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The role of gravitation in thermal physics (and thermo field theory)
Physica A 158 (1989) 437-447 North-Holland, Amsterdam
THE ROLE OF GRAVITATION IN THERMAL PHYSICS (AND THERMO FIELD THEORY) Bernard F. WHITING Institute of Field Physics, Department of Physics and Astronomy, The University of North Carolina, Chapel Hill, NC 27599-3255, USA
With his introduction of quantum mechanical considerations in the treatment of gravitating systems, Hawking indicated a way for the inclusion of general relativity into quantum (and, in particular, thermal) physics. However, probably the single most difficult phenomenon to deal with, subsequent to his work, has been the gravitational attraction of the very feature which his work leads to, viz, the thermal bath surrounding a black hole in equilibrium. It was not until York introduced a finite containing box for the system that certain problematical infinities could be removed and the treatment became well defined, with a sensible, describable, thermodynamic limit, and the possibility of finally realizing a satisfactory canonical ensemble for a black hole in a box. Gravitating systems could then be described thermodynamically, and the problem naturally admitted a path integral formulation which has been used to determine the contribution of non-classical geometries to the partition function for the black hole topological sector of the gravitationa! field. The application of this method to more familiar problems with a fixed space-time suggests an explicit interpretation of the theory of thermo field dynamics when thermal disturbances of the gravitational field can be ignored. This particular understanding may be extended to include sma!' perturbative fluctuations of the geometry (loosely called gravitons) but it seems inappropriate for dcalmg with the actual structure of the space-time manifold, whereas gravitational thermodynamics. as referred to above, would remain perfectly adequate.
1. Introduction
It is precisely because of the traditional difficulty encountered when incorporating gravitation into statistical thermodynamics that general relativity can make such a fundamental and far reaching contribution to thermal physics as it has indeed done in recent years. The very foundations of the mathematically rigorous attempts to validate the heuristic developments of classical t~he~modynamics have excluded the possibility of admitting an unscreened longrange force in the interactions of ordinary matter. Such approaches have thereby suffered an intrinsic restriction, particularly since the thermodynamic limit, as usually considered, is gravitationally non-existent. A simple exz;mple will amply show the striking contrast: e v e n in a finite system of radius r 0, at sufficiently high temperature T ~ mp(rp/ro) 1;2 (where mp and rp are the Planck 0378-4371/89/$03.50 © Elsevier Science Publishers 13.V. (North-Holland Physics Publishing Division)
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units of mass and length), thermal radiation will become unstable and collapse gravitationally [1], whereas the same system containing a black hole is gravitationally stable no matter how high the temperature, and, quantum mechanically [2], the energy will always remain less than would be required for the black hole to completely engulf the system. Even though attention to this particular detail is not normally required in the laboratory, it may well be relevant in the early universe and it will always prevent passage to the usual thermodynamic limit! (Though not the construction of some physically useful heat-bath.) Hawking's deduction that a black hole in empty space would radiate as a black body [3] has had a tremendous effect. It was based on a calculation of collapse for a gravitating system and immediately gave substance to a previous, tentative observation [4] on a similarity between black hole physics and "classical" thermodynamics. More importantly, it stimulated a noticeable resurgence in the already existing study of quantum fields in curved space-lime and led to a local approximation [5] for the RMS quantum fluctuations of a thermal scalar field around a black hole. Eventually, this gave rise to the development of local approximations [6] for the quantum stress tensors of (scalar and spin-l) matter fields [7], in particular on fixed black hole spacetime backgrounds, referring to both in- and out-of-equilibrium situations [8]. The subsequent treatment of back-reaction problems [9] both demonstrated the growing degree to which the effects of quantum fields could be built into the space-time geometry and, at the same time, re-emphasized the everpresent need to provide an adequate quantum treatment of the geometry itself.
2. Gravitational thermodynamics Curiously, it is probable that the most important outcome inspired by Hawking's observation that black holes radiate has not come simply from an extension of quantum field theory techniques to the problematic quantization of gravitational degrees of freedom in order to modify some fixed-background, back-reaction calculation. Rather, it has come from following up the original realization that the occurrence of black hole radiation makes immediate contact with the verv large and well established domain of thermal nhv~ir~ which automatically permits the treatment of quantum systems within the context of a thermodynamic ensemble. The essential step which allowed this dramatic new development [10] came with York's joint realization that (i) the natural context in which to establish a well defined thermodynamic description would be for the gravitating system under consideration to have finite size, and (ii) in that case the boundary data for the canonical ensemble equally represented boundary data for a well defined problem in general relativity. ~'~
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B.F. Whiting / The role of gravitation in thermal physics
439
Originally, in his treatment, York was concerned only with the contribution of classical solutions, in what may bc tcrmcd a zero-loop approximation. However the lull impact of the phenomenon of black hole radiation was not realized until a path integral formulation was developed [11] for the partition function. Whereas, in terms of asymptotic data, the Schwarzschild black hole was normally considered to be both unique and stable, for thermodynamic boundary data it was neither unique nor necessarily stable. Not only were there two classical solutions (if there were any) as was well known to York, but there was a whole free function-of-two-variables worth of gauge-inequivalent, nonclassical geometries, each of which contributed to the partition function as much as did a classical solution [11]. This observation was consistent with the known result that the zero-loop Helmholtz free energy for the Schwarzschild space-time contained substantial entropy for the black hole, and seemed to focus attention on the fact that for gravitational thermodynamics the state of a gravitating system was indeed determined by an ensemble, not by a single space-time. An immediate consequence of working with a gravitating system of finite extent [10] was, finally, settlement concerning the questioned existence of a canonical ensemble for black hole thermodynamics and, eventually, a compatible description of the associated density of states for spacelike three-geometries in the black hole space-time [12] (initially in the zero-loop approximation). In particular, York [10] was able to determine the condition for local thermodynamic stability in the canonical ensemble: with a temperature at which too small a mass occurred for the black hole within a large confining system, the ensemble would be thermodynamically unstable, while the mass in a stable system of infinite extent would itself be infinite (but well defined by a limiting process). Although this criterion was consistent with the condition for gravitational stability of the black hole [13], its relation to the dynamical stability of general relativity was perhaps not fully understood until the construction of the reduced action used in the path integral formulation [11]. The existence of a relation rested on the fact that, for suitable boundary data, the condition for global stability of the thermodynamic ensemble was determined by the condition for local stability of the gravitational field at a cla~sica! stationary point [141. In addition to providing a clear distinction between the conditions for local and global stability, the path integral formulation also allowed a more precisc description of the density of sta~es with remarkably sensible physical consequences [11]. A more detailed account of the formulation and description of gravitational thermodynamics [11] may be found in the accompanying article by York [15] (in these Proceedings) and in the various references cited above [10-14]. As a final remark it is perhaps worth observing that the path integral formulation
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mentioned here is similar, in many of its constructional features, with the formulation given for the "wave function" of the universe [16], but gravitational thermodynamics has drawn more directly on the semi-classical limit in order to yield definite tractable results.
3. Gravitation in thermal physics One of the strongest reasons that gravitation should be properly included in thermal physics, generally, stems from the unphysical nature of the thermodynamic limit. The problem is that, at non-zero temperature, even a massless field (e.g. photon radiation) has energy, and energy gravitates. But there is a further reason, which has been less apparent uptil now but is emphasized in the specific methods [10] first used by York. In the canonical ensemble, for example, the walls of a container of fixed area follow a non-geodesic path in the gravitational field of the matter-plus-radiation contained within the walls, and hence are accelerating; even if the surface pressure is held fixed rather than the surface area, the walls would then be free to move through a gravitational potential well, which will influence their motion. Gravitational effects are always present in thermodynamic systems. Thus, it is indeed fortunate, as York observed, that the boundary data for a thermodynamic ensemble are precisely boundary data for a well posed problem in general relativity. In principle, then, the description of a system in thermal equilibrium, say, necessarily requires treatment of the coupling of the gravitational field to the other matter fields, massive or massless, which determine the system. General relativity always provides the space-time geometry, which then supports the other matter fields, as well as the gravitational radiation which may propagate within the space-time. It is important to realize the general nature of York's observation. Its application is not confined only to the case of pure gravitation. Thus, in computations ~vith a number of collaborators, it has been possible to treat explicitly coupling to a static electric field [17], the field given by a 3-form [18] (which acts as a source for a cosmological constant) and, to some extent, matter such as might occur in a star [i9] given a phenomenological description in terms of an arbitrary equation of state. Coupling to the field strength of a 2-form potential also seems to give useful, non-trivial results, and coupling to a quantum thermal scalar field through a one-loop effective action for a finite system has been almost completed recently [20] and, as discussed below, is expected to lead to a minor modification of the condition for gravitational stability [24]. Even though analysis of the phenomenological system for a star is also incomplete, perhaps it will be fruitful, using it as an example, to illustrate
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a number of general features which emerge, independently of which field it is coupling to gravity, when gravitation is included in thermal physics. Sehutz has introduced an action principle [21], based on velocity potentials for a perfect fluid, which can model a star. The pressure p(m, s) is used as the Lagrangian density, it being expressed in terms of two variables (specific inertial mass and specific entropy per unit rest mass) through an zrbitrary equation of state. In coupling such matter to gravity one needs to deal separately with the density of total mass energy p, which is - T ° of the matter stress tensor, on the one hand, and the sum (p + p) on the other. Using the field equations for a static spherically symmetric system in this phenomenological model, the 4-volume integral of ( p + p) in the action can always be written in terms of at: integral of the proper 3-volume density of a quantity ~ derived from one of the velocity potentials, with no explicit reference to the lapse function (_g00)-~/2 anywhere on the interior of the 4-manifold with boundary. In the case of the constant density star, with a particular choice of the equation of state and solution to the field equations (which permitted the construction of a local Euler relation [22]: p + p = Ts + t~n), this integral can be written in terms of the e n t r o p y ai,u " ' number-density, though not generally. Nevertheless, it always depends functionally on the boundary data of the solution and contributes to the total entropy for the case of a system in thermal equilibrium. There remains the action for general relativity coupled to the total energy density.
Constraint equations As a completely general principle, the components T~° of the matter stress tensor are eliminated from the action by solving the constraint equations of general relativity on a family of spacelike hypersurfaces foliating the manifold [11] (just as constraints in any gauge theory coupled to gravity should be solved to eliminate gauge degrees of freedom from the action). In the case of a static s.tar, only the Too component occurs, and the result of satisfying the superHamiltonian constraint to eliminate it, and of including the boundary, terms for the gravitational part of the action for thermal equilibrium, is the quantity/3 Eh ( n ~ , , m i n g nO black holes within the stnr)~ where [3 = T - l is the inverse temperature maintained at some boundary surface of area A = 47rr~ placed within the star. The total energy of the system is given by
e = M (1-
(1 - 2ce)~/2),
o~
where c~ = GM/r o and
B.F. Whiting / The role of gravitation in thermal physics
442 ro
M = 4~r kl r2p(r) dr. 0
Note that E = M(I + a + ¢r2 + . . . ) when a is small. It is paramountly clear that gravitational effects on the energy for spherical equilibrium configurations are always of the same form, and can never be neglected when the quantity a = G M / r o is not small. It is also clear that the system fails to be physically reasonable whenever a ~> ½; and a more refined analysis [23] for any sensible equation of state shows that one must have a <~ 4~, independent of whatever matter is the source of the energy density. To go beyond this one needs to consider singular manifolds and/or manifolds with different global topology.
4. Quantum fields and back-reaction
Using the understanding gained from the discussion above, one could imagine trying to examine the back-reaction of some quantum thermal field which had been developed in a fixed, flat space-time background. However, when the system is near the state of gravitational collapse, non-linear effects are important and a perturbative analysis is inappropriate. Non-perturbative techniques are needed, of the type used in the path integral formulation for the black hole topological sector [1!], as indicated above. This would require evaluation of the action for the system in configurations far from those which occur at the classical stationary points and, as just mentioned, necessarily will require consideration of the field theory in other topological sectors of the gravitational field. Thus, such interesting problems are not of the nature of a back-reaction calculation, and currently await a comprehensive treatment for arbitrarily large thermodynamic systems. An alternative and more tractable problem concerns back-reaction on a non-trivial background, e.g. a black hole space-time, for which one could successfully examine, at the linearized level, modification of the conditions for thermodynamic stability. This has been partly carried out for a massless scalar field, and is currently under further investigation [24]. It appears that quantum changes in the criterion for local thermodynamic stability are small. In addition, the specification of an action principle for the back-reaction has almost been completed [20], based on an approximation to the one-loop effective action for a quantum scalar field in thermal equilibrium and including all boundary terms resulting from deali~ ". with a finite system. This may be useful for examining contributions given to the partition function by geometries far from the classical solutions of pure gravitation around black holes.
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443
Quantum field theories generally, and back-reaction calculations for them in particular, necessarily are posed originally with respect to a fixed background space-time. Yet even with pure gravitation it has been possible to show that equilibrium configurations at different temperatures may correspond to manifolds with different topology [10]. Quantum corrections due to temporal and spatial fluctuations in the geometry are expected to modify, but not substantially alter, this conclusion. However, these calculations are based on equilibrium configurations only and make no proposal for how the dynamical evolution might occur as the temperature is changed to effect the transition from one equilibrium topology to another. Nevertheless, genuine time dependent formalisms do exist for thermal quantum field theories [25, 26] and techniques from these could, in principle, be invoked to examine whether a system on the brink of instability will manifest an impending transition by a particular behavior either for the matter fields or even for gravitational fluctuations of the space-time geometry. Ia the following section, consequences for one of these methods [26] applied t¢ equilibrium configurations will be examined in relation to the understanding which has been gained from gravitational thermodynamics as discussed above.
5. Thermo field dynamics Before beginning here, it will be useful to stress that, as with quantum field theory in general, thermo field dynamics has been developed for quantum fields on a given space-time background. Thus~ in applying gravitational thermodynamics to the thermal physics of some quantum field coupled to gravity, one may imagine, indeed, that the space-time background has been fixed. In fact, it is usual to assume time independence for the geometry, as occurs for flat space-time. Now the general form of the super-Hamiltonian density is
where Y(M is the proper energy density of the matter and ~G iS the superHamiltonian of general relativity. There is a similar expression for the supermomentum density ~ but it will not be necessary to consider that explicitly here. As was stated above, the constraint equations of general relativity require ~ (and ~ ) to be identically zero on each leaf of a family of spacelike hypersurfaces foliating the manifold. For a time independent geometry the contribution from general relativity can thus be taken as constant. Even if there were gravitational fluctdations the coupling would be so weak ip gene;al that these could be ignored, unless thermal disturbances of the
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gravitational field were indeed substantial. By separating gravitational fluctuations from the background geometry, one might try to include such "gravitons" in a modified matter Hamiltonian describing the system, but one could never include the geometry itself this way. However, the static coupling which actually forms the background geometry can never be ignored, since it must always exactly balance the contribution from the matter energy density. Thus, whenever the gravitational contributions to the super-Hamiltonian constraint are effectively equivalent for any two independent labels of the spacelike hypersurfaces on which the constraints are imposed (i.e. for any two independent instants of "time"), instead of imposing the super-Hamiltonian directly on the geometry-plus-matter, one can equivalently require that the matter Hamiltonian at one time be exactly balanced by the matter Hamiltonian at any other representative time. In effect, one is demanding that the system itself be representative of the heat bath which is imposed gravitationally through the thermodynamic boundary data for the geometry. Therefore, one may use seM - se , = o ,
as is done in thermo field dynamics [26]. In fact it appears that this consequence alone should be sufficient to yield the full development of thermo field dynamics, as it is currently understood. The thermo field condition relating creation and annihilation operators in the system and its thermal bath, a ( t ) = ~*(t-ifl/2), arises also in the real time formalism mentioned above [25]. This condition relies on a discrete isometry of the Euclidean manifold used to represent the state of thermal equilibrium, and it would not in general hold if the background geometry were time dependent. In the particular case of the Kruskal manifold, which is the maximal analytic extension of the Schwarzschild space-time, this isometry can be proved using the Einstein field equations [23]. Although in a general path integral formulation the assumption of such an isometry would be too strong, it may apply for an effective theory in which all the gravitational degrees of freedom have been integrated away. The existence of such an isometry is an essential ingredient in the recent **~rl~,,-.-.~.~,~.-I~...~ r,~l,-Tl ~ .It._ .l__ u,,u,..,ata,,um~; tz.,j U ! t l l t ~ nature of mermo field dynamics on a fixed spacetime background, which has been obtained using techniques from quantum cosmology and, in particular, the formalism for the "wave function" of the universe. This approach refers to the different data residing on two (generally) disjoint surfaces in a global slicing of the periodic Euclidean manifold used for equilibrium thermodynamics (such as occurs for thermal fields in flat space). However, in the case of the black hole (and also Rindler space) the global manifold is simply connected. In this context, l~ usually is taken to correspond
B.F. Whiting I The role of gravitation in thermal physics
445
to matter fields having support only in the left-hand wedge wh~',e the quantum fields defined in the exterior region would be taken to have support only in the fight-hand wedge. Then the relation
would apply, with ~(/3) and ~'(/3) each being defined on a whole section of the global manifold. The fields corresponding to ~(/3) and ~(/3) have been referred to by Israel [28], in an application of the methods of thermo field dynamics, and global representations in a particular basis have been constructed explicitly [29]. Both sets of fields have positive norm on a whole spacelike section of the global manifold (and neither has support restricted to only the left or fight-hand wedge). Thus, the Hamiltonian for evolution forward in the global Kruskal time variable would be ~(/3) + ~(/3), not zero, in contrast to the right-hand side above. There, the evolution is partly forward and partly backward in the global time, even though neither the fields for ~ ( / 3 ) nor the fields for ~'(/3) have total support in either part (the fields for M and M do, however). No apparent conflict of this kind arises in deriving thermo field dynamics from gravitational thermodynamics, since 1VI need not even refer to fields anywhere in the actual space-time of the original field theory; reference to another appropriate element of the thermodynamic ensemble would be sufficient. In this way, time dependent geometries could also be included, with 1Vl(.p,x being representative of any other matter configuration which was compatible with the geometry at time t. Nevertheless, it would appear that only gravitational thermodynamics can properly describe the full contribution of the geometry (as has been done for the black hole topological sector) to, say, the partition function, through its initial inclusion in the path integral formulation.
6. Discussion
Gravity introduces, an additional scale into thermal physics. Thus a sphericaliy symmetric system of "~"",,,~,~,... ..... ,,;the,...-n~ M ~ncl ~nrface ..... area A = 4rrr?,.. must satisfy a condition for gravitational stability, e.g. GM/r,~ <~ 4~. On the other hand thermodynamics introduces an extra scale into general relativity. Thus. in the canonical ensemble, a black hole must satisfy the condition for thermodynamic stability [11], i.e. 3~h/8rrr o<~ ~~. The two scale~ of D avitational thermodynamics can become mixed (but not identified) as for example when thermodynamics determines the energy density of radiation for a finite system in thermal equilibrium. In fact, these two conditions listed above are ccm-
446
B.F. Whiting / The role of gravitation in thermal physics
plementary, in that the former requires G(E)/ro<~ 3, where ( E ) is the expectation value of the energy in an equilibrium configuration, while the latter requires G ( E ) Iro ~> ], whe~'e here only the zero-loop value of ( E ) has been used for the black hole. Nevertheless, these conditions can be satisfied together for certain boundary data, since energy scales differently for radiation than for the gravitational field of an equilibrium black hole. The interplay between the different scales combined with the topological transformations which they herald, as seen in gravitational thermodynamics, represent rich new developments in thermal physics. In particular, there is an unexplored Planck scaleregime, related to early universe quantum cosmology, in which both the above conditions would be violated, and for which higher order corrections require investigation.
Acknowledgments I have benefited from conversations with N.J. Papastamatiou and R. Laflamme. I especially thank J.W. York for his collaboration on part of this work, and I am grateful to my various colleagues at Chapel Hill for the opportunity to refer to their work prior to publication. This research has been supported by the National Science Foundation.
References [1] J.W. York, Black holes and partition functioas, in: Proc. Osgood Hill Conf. on Conceptual Problems in Quantum Gravity, to be pu01ished. [2] B.F. Whiting, Non-classical geometries in gravitational thermodynamics, in: Proc. fifth Marcel Grassman meeting, held in Perth, Western Australia, August 8-13, 1988, D.G. Blair and M.J. Buckingham, eds. (Cambridge Univ. Press, Cambridge), to be published. [3] S.W. Hawking, Commun. Math. Phys. 43 (1975) 199. 14] I.D. Bekenstein, Phys. Rev. D 7 (1973) 2333. [5] M.S. Fawcett and B.F. Whiting, Spontaneous symmetry breaking near a black hole, in: The Quantum Structure of Space and Time, M.J. Duff and C.J. Isham, eds. (Cambridge Univ. Press, Cambridge, 19,'42). [6] D.N. Page, Phys. Rev. D 25 (1982) 1499. [7] K.W. Howard, Phys. Rev. D 30 (1984) 2532. [8] M.R. Bxown, A.C. Ottcw'ill and D.N. Page, Phys. Rev. D 33 (1986) 2840. [9] J.W. York, Phys. Rev. D 31 (1985) 775; K.W. Howard, B. Jensen and B.F. Whiting, Back-reaction for a lhermal scalar field in anti de Sitter space-time, Meudon preprint (1985). [l(I] J.W. York, Phys. Rev. D 33 (1986) 2(102. [ill B.F. Whiting and J.W. York, Phys. Rev. Lett. 61 (1988) 1336. [12] H.W. Braden, B.F. Whiting and J.W. York, Phys. Rev. D 36 (1987) 3614. [13] B.F. Whiting, Gravitational constraints and thermodynami~ ~tability, to be published (1988).
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1141 B.F. Whiting, Black holcs and thcrmodynamics, in: Proc. ninth lAMP ct,:i~rc~,,,, held in Swansea, Wales, July 12-27, 1988, B. Simon, I.M. Davtes and A. 'lrurpan, eds. (Adam Hilger, Bristol), to be published. I151 J.W. York, Physica A 158 (1989) 425, these Proceedings. [161 J.B. Hartle and S.W. Hawking, Phys. Rev. D 28 (1983) 2960. [171 H.W. Braden, B.F. Whiting and J.W. York, Charged black holes in the grand canonical ensemble, to be published (1988). [18] J.D. Brown, B.F. Whiting+ and J.W. York, Antisymmetric tensor source field in gravitational thermodynamics, to be published (1988). [191 J.D. Brown, G.L. Comer and B.F. Whiting, unpublished notes, Chapel Hill (1988). 1201 J. Melmed, unpublished notes, Chapel Hill (1988). [211 B.F. Schutz, Phys. Rev. D 2 (1970) 2762. I221 B.F. Whiting, unpublished notes, Chapel Hill (1988). [23] R.M. Wald, General Relativity (Chicago Univ. Press, Chicago, 19J:~4). [24] G.L. Comer and J.W. York, unpublished notes+ Chapel Hill (19~S). [251 A.J. Niemi and G.W. Semenoff, Ann. Phys. 152 (1984) 105; Nuci. Phys. B231J (19~4) 1,~1. [26] Y. Takahashi and H. Umezawa, Collective Phenomena 2 (1975) 55. [271 R. Laflamme, Geometry and thermo felds, DAMTP preprint (1988); See also Physica A 158 (1989) 58, these Proceedings. [28] W. Israel, Thermo-field dynamics of black holes, Phys. Lett. A 57 (1976) 107. [29] N. Sanchez and B.F. Whiting, Nuci. Phys. B283 (1987) 605.
THE ROLE OF GRAVITATION IN THERMAL PHYSICS (AND THERMO FIELD THEORY) Bernard F. WHITING Institute of Field Physics, Department of Physics and Astronomy, The University of North Carolina, Chapel Hill, NC 27599-3255, USA
With his introduction of quantum mechanical considerations in the treatment of gravitating systems, Hawking indicated a way for the inclusion of general relativity into quantum (and, in particular, thermal) physics. However, probably the single most difficult phenomenon to deal with, subsequent to his work, has been the gravitational attraction of the very feature which his work leads to, viz, the thermal bath surrounding a black hole in equilibrium. It was not until York introduced a finite containing box for the system that certain problematical infinities could be removed and the treatment became well defined, with a sensible, describable, thermodynamic limit, and the possibility of finally realizing a satisfactory canonical ensemble for a black hole in a box. Gravitating systems could then be described thermodynamically, and the problem naturally admitted a path integral formulation which has been used to determine the contribution of non-classical geometries to the partition function for the black hole topological sector of the gravitationa! field. The application of this method to more familiar problems with a fixed space-time suggests an explicit interpretation of the theory of thermo field dynamics when thermal disturbances of the gravitational field can be ignored. This particular understanding may be extended to include sma!' perturbative fluctuations of the geometry (loosely called gravitons) but it seems inappropriate for dcalmg with the actual structure of the space-time manifold, whereas gravitational thermodynamics. as referred to above, would remain perfectly adequate.
1. Introduction
It is precisely because of the traditional difficulty encountered when incorporating gravitation into statistical thermodynamics that general relativity can make such a fundamental and far reaching contribution to thermal physics as it has indeed done in recent years. The very foundations of the mathematically rigorous attempts to validate the heuristic developments of classical t~he~modynamics have excluded the possibility of admitting an unscreened longrange force in the interactions of ordinary matter. Such approaches have thereby suffered an intrinsic restriction, particularly since the thermodynamic limit, as usually considered, is gravitationally non-existent. A simple exz;mple will amply show the striking contrast: e v e n in a finite system of radius r 0, at sufficiently high temperature T ~ mp(rp/ro) 1;2 (where mp and rp are the Planck 0378-4371/89/$03.50 © Elsevier Science Publishers 13.V. (North-Holland Physics Publishing Division)
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B.F. Whiting / The role of gravitation in thermal physics
units of mass and length), thermal radiation will become unstable and collapse gravitationally [1], whereas the same system containing a black hole is gravitationally stable no matter how high the temperature, and, quantum mechanically [2], the energy will always remain less than would be required for the black hole to completely engulf the system. Even though attention to this particular detail is not normally required in the laboratory, it may well be relevant in the early universe and it will always prevent passage to the usual thermodynamic limit! (Though not the construction of some physically useful heat-bath.) Hawking's deduction that a black hole in empty space would radiate as a black body [3] has had a tremendous effect. It was based on a calculation of collapse for a gravitating system and immediately gave substance to a previous, tentative observation [4] on a similarity between black hole physics and "classical" thermodynamics. More importantly, it stimulated a noticeable resurgence in the already existing study of quantum fields in curved space-lime and led to a local approximation [5] for the RMS quantum fluctuations of a thermal scalar field around a black hole. Eventually, this gave rise to the development of local approximations [6] for the quantum stress tensors of (scalar and spin-l) matter fields [7], in particular on fixed black hole spacetime backgrounds, referring to both in- and out-of-equilibrium situations [8]. The subsequent treatment of back-reaction problems [9] both demonstrated the growing degree to which the effects of quantum fields could be built into the space-time geometry and, at the same time, re-emphasized the everpresent need to provide an adequate quantum treatment of the geometry itself.
2. Gravitational thermodynamics Curiously, it is probable that the most important outcome inspired by Hawking's observation that black holes radiate has not come simply from an extension of quantum field theory techniques to the problematic quantization of gravitational degrees of freedom in order to modify some fixed-background, back-reaction calculation. Rather, it has come from following up the original realization that the occurrence of black hole radiation makes immediate contact with the verv large and well established domain of thermal nhv~ir~ which automatically permits the treatment of quantum systems within the context of a thermodynamic ensemble. The essential step which allowed this dramatic new development [10] came with York's joint realization that (i) the natural context in which to establish a well defined thermodynamic description would be for the gravitating system under consideration to have finite size, and (ii) in that case the boundary data for the canonical ensemble equally represented boundary data for a well defined problem in general relativity. ~'~
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Originally, in his treatment, York was concerned only with the contribution of classical solutions, in what may bc tcrmcd a zero-loop approximation. However the lull impact of the phenomenon of black hole radiation was not realized until a path integral formulation was developed [11] for the partition function. Whereas, in terms of asymptotic data, the Schwarzschild black hole was normally considered to be both unique and stable, for thermodynamic boundary data it was neither unique nor necessarily stable. Not only were there two classical solutions (if there were any) as was well known to York, but there was a whole free function-of-two-variables worth of gauge-inequivalent, nonclassical geometries, each of which contributed to the partition function as much as did a classical solution [11]. This observation was consistent with the known result that the zero-loop Helmholtz free energy for the Schwarzschild space-time contained substantial entropy for the black hole, and seemed to focus attention on the fact that for gravitational thermodynamics the state of a gravitating system was indeed determined by an ensemble, not by a single space-time. An immediate consequence of working with a gravitating system of finite extent [10] was, finally, settlement concerning the questioned existence of a canonical ensemble for black hole thermodynamics and, eventually, a compatible description of the associated density of states for spacelike three-geometries in the black hole space-time [12] (initially in the zero-loop approximation). In particular, York [10] was able to determine the condition for local thermodynamic stability in the canonical ensemble: with a temperature at which too small a mass occurred for the black hole within a large confining system, the ensemble would be thermodynamically unstable, while the mass in a stable system of infinite extent would itself be infinite (but well defined by a limiting process). Although this criterion was consistent with the condition for gravitational stability of the black hole [13], its relation to the dynamical stability of general relativity was perhaps not fully understood until the construction of the reduced action used in the path integral formulation [11]. The existence of a relation rested on the fact that, for suitable boundary data, the condition for global stability of the thermodynamic ensemble was determined by the condition for local stability of the gravitational field at a cla~sica! stationary point [141. In addition to providing a clear distinction between the conditions for local and global stability, the path integral formulation also allowed a more precisc description of the density of sta~es with remarkably sensible physical consequences [11]. A more detailed account of the formulation and description of gravitational thermodynamics [11] may be found in the accompanying article by York [15] (in these Proceedings) and in the various references cited above [10-14]. As a final remark it is perhaps worth observing that the path integral formulation
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mentioned here is similar, in many of its constructional features, with the formulation given for the "wave function" of the universe [16], but gravitational thermodynamics has drawn more directly on the semi-classical limit in order to yield definite tractable results.
3. Gravitation in thermal physics One of the strongest reasons that gravitation should be properly included in thermal physics, generally, stems from the unphysical nature of the thermodynamic limit. The problem is that, at non-zero temperature, even a massless field (e.g. photon radiation) has energy, and energy gravitates. But there is a further reason, which has been less apparent uptil now but is emphasized in the specific methods [10] first used by York. In the canonical ensemble, for example, the walls of a container of fixed area follow a non-geodesic path in the gravitational field of the matter-plus-radiation contained within the walls, and hence are accelerating; even if the surface pressure is held fixed rather than the surface area, the walls would then be free to move through a gravitational potential well, which will influence their motion. Gravitational effects are always present in thermodynamic systems. Thus, it is indeed fortunate, as York observed, that the boundary data for a thermodynamic ensemble are precisely boundary data for a well posed problem in general relativity. In principle, then, the description of a system in thermal equilibrium, say, necessarily requires treatment of the coupling of the gravitational field to the other matter fields, massive or massless, which determine the system. General relativity always provides the space-time geometry, which then supports the other matter fields, as well as the gravitational radiation which may propagate within the space-time. It is important to realize the general nature of York's observation. Its application is not confined only to the case of pure gravitation. Thus, in computations ~vith a number of collaborators, it has been possible to treat explicitly coupling to a static electric field [17], the field given by a 3-form [18] (which acts as a source for a cosmological constant) and, to some extent, matter such as might occur in a star [i9] given a phenomenological description in terms of an arbitrary equation of state. Coupling to the field strength of a 2-form potential also seems to give useful, non-trivial results, and coupling to a quantum thermal scalar field through a one-loop effective action for a finite system has been almost completed recently [20] and, as discussed below, is expected to lead to a minor modification of the condition for gravitational stability [24]. Even though analysis of the phenomenological system for a star is also incomplete, perhaps it will be fruitful, using it as an example, to illustrate
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a number of general features which emerge, independently of which field it is coupling to gravity, when gravitation is included in thermal physics. Sehutz has introduced an action principle [21], based on velocity potentials for a perfect fluid, which can model a star. The pressure p(m, s) is used as the Lagrangian density, it being expressed in terms of two variables (specific inertial mass and specific entropy per unit rest mass) through an zrbitrary equation of state. In coupling such matter to gravity one needs to deal separately with the density of total mass energy p, which is - T ° of the matter stress tensor, on the one hand, and the sum (p + p) on the other. Using the field equations for a static spherically symmetric system in this phenomenological model, the 4-volume integral of ( p + p) in the action can always be written in terms of at: integral of the proper 3-volume density of a quantity ~ derived from one of the velocity potentials, with no explicit reference to the lapse function (_g00)-~/2 anywhere on the interior of the 4-manifold with boundary. In the case of the constant density star, with a particular choice of the equation of state and solution to the field equations (which permitted the construction of a local Euler relation [22]: p + p = Ts + t~n), this integral can be written in terms of the e n t r o p y ai,u " ' number-density, though not generally. Nevertheless, it always depends functionally on the boundary data of the solution and contributes to the total entropy for the case of a system in thermal equilibrium. There remains the action for general relativity coupled to the total energy density.
Constraint equations As a completely general principle, the components T~° of the matter stress tensor are eliminated from the action by solving the constraint equations of general relativity on a family of spacelike hypersurfaces foliating the manifold [11] (just as constraints in any gauge theory coupled to gravity should be solved to eliminate gauge degrees of freedom from the action). In the case of a static s.tar, only the Too component occurs, and the result of satisfying the superHamiltonian constraint to eliminate it, and of including the boundary, terms for the gravitational part of the action for thermal equilibrium, is the quantity/3 Eh ( n ~ , , m i n g nO black holes within the stnr)~ where [3 = T - l is the inverse temperature maintained at some boundary surface of area A = 47rr~ placed within the star. The total energy of the system is given by
e = M (1-
(1 - 2ce)~/2),
o~
where c~ = GM/r o and
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442 ro
M = 4~r kl r2p(r) dr. 0
Note that E = M(I + a + ¢r2 + . . . ) when a is small. It is paramountly clear that gravitational effects on the energy for spherical equilibrium configurations are always of the same form, and can never be neglected when the quantity a = G M / r o is not small. It is also clear that the system fails to be physically reasonable whenever a ~> ½; and a more refined analysis [23] for any sensible equation of state shows that one must have a <~ 4~, independent of whatever matter is the source of the energy density. To go beyond this one needs to consider singular manifolds and/or manifolds with different global topology.
4. Quantum fields and back-reaction
Using the understanding gained from the discussion above, one could imagine trying to examine the back-reaction of some quantum thermal field which had been developed in a fixed, flat space-time background. However, when the system is near the state of gravitational collapse, non-linear effects are important and a perturbative analysis is inappropriate. Non-perturbative techniques are needed, of the type used in the path integral formulation for the black hole topological sector [1!], as indicated above. This would require evaluation of the action for the system in configurations far from those which occur at the classical stationary points and, as just mentioned, necessarily will require consideration of the field theory in other topological sectors of the gravitational field. Thus, such interesting problems are not of the nature of a back-reaction calculation, and currently await a comprehensive treatment for arbitrarily large thermodynamic systems. An alternative and more tractable problem concerns back-reaction on a non-trivial background, e.g. a black hole space-time, for which one could successfully examine, at the linearized level, modification of the conditions for thermodynamic stability. This has been partly carried out for a massless scalar field, and is currently under further investigation [24]. It appears that quantum changes in the criterion for local thermodynamic stability are small. In addition, the specification of an action principle for the back-reaction has almost been completed [20], based on an approximation to the one-loop effective action for a quantum scalar field in thermal equilibrium and including all boundary terms resulting from deali~ ". with a finite system. This may be useful for examining contributions given to the partition function by geometries far from the classical solutions of pure gravitation around black holes.
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Quantum field theories generally, and back-reaction calculations for them in particular, necessarily are posed originally with respect to a fixed background space-time. Yet even with pure gravitation it has been possible to show that equilibrium configurations at different temperatures may correspond to manifolds with different topology [10]. Quantum corrections due to temporal and spatial fluctuations in the geometry are expected to modify, but not substantially alter, this conclusion. However, these calculations are based on equilibrium configurations only and make no proposal for how the dynamical evolution might occur as the temperature is changed to effect the transition from one equilibrium topology to another. Nevertheless, genuine time dependent formalisms do exist for thermal quantum field theories [25, 26] and techniques from these could, in principle, be invoked to examine whether a system on the brink of instability will manifest an impending transition by a particular behavior either for the matter fields or even for gravitational fluctuations of the space-time geometry. Ia the following section, consequences for one of these methods [26] applied t¢ equilibrium configurations will be examined in relation to the understanding which has been gained from gravitational thermodynamics as discussed above.
5. Thermo field dynamics Before beginning here, it will be useful to stress that, as with quantum field theory in general, thermo field dynamics has been developed for quantum fields on a given space-time background. Thus~ in applying gravitational thermodynamics to the thermal physics of some quantum field coupled to gravity, one may imagine, indeed, that the space-time background has been fixed. In fact, it is usual to assume time independence for the geometry, as occurs for flat space-time. Now the general form of the super-Hamiltonian density is
where Y(M is the proper energy density of the matter and ~G iS the superHamiltonian of general relativity. There is a similar expression for the supermomentum density ~ but it will not be necessary to consider that explicitly here. As was stated above, the constraint equations of general relativity require ~ (and ~ ) to be identically zero on each leaf of a family of spacelike hypersurfaces foliating the manifold. For a time independent geometry the contribution from general relativity can thus be taken as constant. Even if there were gravitational fluctdations the coupling would be so weak ip gene;al that these could be ignored, unless thermal disturbances of the
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gravitational field were indeed substantial. By separating gravitational fluctuations from the background geometry, one might try to include such "gravitons" in a modified matter Hamiltonian describing the system, but one could never include the geometry itself this way. However, the static coupling which actually forms the background geometry can never be ignored, since it must always exactly balance the contribution from the matter energy density. Thus, whenever the gravitational contributions to the super-Hamiltonian constraint are effectively equivalent for any two independent labels of the spacelike hypersurfaces on which the constraints are imposed (i.e. for any two independent instants of "time"), instead of imposing the super-Hamiltonian directly on the geometry-plus-matter, one can equivalently require that the matter Hamiltonian at one time be exactly balanced by the matter Hamiltonian at any other representative time. In effect, one is demanding that the system itself be representative of the heat bath which is imposed gravitationally through the thermodynamic boundary data for the geometry. Therefore, one may use seM - se , = o ,
as is done in thermo field dynamics [26]. In fact it appears that this consequence alone should be sufficient to yield the full development of thermo field dynamics, as it is currently understood. The thermo field condition relating creation and annihilation operators in the system and its thermal bath, a ( t ) = ~*(t-ifl/2), arises also in the real time formalism mentioned above [25]. This condition relies on a discrete isometry of the Euclidean manifold used to represent the state of thermal equilibrium, and it would not in general hold if the background geometry were time dependent. In the particular case of the Kruskal manifold, which is the maximal analytic extension of the Schwarzschild space-time, this isometry can be proved using the Einstein field equations [23]. Although in a general path integral formulation the assumption of such an isometry would be too strong, it may apply for an effective theory in which all the gravitational degrees of freedom have been integrated away. The existence of such an isometry is an essential ingredient in the recent **~rl~,,-.-.~.~,~.-I~...~ r,~l,-Tl ~ .It._ .l__ u,,u,..,ata,,um~; tz.,j U ! t l l t ~ nature of mermo field dynamics on a fixed spacetime background, which has been obtained using techniques from quantum cosmology and, in particular, the formalism for the "wave function" of the universe. This approach refers to the different data residing on two (generally) disjoint surfaces in a global slicing of the periodic Euclidean manifold used for equilibrium thermodynamics (such as occurs for thermal fields in flat space). However, in the case of the black hole (and also Rindler space) the global manifold is simply connected. In this context, l~ usually is taken to correspond
B.F. Whiting I The role of gravitation in thermal physics
445
to matter fields having support only in the left-hand wedge wh~',e the quantum fields defined in the exterior region would be taken to have support only in the fight-hand wedge. Then the relation
would apply, with ~(/3) and ~'(/3) each being defined on a whole section of the global manifold. The fields corresponding to ~(/3) and ~(/3) have been referred to by Israel [28], in an application of the methods of thermo field dynamics, and global representations in a particular basis have been constructed explicitly [29]. Both sets of fields have positive norm on a whole spacelike section of the global manifold (and neither has support restricted to only the left or fight-hand wedge). Thus, the Hamiltonian for evolution forward in the global Kruskal time variable would be ~(/3) + ~(/3), not zero, in contrast to the right-hand side above. There, the evolution is partly forward and partly backward in the global time, even though neither the fields for ~ ( / 3 ) nor the fields for ~'(/3) have total support in either part (the fields for M and M do, however). No apparent conflict of this kind arises in deriving thermo field dynamics from gravitational thermodynamics, since 1VI need not even refer to fields anywhere in the actual space-time of the original field theory; reference to another appropriate element of the thermodynamic ensemble would be sufficient. In this way, time dependent geometries could also be included, with 1Vl(.p,x being representative of any other matter configuration which was compatible with the geometry at time t. Nevertheless, it would appear that only gravitational thermodynamics can properly describe the full contribution of the geometry (as has been done for the black hole topological sector) to, say, the partition function, through its initial inclusion in the path integral formulation.
6. Discussion
Gravity introduces, an additional scale into thermal physics. Thus a sphericaliy symmetric system of "~"",,,~,~,... ..... ,,;the,...-n~ M ~ncl ~nrface ..... area A = 4rrr?,.. must satisfy a condition for gravitational stability, e.g. GM/r,~ <~ 4~. On the other hand thermodynamics introduces an extra scale into general relativity. Thus. in the canonical ensemble, a black hole must satisfy the condition for thermodynamic stability [11], i.e. 3~h/8rrr o<~ ~~. The two scale~ of D avitational thermodynamics can become mixed (but not identified) as for example when thermodynamics determines the energy density of radiation for a finite system in thermal equilibrium. In fact, these two conditions listed above are ccm-
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B.F. Whiting / The role of gravitation in thermal physics
plementary, in that the former requires G(E)/ro<~ 3, where ( E ) is the expectation value of the energy in an equilibrium configuration, while the latter requires G ( E ) Iro ~> ], whe~'e here only the zero-loop value of ( E ) has been used for the black hole. Nevertheless, these conditions can be satisfied together for certain boundary data, since energy scales differently for radiation than for the gravitational field of an equilibrium black hole. The interplay between the different scales combined with the topological transformations which they herald, as seen in gravitational thermodynamics, represent rich new developments in thermal physics. In particular, there is an unexplored Planck scaleregime, related to early universe quantum cosmology, in which both the above conditions would be violated, and for which higher order corrections require investigation.
Acknowledgments I have benefited from conversations with N.J. Papastamatiou and R. Laflamme. I especially thank J.W. York for his collaboration on part of this work, and I am grateful to my various colleagues at Chapel Hill for the opportunity to refer to their work prior to publication. This research has been supported by the National Science Foundation.
References [1] J.W. York, Black holes and partition functioas, in: Proc. Osgood Hill Conf. on Conceptual Problems in Quantum Gravity, to be pu01ished. [2] B.F. Whiting, Non-classical geometries in gravitational thermodynamics, in: Proc. fifth Marcel Grassman meeting, held in Perth, Western Australia, August 8-13, 1988, D.G. Blair and M.J. Buckingham, eds. (Cambridge Univ. Press, Cambridge), to be published. [3] S.W. Hawking, Commun. Math. Phys. 43 (1975) 199. 14] I.D. Bekenstein, Phys. Rev. D 7 (1973) 2333. [5] M.S. Fawcett and B.F. Whiting, Spontaneous symmetry breaking near a black hole, in: The Quantum Structure of Space and Time, M.J. Duff and C.J. Isham, eds. (Cambridge Univ. Press, Cambridge, 19,'42). [6] D.N. Page, Phys. Rev. D 25 (1982) 1499. [7] K.W. Howard, Phys. Rev. D 30 (1984) 2532. [8] M.R. Bxown, A.C. Ottcw'ill and D.N. Page, Phys. Rev. D 33 (1986) 2840. [9] J.W. York, Phys. Rev. D 31 (1985) 775; K.W. Howard, B. Jensen and B.F. Whiting, Back-reaction for a lhermal scalar field in anti de Sitter space-time, Meudon preprint (1985). [l(I] J.W. York, Phys. Rev. D 33 (1986) 2(102. [ill B.F. Whiting and J.W. York, Phys. Rev. Lett. 61 (1988) 1336. [12] H.W. Braden, B.F. Whiting and J.W. York, Phys. Rev. D 36 (1987) 3614. [13] B.F. Whiting, Gravitational constraints and thermodynami~ ~tability, to be published (1988).
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1141 B.F. Whiting, Black holcs and thcrmodynamics, in: Proc. ninth lAMP ct,:i~rc~,,,, held in Swansea, Wales, July 12-27, 1988, B. Simon, I.M. Davtes and A. 'lrurpan, eds. (Adam Hilger, Bristol), to be published. I151 J.W. York, Physica A 158 (1989) 425, these Proceedings. [161 J.B. Hartle and S.W. Hawking, Phys. Rev. D 28 (1983) 2960. [171 H.W. Braden, B.F. Whiting and J.W. York, Charged black holes in the grand canonical ensemble, to be published (1988). [18] J.D. Brown, B.F. Whiting+ and J.W. York, Antisymmetric tensor source field in gravitational thermodynamics, to be published (1988). [191 J.D. Brown, G.L. Comer and B.F. Whiting, unpublished notes, Chapel Hill (1988). 1201 J. Melmed, unpublished notes, Chapel Hill (1988). [211 B.F. Schutz, Phys. Rev. D 2 (1970) 2762. I221 B.F. Whiting, unpublished notes, Chapel Hill (1988). [23] R.M. Wald, General Relativity (Chicago Univ. Press, Chicago, 19J:~4). [24] G.L. Comer and J.W. York, unpublished notes+ Chapel Hill (19~S). [251 A.J. Niemi and G.W. Semenoff, Ann. Phys. 152 (1984) 105; Nuci. Phys. B231J (19~4) 1,~1. [26] Y. Takahashi and H. Umezawa, Collective Phenomena 2 (1975) 55. [271 R. Laflamme, Geometry and thermo felds, DAMTP preprint (1988); See also Physica A 158 (1989) 58, these Proceedings. [28] W. Israel, Thermo-field dynamics of black holes, Phys. Lett. A 57 (1976) 107. [29] N. Sanchez and B.F. Whiting, Nuci. Phys. B283 (1987) 605.