Red-shifts near black holes

TL;DR

This paper derives a differential equation describing red-shifts near non-stationary black holes, revealing universal behaviors and extending concepts like surface gravity, with implications for Hawking radiation predictions.

Contribution

It introduces a simple ODE model for red-shifts in dynamic black hole spacetimes and explores its implications for black hole thermodynamics and Hawking radiation.

Findings

Asymptotic red-shift behavior near horizons resembles phase transition universality.

The validity of the Unruh vacuum is explained through this universality.

Hawking radiation flux is modified in non-stationary black hole scenarios.

Abstract

A simple ordinary differential equation is derived governing the red-shifts of wave-fronts propagating through a non-stationary spherically symmetric space-time. Approach to an event horizon corresponds to approach to a fixed point; in general, the phase portrait of the equation illuminates the qualitative features of the geometry. In particular, the asymptotics of the red-shift as a horizon is approached, a critical ingredient of Hawking's prediction of radiation from black holes, are easily brought out. This asympotic behavior has elements in common with the universal behavior near phase transitions in statistical physics. The validity of the Unruh vacuum for the Hawking process can be understood in terms of this universality. The concept of surface gravity is extended to to non-stationary spherically symmetric black holes. Finally, it is shown that in the non-stationary case, Hawking's predicted flux of radiation from a black hole would be modified.