Black hole kinematics: The “in”-vacuum energy density and flux for different observers

Abstract

We have investigated the local invariant scalar observables---energy density and flux---which explicitly depend on the kinematics of the concerned observers in the thin null shell gravitational collapse geometry. The use of globally defined null coordinates allows for the definition of a unique in-vacuum for the scalar field propagating in this background. Computing the stress-energy tensor for this scalar field, we work out the energy density and flux for the static observers outside the horizon and then consider the radially in-falling observers who fall in from some specified initial radius all the way through the horizon and inside to the eventual singularity. Our results confirm the thermal Tolman-shifted energy density and fluxes for the static observers which diverge at the horizon. For the in-falling observer starting from far off, both the quantities---energy density and flux---at the horizon crossing are regular and finite. For example, the flux at the horizon for the in-falling observer from infinity is approximately 24 times the flux for the observer at infinity. Compared with the static observers in the near-horizon region, this is quite small. Both the quantities grow as the in-fall progresses inside the horizon and diverge at the singularity.