When is g_{tt} g_{rr} = -1?

TL;DR

This paper explains why in Schwarzschild-like coordinates, the metric components satisfy g_{tt} g_{rr} = -1, linking it to the properties of the Ricci tensor and the affine parameterization of radial null geodesics.

Contribution

It establishes a fundamental geometric condition that characterizes when g_{tt} g_{rr} = -1 in spherically symmetric spacetimes.

Findings

g_{tt} g_{rr} = -1 holds when Ricci tensor's radial null-null component vanishes.

This condition is equivalent to the area-radius coordinate being an affine parameter on radial null geodesics.

The result applies to Schwarzschild, Reissner-Nordstrom-de Sitter, and related metrics.

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.