When is g_{tt} g_{rr} = -1?
Abstract
The Schwarzschild metric, its Reissner-Nordstrom-de Sitter generalizations to higher dimensions, and some further generalizations all share the feature that g_{tt} g_{rr}=-1 in Schwarzschild-like coordinates. In this pedagogical note we trace this feature to the condition that the Ricci tensor (and stress-energy tensor in a solution to Einstein's equation) has vanishing radial null-null component, i.e. is proportional to the metric in the t-r subspace. We also show this condition holds if and only if the area-radius coordinate is an affine parameter on the radial null geodesics.
Cite
@article{arxiv.0707.3222,
title = {When is g_{tt} g_{rr} = -1?},
author = {Ted Jacobson},
journal= {arXiv preprint arXiv:0707.3222},
year = {2008}
}
Comments
3 pages; v2: references, and discussion of Born-Infeld solutions and O(3) and string hedgehogs added; 4 pages; v3: slight editing; comment added that condition implies radial pressure is negative of energy density; version to appear in CQG