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

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

2007-07-21

arxiv.org/abs/0707.3222v3

Class.Quant.Grav.24:5717-5719,2007

Astronomy and Astrophysics

Astrophysics

General Relativity and Quantum Cosmology

3 pages; v2: references, and discussion of Born-Infeld solutions and O(3) and string hedgehogs added; 4 pages; v3: slight edit

Scientific paper

10.1088/0264-9381/24/22/N02

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.

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