Abstract
We describe a unitary scattering process, as observed from spatial infinity, of massless scalar particles on an asymptotically flat Schwarzschild black hole background. In order to do so, we split the problem in two different regimes governing the dynamics of the scattering process. The first describes the evolution of the modes in the region away from the horizon and can be analysed in terms of the effective Regge-Wheeler potential. In the near horizon region, where the Regge-Wheeler potential becomes insignificant, the WKB geometric optics approximation of Hawking’s is replaced by the near-horizon gravitational scattering matrix that captures non-perturbative soft graviton exchanges near the horizon. We perform an appropriate matching for the scattering solutions of these two dynamical problems and compute the resulting Bogoliubov relations, that combines both dynamics. This allows us to formulate an S-matrix for the scattering process that is manifestly unitary. We discuss the analogue of the (quasi)-normal modes in this setup and the emergence of gravitational echoes that follow an original burst of radiation as the excited black hole relaxes to equilibrium.
Highlights
Since such degrees of freedom predominantly affect ultraviolet physics, and the asymptotic observer may choose to perform low energy experiments exclusively, why are they relevant in the first place?
We present a construction of the complete scattering matrix that incorporates both the effects alluded to above, that are associated to the near horizon gravitational interactions and to the Regge-Wheeler gravitational potential
This is done by imposing appropriate matching conditions for the modes entering the Regge-Wheeler potential and those arising in the near horizon gravitational dynamics
Summary
To set the stage for an analysis of the S-matrix, we first review Hawking’s original approach [6, 7] which suggested that black holes radiate in a thermal fashion. The WKB approximation employed, and the resulting Bogoliubov transformations that relate the modes on past null infinity I− to those on I+, hold only long after the black hole has formed Such an approach cannot model the complete collapse and evaporation process. It is that this calculation does not take into account the gravitational backreaction of ingoing modes on the outgoing ones and on the geometry itself This is a substantial effect from the point of view of the late time asymptotic external observer due to the blueshift of the modes as their evolution is traced backwards in time towards the horizon. In contrast to Hawking’s WKB result (2.2) that gave rise to a Bogoliubov map for asymptotic states, the amplitude (2.5) provides for a near horizon mode relation that can be used to describe the scattering of a massless scalar field.
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