Null Surfaces in Static Space-times

TL;DR

This paper clarifies the criteria for null surfaces in static space-times, demonstrating that the condition $g_{tt}=0$ correctly identifies null surfaces, correcting previous misconceptions involving $g^{rr}=0$ and coordinate singularities.

Contribution

The paper critically analyzes and corrects the criteria for null surfaces in static space-times, emphasizing the proper use of coordinate systems and induced metrics.

Findings

Surfaces with $g_{tt}=0$ are null in static space-times.

The condition $g^{rr}=0$ is not the correct criterion for null surfaces.

Coordinate singularities can mislead the identification of null surfaces.

Abstract

In this paper I consider surfaces in a space-time with a Killing vector $\xi^{\alpha}$ that is time-like and hypersurface orthogonal on one side of the surface. The Killing vector may be either time-like or space-like on the other side of the surface. It has been argued that the surface is null if $\xi_{\alpha}\xi^{\alpha}\rightarrow 0$ as the surface is approached from the static region. This implies that, in a coordinate system adapted to $\xi$, surfaces with $g_{tt}=0$ are null. In spherically symmetric space-times the condition $g^{rr}=0$ instead of $g_{tt}=0$ is sometimes used to locate null surfaces. In this paper I examine the arguments that lead to these two different criteria and show that both arguments are incorrect. A surface $\xi=$ constant has a normal vector whose norm is proportional to $\xi_{\alpha}\xi^{\alpha}$. This lead to the conclusion that surfaces with $\xi_{\alpha}\xi^{\alpha}=0$ are null. However, the proportionality factor generally diverges when $g_{tt}=0$, leading to a different condition for the norm to be null. In static spherically symmetric space-times this condition gives $g^{rr}=0$, not $g_{tt}=0$. The problem with the condition $g^{rr}=0$ is that the coordinate system is singular on the surface. One can either use a nonsingular coordinate system or examine the induced metric on the surface to determine if it is null. By using these approaches it is shown that the correct criteria is $g_{tt}=0$. I also examine the condition required for the surface to be nonsingular.