A note on the thermodynamics of gravitational radiation

It is shown that linearized gravitational radiation confined in a cavity can achieve thermal equilibrium if the mean density of the radiation and the size of the cavity satisfy certain constraints. PACS numbers: 04.30, 05.70.−a, 05.40.Jc Is it possible, in principle, to confine gravitational radiation in a box, for a time long enough for it to achieve thermal equilibrium? This issue has been investigated earlier by various authors [1–3], not all of whom have arrived at the same conclusion. The purpose of this paper is to pursue a particular line of thought which does not seem to have been looked at in earlier work. We argue that it is possible to thermalize gravitational radiation, starting from a non-equilibrium configuration, so long as certain constraints (discussed below) are satisfied. Let us begin by summarizing the arguments given by Smolin [1]. He showed, by considering various mechanisms of absorption of radiation by matter, that no realistic material is an efficient absorber of gravitational radiation, except over a narrow band of frequencies. These mechanisms include absorption of classical gravitational radiation by classical matter, ionization of atoms by gravitons and phonon excitation. He concluded that a state of thermal equilibrium between radiation and matter cannot be reached in a finite time. Garfinkle and Wald [2] presented a counter example to the result by using a shell of charged matter balanced just outside its Schwarzschild radius. However, Dell [3] later argued that such a shell is unstable in the presence of electromagnetic radiation. Whatever the nature of the confining cavity, the absorption coefficient f will always be smaller than 1. If the box is made thick enough to raise f to 1, it undergoes gravitational collapse. However, it is not absolutely essential, for the purpose of attaining equilibrium, to have f close to unity. For any cavity, it is possible to define a leakage time scale tL over which, say, a fraction 1/e of the initial radiation escapes. Also, for an initial configuration ρ(ν)—the energy density of gravitational radiation in the cavity—an equilibrium time scale tE can be defined, 0264-9381/03/204419+05$30.00 © 2003 IOP Publishing Ltd Printed in the UK 4419 4420 T Padmanabhan and T P Singh over which the radiation may achieve equilibrium. What needs to be settled is whether tE can be smaller than tL, for a general initial configuration ρ(ν). We argue here that this is possible, and tE < tL need not imply gravitational collapse of the cavity or of the radiation inside it. Moreover, we have no need to concern ourselves with any specific mechanism for confining radiation. Our argument is analogous to the standard discussion of Brownian motion of a molecule in a fluid, where one shows that as a consequence of the random collisions and viscous drag, the molecule attains a Maxwellian velocity distribution. A variant of this argument was used by Einstein [4] to derive the Planckian distribution for electromagnetic radiation in equilibrium in a cavity. The argument went as follows: He assumed that the radiation was in equilibrium with a molecule (say, with two energy levels). The molecule absorbs and re-emits at a characteristic frequency ν0. The momentum transfers to the molecule take place through random collisions with the quanta of radiation and through the systematic drag force it experiences because it sees a Doppler-shifted radiation while in motion. Einstein showed that if the molecule has a Maxwellian velocity distribution, then the exchange of momenta with radiation preserves this distribution only if the energy density of the radiation obeys the Planck law. In his paper, Einstein concerned himself with the equilibrium situation. The discussion of momentum transfers to the molecule assumes greater significance in a study of approach to equilibrium. Now, there is a systematic evolution of the rms velocity of the molecule, which may be approximately described by the Langevin equation