Null surfaces in static space-times

Abstract

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 _α ξ ^α → 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 = const has a normal vector whose norm is proportional to ξ _α ξ ^α. This lead to the conclusion that surfaces with ξ _α ξ ^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.