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.

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