Spatial metrics which are static in many ways
Given a spatial (i.e. 3-dimensional and positive definite) metric \(h_{ab}(x^1,x^2,x^3) dx^a dx^b\), one may look for a real-valued function \(N(x^1,x^2,x^3)\) such that the 4-dimensional static Lorentzian metric \(-N(x^1,x^2,x^3)^2 dt^2+ h_{ab}(x^1,x^2,x^3) dx^a dx^b\) satisfies Einstein's vacuum field equation, possibly with a cosmological constant. In this paper the author determines all cases in which this problem has two different solutions \(N=N_1\) and \(N=N_2\). The resulting Lorentzian metrics turn out to be all of Petrov type \(D\). The locally rotationally symmetric Kasner metric is an example.
locally rotationally symmetric Kasner metric
scientific article; zbMATH DE number 1562777