Hawking radiation in the Swiss-cheese universe

The Hawking radiation forms the essential basis of black-hole thermodynamics. Black-hole thermodynamics denotes a good correspondence between black-hole kinematics and the laws of ordinary thermodynamics, but has so far been considered only in an asymptotically flat case. Does such correspondence rely strongly on the feature of gravity vanishing at infinity? In order to resolve this question, extending the Hawking radiation to a case with a dynamical boundary condition like an expanding universe should be considered. Therefore, the Hawking radiation in an expanding universe is discussed in this paper. As a concrete model of a black hole in an expanding universe, we use the 'Swiss-cheese' universe which is a spacetime including a Schwarzschild black hole in the Friedmann-Robertson-Walker universe. Further, for simplicity, our calculation is performed in two dimensions. The resultant spectrum of the Hawking radiation measured by a comoving observer is generally different from a thermal one. We find that the qualitative behaviour of the non-thermal spectrum is of dumping oscillation as a function of the frequency measured by the observer, and that the intensity of the Hawking radiation is enhanced by the presence of a cosmological expansion. It is appropriate to say that a black hole with an asymptotically flat boundary condition stays in a lowest energy thermal equilibrium state, and that once a black hole is put into an expanding universe, it is excited to a non-equilibrium state and emits its mass energy with stronger intensity than a thermal one.