Null Surfaces in Static Space-times

In this paper I consider surfaces in a space-time with a Killing vector $ξ^α$ 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 $ξ_αξ^α\rightarrow 0$ as the surface is approached from the static region. This implies that, in a coordinate system adapted to $ξ$, 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 $ξ=$ constant has a normal vector whose norm is proportional to $ξ_αξ^α$. This lead to the conclusion that surfaces with $ξ_αξ^α=0$ 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.

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