Asymptotic behavior of solutions and spectrum of states in the quantum scalar field theory in the Schwarzschild spacetime

TL;DR

This paper analyzes the asymptotic behavior and spectral properties of a quantized scalar field in Schwarzschild spacetime, revealing a split in the spectrum related to the spacetime's topology and black hole proximity.

Contribution

It introduces a new spectral decomposition for scalar fields in Schwarzschild spacetime, highlighting the influence of topology on state behavior near the black hole.

Findings

Spectrum splits into two parts for energies above the field mass

States far from the black hole resemble plane waves

States near the horizon decay away from the black hole

Abstract

In this paper, the study of canonical quantization of a free real massive scalar field in the Schwarzschild spacetime is continued. The normalization constants for the eigenfunctions of the corresponding radial equation are calculated, providing the necessary coefficients for the doubly degenerate scatteringlike states that are used in the expansion of the quantum field. It is shown that one can pass to a new type of states such that the spectrum of states with energies larger than the mass of the field splits into two parts. The first part consists of states that resemble properly normalized plane waves far away from the black hole, so they just describe the theory for an observer located in that area. The second part consists of states that live relatively close to the horizon and whose wave functions decrease when one goes away from the black hole. The appearance of the second part of the spectrum, which follows from the initial degeneracy of the scatteringlike states, is a consequence of the topological structure of the Schwarzschild spacetime.