2016/12/17 by Dan N. Vollick · 1 citation
Physics and Astronomy · #gr-qc
paper · pdf · doi:10.1140/epjp/i2015-15157-6
published as European Physics Journal Plus 130, 157 (2015)
arxiv created 2016/12/17 · arxiv updated 2016/12/20
In this paper I consider surfaces in a space-time with a Killing vector ξα 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 ξαξα→ 0 as the surface is approached from the static region. This implies that, in a coordinate system adapted to ξ, surfaces with gtt=0 are null. In spherically symmetric space-times the condition grr=0 instead of gtt=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 ξαξα. This lead to the conclusion that surfaces with ξαξα=0 are null. However, the proportionality factor generally diverges when gtt=0, leading to a different condition for the norm to be null. In static spherically symmetric space-times this condition gives grr=0, not gtt=0. The problem with the condition grr=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 gtt=0. I also examine the condition required for the surface to be nonsingular.