When is gtt grr = −1?

The Schwarzschild metric, its Reissner-Nordstrom-de Sitter generalizations to higher dimensions, and some further generalizations all share the feature that gtt grr = −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. A notable feature of the Schwarzschild solution to Einstein’s equation, which generalizes to a rather wide class of static solutions of the form ds = −f(r) dt + f(r) dr + r hij(x)dxdx, (1) is the fact that the metric component grr is the reciprocal of −gtt. The purpose of this pedagogical note is to point out that this commonly occuring feature arises if and only if the radial null-null components of the Ricci tensor (which are equal) vanish; equivalently, if the restriction of the Ricci tensor to the t-r subspace is proportional to gμν . When the Einstein equation is satisfied, this is a condition on the stress-energy tensor. An equivalent condition is that the coordinate r is an affine parameter on the radial null geodesics. Oddly, the following elementary remarks are not easy to find in the literature, so far as I have been able to determine. However, subsequent to posting the first version of this paper I learned that in Ref. [1] it was noted explicitly that the metric form (1) follows from this condition on the stress-energy tensor. E-mail: [email protected]