THE THERMODYNAMICS O F BLACK HOLES

D. W . Sciama

Department of A strophj~sics Oxjord University Oxford. England

I N T R O D I J C T I O N

It has now become clear that we are attending the birth of a major new un i - fication ofphysics, i n which general relativity, quantum tield theory, and thermo- dynamics are all equally involved. It is my task to emphasize the thermodynamic aspects of this unification and. since thermodynamics is easier to understand than quantum field theory, I shall endeavor to do so i n terms of the simplest possible physical pictures. By so doing I shall neglect many deep and subtle aspects of the problem, and for this I apologize. My excuse is that i t is often helpful to under- stand the basic physics first. and only then to go on to more sophisticated treat men ts.

U K I F O R M ACCELERATION A N D THE

~ L ~ C T U A T I ~ N - D I S S I ~ A T I ~ ~ T H E O R E M

Our starting-point is the fact emphasized by Rindler' that some aspects of black holes are mimicked by uniformly accelerated motions in Minkowski space- time. Now we know from the work of Mould' that a radiation detector acceler- ating uniformly through a classical Coulomb field can absorb radiant energy and so be said to have detected a radiation field. Similarly, when a detector accelerates uniformly through the zero-point quantum fluctuations of the vacuum electro- magnetic field, it again absorbs radiant energy.3~6 A simple picture of this process can he derived from the discussion of radiative transitions given by Fain and Khanin.' A detector in its ground state absorbs energy from each mode of the zero-point fluctuations of'the field a t a rate proportional to the noise power i n this mode at the detector. I t also radiates energy i n this mode at a rate proportional to the noise power associated wi th the rero-point fluctuations of its own dipole moment. I f the detector were inertial, these two noise powers would be equal (in, per mode). and there would be no net transfer of energy between the de- tector and the field.

I f the detector were accelerated, we could compute the noise power of the field by using the Wiener-Khinchin theorem, which states that this noise power is the Fourier transform of the autocorrelation function of the field. evaluated along the world-line of the detector. So long as the detector is robust enough to with- stand the dynamic efrects of the acceleration. the o n l y change from the previous calculation would he that the noise power is evaluated for a dilyerent (namely, nongeodesic) world-line. It turns out that the noise power would be increased, and so the detector would gain energy and become excited.

As a result of this excitation, the rate a t which the detector emits energy would 161

162 A n n a l s New York Academy of Sciences

be increased by its spontaneous decay, and i f the acceleration were uniform we would expect the system to attain a steady-state, in which the heating and cooling rates would be in balance. We could describe this situation by saying that par t of the work done by the force inducing the acceleration is being steadily dis- sipated into heat. Moreover, since the heating rate is proportional t o the noise power in the field, we would have here an example of the fluctuation-dissipa- tio n the or em .8.9

While we thus make natural contact with ideas based o n irreversible thermo- dynamics, nothing that has been said so far would have prepared us for the most remarkable discovery of Davies4 and UnruhsV6 (inspired by the even more remark- able work of Hawking'"' ' '), that the noise power of the vacuum field evaluated along the world-line of'a unijorni1.v accelerating bodv includes a thermal Planck spectrurn. with the temperature T being related to the acceleration a by

( 1 )

This result implies that when a steady state is set up, the detector would also come into thermal equilibrium a t the temperature T . A lower limit on the time it would take for this equilibrium to be set u p is given by the requirement that the detector should interact with several oscillation cycles of the electromagnetic field a t fre- quencies where the Planck spectrum has its maximum. It then follows immediately from Equation I that in this t ime interval the accelerated detector would be boosted to a velocity close to that of light, a s measured i n those inertial frames For which the initial velocity of the detector is subrelativistic.

The essential reason for this thermal result seems to lie in a combination of the stationary property of uniform acceleration and the fact that asymptotically the velocity in such a motion tends t o the velocity of light. This reason can be ele- gantly reexpressed by saying that in the rest frame of the detector there is a sta- tionary event horizon.'*

The question now arises, is i t possible to test experimentally the prediction that a system moving with uniform acceleration 01 is raised to a temperature T =

f fa/Z*ck? This temperature is extremely small for most experimentally realizable accelerations, a s U n r ~ h ~ . ~ has pointed out . For instance, the acceleration required to produce a temperature of 3°K would be about 3 x loz4 cm s e c 2 . There is, however, a t least one situation in physics where even larger accelerations are achieved and that is during a high-energy collision between two elementary particles. This was realized by de Witt, who was worried that the resulting fireball might be energetic enough to close space gravitationally around itself. Of course, the deceleration occurring during a high-energy collision would not be exactly uniform. Nevertheless, I conjecture that, so long as the velocity change involved is of order c, it would be a good approximation to suppose that a fireball would be formed whose state of excitation is not far from being exactly thermal. I f the dis- tance over which the particles are slowed from relativistic to subrelativistic ve- locities is determined by strong interactions mediated by virtual n-mesons, then a simple estimate shows that the fireball temperature 7'' would be given by

+a k T = -. 2nc

Sciama: Thermodynamics of Black Holes I63

where M , is the mass of a *-meson. There is some experimental evidence that a fireball of this temperature is indeed produced, but of course more conventional physical arguments have been given for expecting such a result. An attempt is being made by Cade i to test the conjecture and to see whether, i f i t is correct, the Rindler-de Witt fireball can be distinguished from its more conventional brethren.

THL Tiit K M O D Y N A M I C \ oi BI A C K H o i t\

There is one feature of o u r discussion i n the last section that was not im- portant there but becomes so in the context of black holes. According to quantum field theory, the zero-point fluctuations of the electromagnetic field can be re- garded as a series of creations and annihilations of pairs of virtual photons. Since an accelerating system in its ground state would absorb virtual photons faster than it would emit them, some of the unabsorbed virtual photons would have n o part- ners with which to annihilate, and so would become real. The energy source for these real photons would then be the work done by the force that is accelerating the system

I n a similar way, according to Hawking,'"." a black hole can absorb virtual photons, leaving their partners free to propagate i n some cases to arbitrarily great distances from the black hole. The main difference is that the energy needed for this process to operate is provided by the mass o f the black hole itself, since the captured photons move on to negative total energy orbits ( that is, orbits [or which the gravitational binding exceeds the local kinetic energy).

As is now very well known, Hawking'" showed that such a photon f lux should actually be produced when a star collapses to the black hole condition. As all transients die ou t , the flux would approach that produced by a hot body of tem- perature Tgiven by

-kh 2ac '

k T = -

where K is the surface gravity" of the black hole.

hole by the relationI3 I s

It is also well known that one can define an entropy S for a stationary black

S = L A k 4-K

where A is the surface area of the event horizon. With these definitions of their temperature and entropy, black holes would obey the zeroth. first, and second laws of thermodynamics." l 6 Even the third law would be obeyed classically i f the Cosmic Censorship hypothesis were correct ( that is. roughly speaking, if the singularities resulting from the gravitational collapse of a realistic, but classical. star would always be hidden from a distant observer by an event horizon)."

I have given elsewhere'' an elementary discussion of this result that black holes obey the usual laws of thermodynamics, and I d o not want to repeat that discus- sion here. Instead, I will mention that one can also show that black holes obey the laws of irreversible thermodynamics, which relate the rates of irreversible (dissipa-

164 Annals New York Academy of Sciences

tive) processes to the microscopic (fluctuation) properties of the systems con- cerned. The motivation for this work was the observation that when ordered energy is absorbed by a black hole and then reemitted a s Hawking radiation, it would emerge in a completely random state.18 *’ A black hole is thus a perfect dissipator. It is natural, then, to enquire whether this process satisfies a fluctua- tion-dissipation theorem, with the fluctuations arising, as in the case of uniform acceleration. entirely from the zero-point quantum fluctuations of the fields con- cerned. In particular, one would want to show this for the quantum fluctuations of the gravitational field itself, occurring jus t outside the event horizon.

Recently Candelas and 1’’ succeeded in showing that such a fluctuation- dissipation theorem is indeed satisfied by the quantum fluctuations of the Weyl conformal curvature tensor just outside the event horizon of a Kerr black hole. One can thus understand the Hawking dissipation in terms of standard quantum field theory and general relativity, but only if one takes into account the quantum deviations of the geometry of the black hole from the stationary, classical, Kerr con figuration ,

I find it encouraging that one can understand so much of the apparently bizarre Hawking radiation phenomenon in terms of “ordinary” physics. It tempts me to strengthen and extend my opening remarks by asserting that before in- venting radically new physics, one should attempt to unify further general rela- tivity, quantum field theory, and irreversible thermodynamics. I believe that important new insights would be gained by doing this, and that it would help us to take a further step towards the solution of that apparently intractable problem the complete and consistent quantization of the gravitational field.

I a m grateful to many colleagues for helpful discussions, to my collaborator M r . P. Candelas, and to Professor B. de Witt and Dr. A . Cadez for helping m e to realize the possible relevance of the uniform acceleration approximation for the high-energy collisions of elementary particles.

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DISCUSSION

DR. W . KUNDT (University of’ Hamburg): H o w soon after formation of the black hole d o y o u want to apply Hawking’s equilibrium thermodynamics; after some Schwarzschild times, or after some evaporation times‘?

D R . SCIAMA: I would expect a collapsing system to settle down t o its asymptotic form after a few Schwarrschild times since this is the timescale for the gravitational field to settle down t o this asymptotic value as shown by Price’s calculations.