Geometry of Vaidya spacetimes

TL;DR

This paper analyzes the geometric structure of Vaidya spacetimes with decreasing mass, focusing on null geodesics and singularities, and establishes a relation between null singularity strength and mass decay.

Contribution

It provides a detailed analysis of null geodesics and singularities in Vaidya spacetimes, including an explicit formula linking null singularity strength to mass decay rate.

Findings

Existence of an asymptotic light-like singularity in complete evaporation cases.

Null singularity strength is independent of mass decay rate.

Explicit formula relating null singularity to mass function behavior.

Abstract

We investigate the geometrical structure of Vaidya's spacetime in the case of a white hole with decreasing mass, stabilising to a black hole in finite or infinite time or evaporating completely. Our approach relies on a detailed analysis of the ordinary differential equation describing the incoming principal null geodesics, among which are the generators of the past horizon. We devote special attention to the case of a complete evaporation in infinite time and establish the existence of an asymptotic light-like singularity of the conformal curvature, touching both the past space-like singularity and future time-like infinity. This singularity is present independently of the decay rate of the mass. We derive an explicit formula that relates directly the strength of this null singularity to the asymptotic behaviour of the mass function.