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2007

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

Jacobson, Ted

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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.

When is g_{tt} g_{rr} = -1? · Around

Built on

M. Demianski, “Static electromagnetic geon,” Found. Phys. 16
1986
Earlier work this paper cites.
M. Barriola and A. Vilenkin, “Gravitational Field of a Global Monopole,” Phys. Rev. Lett. 63
1989
Earlier work this paper cites.

Similar

E. I. Guendelman and A. Rabinowitz, “The Gravitational field of a hedgehog and the evolution of vacuum bubbles,” Phys. Rev. D 44
1991
Cited alongside, same era.
E. Guendelman and A. Rabinowitz, “Linearity, nonself interacting spherically symmetric gravitational fields, the sphereland equivalence principle and Hamiltonian bubbles,” Gen. Rel. Grav. 28
1996
Cited alongside, same era.

Then

D. Birmingham, “Topological black holes in anti-de Sitter space,” Class. Quant. Grav. 16
1999
Later among the works it cites.

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