Schwarzschild horizon and the gravitational redshift formula

TL;DR

This paper investigates the conditions under which the gravitational redshift formula applies exactly in Schwarzschild space-time, revealing that it holds for short wavelengths but can differ for long waves due to backscattering effects.

Contribution

It provides an exact formulation of gravitational redshift in Schwarzschild space-time, clarifying when the standard formula is valid and how backscattering influences energy flux scaling.

Findings

Short wavelength radiation obeys the standard redshift formula.

Long wavelength radiation can deviate from the standard redshift due to backscattering.

Redshift dependence on frequency varies for different wave regimes.

Abstract

The gravitational redshift formula is usually derived in the geometric optics approximation. In this note we consider an exact formulation of the problem in the Schwarzschild space-time, with the intention to clarify under what conditions this redshift law is valid. It is shown that in the case of shocks the radial component of the Poynting vector can scale according to the redshift formula, under a suitable condition. If that condition is not satisfied, then the effect of the backscattering can lead to significant modifications. The obtained results imply that the energy flux of the short wavelength radiation obeys the standard gravitational redshift formula while the energy flux of long waves can scale differently, with redshifts being dependent on the frequency.