily large deviations from the Second Law, whereas nothing in the26

Classical Black Holes Are Hot

11. Sept 2023of the manipulations required would seem to admit of the construction of a perpetuum mobile of thesecond kind, contrary to the Kelvin Postulate: if we could perform Geroch’s experiment, then wewould eo ipso have the fineness of control to extract the work performed on individual Brownianmotion particles, which would also be a perfect transformation of heat into work. Therefore, it
seems that the lesson here should be: hold on to the Kelvin Postulate, and it follows that one must
deny the infinite precision the thought-experiment requires. That response, however, does not infact work in the end. Infinite precision is a red herring, because one needs only to lower the box toa proper distance from the black hole less than S/2πE (where S and E are the entropy and initialenergy of the box) in order to ensure that the change in black hole area is less than S (Unruh andWald 1982), so violating the Generalized Second Law.47
A stronger response is that, even if one thinks of these as violations of the letter of the classicalSecond Law, they cannot be used in the purely classical case to construct a perpetuum mobile ofthe second kind, as one cannot return the black hole to its original state, so they do not violate its
spirit.
There is, however, another possible mechanism for producing arbitrarily large violations of theGeneralized Second Law if one treats classical black holes as truly thermodynamical objects. Put a
Kerr black hole in a reflecting box and pervade the box with thermal electromagnetic radiation at alower (Planck) temperature than the classical Bekenstein-Hawking temperature of the black hole.
The black hole will eventually absorb the thermal radiation: heat, it seems, would spontaneously
flow from a system at a lower temperature to one at a higher temperature, a seeming violationof the standard Clausius Postulate.48 First, one should note that this is not a violation of theGeneralized Clausius Postulate for black holes, as the irreducible mass, and so the area of the blackhole, increases after absorption. If one takes the Generalized Clausius Postulate as the appropriate
formulation of the Generalized Second Law in the context of classical black-hole mechanics andthermodynamics, as the ordinary Clausius Postulate is in classical thermodynamics alone (Maxwell
1891; Fermi 1937), then there is no violation of the Generalized Second Law.
Arguably, moreover, a prohibition on spontaneous heat flow from a body at higher temperature
to one at a lower temperature is not a violation even of the ordinary Second Law. As I discuss inmore detail below, nothing in even ordinary thermodynamics prohibits two bodies with different
temperatures in thermal contact gaining and losing heat from each other simultaneously. This
point, in fact, points to the most decisive reason to think Wald’s proposed counter-example is
not truly one. As I emphasized in §5, and especially in §5.1, the appropriate notion of “thermal
coupling” for interactions involving black holes is the transfer of gravitational heat, not just thetransfer of ordinary thermodynamical or electromagnetic heat. Consider an analogous case, withelectromagnetic heat and ordinary heat: that a colder liquid flows into an area of hotter electromagnetic radiation does not imply that heat has flowed from a colder to a hotter system. What
black-hole case thermalizes.
47. One still has to verify that there are in fact substances such that the required entropy can be squeezed intoa small enough container for the given mass-energy. It is easy to construct examples of such substances in quantum theory (Unruh and Wald 1982), but I do not know of explicit examples for classical systems. Nonetheless, it
should be noted that nothing in classical thermodynamics prohibits such systems either. This fact, indeed, wasthe motivation behind Bekenstein’s (1981) proposal for a universal upper bound on the ratio of entropy to energy
for all possible physical systems.
48. I thank Robert Wald for proposing this case to me. A related one is described in Unruh and Wald (1982).

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Classical Black Holes Are Hot

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is relevant is the exchange of heat appropiately understood between the two systems, and in particular the way that the energy of the electromagnetic radiation is transformed into the liquid’s
ordinary heat. In the same way, in the case at issue for us, the mere fact that thermalized radiation
falls into the black hole does not by itself imply that it is carrying an excess of gravitational heat
into the black hole. As the radiation falls into the black hole, the gravitational radiation emitted
by the black hole from the disturbances caused by the in-falling radiation—“gravitational heat”—isabsorbed by the radiation and transformed into ordinary electromagnetic heat. Even in this case,therefore, heat isn’t flowing from the colder to the hotter body, but rather from the hotter to thecolder.
Before turning to what we may learn from my conclusions, if they are correct, I consider a few
more possible problems, none of which I consider severe. Indeed, the resolution of all of them lies
in showing that, as with the issue just discussed, the proposed problem really is a problem fortreating classical black holes as thermodynamical systems if and only if it is also a problem forordinary classical thermodynamical systems—which in fact strengthens the overall conclusion ofthis paper, viz., that one can give a classical theory of black-hole thermodynamics that behaves inevery way like the theory of ordinary thermodynamical systems, warts and all, which is the only
conclusion I am arguing for.
First, it is easy to see that ordinary thermodynamical systems can violate both the First andthe Second Laws of thermodynamics if quantum effects are not accounted for. Consider water in
equilibrium, just above freezing temperature in a transparent, insulating container surrounded bya spatiotemporally extended vacuum. In particular, there is no ambient, external electromagnetic
field. According to classical thermodynamics, the water should stay as is. In fact, it is emitting
blackbody radiation, and so, losing energy and lowering in temperature. Eventually it will freeze,
spontaneously evolving into a state of lower entropy, a violation of the Second Law. Thus, nottaking quantum effects into account can lead to “inconsistencies” in ordinary thermodynamics aswell, exactly as with black hole thermodynamics. Another example is given by black asphalt inthe sun: its temperature “spontaneously” rises to a higher value than the ambient air and theEarth underneath, the only two other ordinary thermodynamical substances in direct contact with
it. The solution, of course, is to include the sun’s thermal radiation. Ordinary thermodynamics
and classical black hole thermodynamics are in the same boat in this regard: in so far as one canarrange what look to be violations of the First or Second Law for classical black holes by couplingthem to quantum systems, one can do as well for ordinary thermodynamical systems.
As is well known, the surface gravity κ is well defined only for stationary black holes; does thismean that my analysis cannot apply to non-stationary black holes? Yes, it is the case that myanalysis cannot apply to non-stationary black holes. Non-stationary black holes are ones out of
equilibrium, and so this presents the same situation as obtains in classical equilibrium thermodynamics. I think we often forget that, strictly speaking, temperature in ordinary thermodynamics
is well defined only for bodies in (or quite close to) thermal equilibrium. One way to see this is tonote that, for systems far from equilibrium, different kinds of thermometric device will return very
different readings, as fine details of their different couplings to the system which are negligible for
equilibrium systems become non-trivial, in particular due to phenomena manifesting themselvesat temporal and spatial scales below the hydrodynamic scale.49
49. See, e.g., Benedict (1969, §§4.1–4.4, pp. 24–9). This reference is not the most up-to-date with regard to the28Classical Black Holes Are Hot

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Another problem is that it seems as though we can attribute heat to a Schwarzschild blackhole only when it is being perturbed. Again, the situation is in fact the same as in classicalthermodynamics, wherein it never makes sense to attribute a definite quantity of heat to an isolated
system in equilibrium. The only definite claims one can make, as Maxwell (1891, chs. i, iii, iv,viii, xii) himself so insightfully and eloquently pointed out, are about the quantification of heattransfer. And one can extract both “heat” and work from a Schwarzschild black hole, by perturbing
it; indeed, this is in perfect analogy with ordinary thermodynamical systems that have reachedheat death, from which heat and work can be extracted only if one perturbs them properly. Thisline of thought is strongly buttressed by the results of Kiefer (2004), as I discussed in §5.4.
In fact, the analogy is even better than those brief remarks suggest: stationary classical black
holes do not “radiate heat”, but neither do ordinary classical thermodynamical systems in equilibrium; they exchange heat only when they are in direct contact (contiguous) with another system
at a different temperature, but the same holds for stationary classical black holes, in so far as their
immediately contiguous environment is “at the same temperature”, viz., has essentially the same
effective surface gravity as measured at infinity as the black hole does. Still, one may protest, inthe construction of the Carnot-Geroch Cycle I ignored perturbations to the black hole from thelowering of the box, so how can one say, given my definitions and arguments, that energy was
extracted from it? Given the assumption that the total energy of the box is negligible comparedto the mass of the black hole, I claim it is a good approximation to ignore any perturbations tothe black hole while still accounting for the (relatively negligible) amount of energy the box gains
by being lowered through the black hole’s “gravitational field”. This approximation is of exactly
the same kind as one uses for ordinary Carnot cycles, where one does not treat the heat sink as
having lost heat (changed its internal energy), only the body moving through the cycle as having
gained it.
Another potential problem: it is clear that black holes have, by the standard definition, negative specific heat, since their surface gravity decreases as their mass-energy increases. Standard
arguments, however, conclude that two bodies with negative specific heat cannot thermally equilibrate. There is, though, a hidden assumption in the standard arguments, to wit, “conservation ofheat”—that is to say, it is always assumed that, for two bodies in thermal contact, one can gain
heat only if the other loses it, and that in the same amount. Heat, however, is not a substance, as
everyone from Maxwell (1891) to Planck (1926) to Sommerfeld (1964) is at pains to emphasize, andso obeys no conservation law. There is no reason why two bodies with different temperatures inthermal contact cannot both “gain or lose heat from or to each other” at the same time. When two
black holes in quasi-stationary orbit50 about each other equilibrate, the temperatures of both bodies simultaneously decrease as they both gain heat from the other, the one of higher temperature
decreasing more quickly than the other, so they will eventually reach the same temperature.51
international agreement on defining the standard, practical methods for the determination of temperature, but I
have found no better reference for the nuts and bolts of thermometry. See Curiel (2010, §3.4) for a discussion of
the details.
50. There are no solutions to the Einstein field equation representing two Kerr black holes in stable orbit about
each other (Manko and Ruiz 2001).
51. I have no rigorous calculation for this kind of process. Heuristic and very crude symmetry considerations,however, militate in favor of it. If two black holes are very far apart—very weakly coupled—and interact only
through exchange of energy by emission and absorption of gravitational radiation (driven by their relative or-29Classical Black Holes Are Hot

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A potentially more serious problem with my analysis is that it is difficult to see what sense canbe made of “exchange” between a global energetic quantity (in the case of stationary, asymptoticallyflat black holes, ADM mass) on the one hand, and localized stress-energy of ordinary systems on
the other. A more poignant way of posing the problem is to note that gravitational energy isstrictly non-local in the precise sense that there is no such thing as a gravitational stress-energy
tensor (Curiel 2019a), and so it satisfies no general conservation law. How, then, can one talk about
exchange for such a recherché quantity? There are, I think, two responses to this problem, one
stronger than the other. The first, weaker, response is that one always has in place a quasi-local
notion of mass-energy in stationary and axisymmetric spacetimes (Szabados 2009), which suffices
for the purposes of my arguments, just as it does in Newtonian gravitational theory (à la the“Poynting integral” of Bondi 1962).
The stronger response, which is more to the point, is that neither is heat a localized form ofenergy in classical thermodynamics—it is not a perfect differential (as the discussion of Sommerfeld
1964 makes particularly clear), and so it also has no corresponding conservation law—just likegravitational energy—and yet we feel no inconsistency in talking there about exchange of energy
for a quantity that can be represented only as a total magnitude, with no corresponding localized
density. Sauce for the goose, again, is sauce for the gander.
Another possible problem for my arguments: if the classical constant α is “really” based on ℏ,
then in the classical limit won’t α → 0 and so temperature go to absolute zero for all black holes?
No. α is the analog of Boltzmann’s constant for black holes, which, though it is “really” quantum
in origin, is used in a consistent and fruitful way in classical thermodynamics; one does not set it tozero just because one is dealing with classical thermodynamical systems in a classical framework.
My arguments, I think, have not only residual possible problems; they also open the possibility
for real insight into existing questions about black-hole mechanics and thermodynamics. Although
the following is not a problem peculiar to my analysis, it is a general one in the field I believe
my analysis can give some insight into. Black holes have enormous entropy, far more than any
reasonably conceivable material system that could result in one from gravitational collapse (Penrose1979) There must, therefore, be a correspondingly enormous and discontinuous jump in entropy
when a collapsing body passes its Schwarzschild radius, of order at least 1010 . How can one explain
that? It is here that I believe my old-fashioned approach to entropy bears some of its sweetest fruit.
More modern characterizations of entropy, whether of a Boltzmannian, Gibbsian, von-Neumannlike, or Shannon-like form, have no explanation for this jump. If, however, one conceives of entropy
as a measure of how much work it takes to extract energy from a system, how much free energy asystem has, what forms its internal energy (as opposed to free energy) are in, then black holes haveenormous energy, only a very small amount of which is extractible, and there is a clear physicaldiscontinuity in extractability of energy when an event horizon forms. Indeed, one can view theenormously greater entropy of a black hole, compared to the matter that formed it and any matterthat later falls into it, as a marker of the extreme irreversibility of such processes—the 1-way causal
structure of the event horizon is encoded in the enormity of the black hole’s entropy.
I leave the reader with a question concerning this entire field that, though not peculiar to mybital motion), then, once they reach the same surface gravity and so have the same mass and the same size (crosssection for emission and absorption), there is no reason why one should gain or absorb more gravitational radiation than the other.

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arguments here, I feel strongly needs to be investigated further by both philosophers and physicists.
The Laws of thermodynamics are empirical generalizations, indeed, the paradigm of such. I knowof no other fundamental propositions in physics whose support comes entirely from experimental
evidence, with not even the suggestion of the possibility of a formal derivation from “deeper”
physical principles. Also, I know of no other propositions, with the possible exception of the
Newtonian inverse-squared distance dependence of gravitational attraction between two proximate
bits of matter, that are more deeply entrenched empirically than the Laws of thermodynamics.
But, entirely to the contrary, and with the exception only of the Third Law (which is also themost weakly supported by experimental evidence in classical thermodynamics), all the Laws of
black-hole mechanics are theorems of differential geometry. They require no input from physicaltheory at all. One will sometimes see the claim that one or the other of the Laws requires theassumption of the Einstein field equation, but this is not true: all the Laws are independent of theEinstein field equation in the strong sense that one can assume its negation and still derive theLaws; the Einstein field equation enters only when one wants to give a physical interpretation of thequantities involved by way of its asserted relation between the Ricci tensor and the stress-energy
tensor of matter.52 So how can laws that, in one context, are nothing but empirical generalizations,magically transform into mathematical theorems when extended into a new context?

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