Reflection and transmission at the apparent horizon during gravitational collapse

Abstract

We examine the wave-functionals describing the collapse of a self-gravitating dust ball in an exact quantization of the gravity-dust system. We show that ingoing (collapsing) dust shell modes outside the apparent horizon must necessarily be accompanied by outgoing modes inside the apparent horizon, whose amplitude is suppressed by the square root of the Boltzmann factor at the Hawking temperature. Likewise, ingoing modes in the interior must be accompanied by outgoing modes in the exterior, again with an amplitude suppressed by the same factor. A suitable superposition of the two solutions is necessary to conserve the dust probability flux across the apparent horizon, thus each region contains both ingoing and outgoing dust modes. If one restricts oneself to considering only the modes outside the apparent horizon then one should think of the apparent horizon as a partial reflector, the probability for a shell to reflect being given by the Boltzmann factor at the Hawking temperature determined by the mass contained within it. However, if one considers the entire wave function, the outgoing wave in the exterior is seen to be the transmission through the horizon of the interior outgoing wave that accompanies the collapsing shells. This transmission could allow information from the interior to be transferred to the exterior.