Cosmological and black hole horizon fluctuations

The quantum fluctuations of horizons in Robertson-Walker universes and in Schwarzschild spacetime are discussed. The source of the metric fluctuations is taken to be quantum linear perturbations of the gravitational field. Light cone fluctuations arise when the retarded Green’s function for a massless field is averaged over these metric fluctuations. This averaging replaces the $\delta$ function on the classical light cone with a Gaussian function, the width of which is a measure of the scale of the light cone fluctuations. Horizon fluctuations are taken to be measured in the frame of a geodesic observer falling through the horizon. In the case of an expanding universe, this is a comoving observer either entering or leaving the horizon of another observer. In the black hole case, we take this observer to be one who falls freely from rest at infinity. We find that cosmological horizon fluctuations are typically characterized by the Planck length. However, black hole horizon fluctuations in this model are much smaller than Planck dimensions for black holes whose mass exceeds the Planck mass. Furthermore, we find black hole horizon fluctuations which are sufficiently small as not to invalidate the semiclassical derivation of the Hawking process.