Extrinsically flat static spacetimes

Abstract

We consider static spacetimes whose spatial part admits foliations with the extrinsic curvature tensor K_{ab}=0. There are two complementary cases when the gradient of the lapse function points 1) to the direction of foliation or 2) orthogonally to it. Case 1) gives generalization of metrics like Bertotti-Robinson or Nariai. In case 2) the matter source violates the null energy condition at least on the part of the manifold, having in this sense phantom nature. We also demonstrate that for the manifolds under discussion the horizon can be naked in the sense that certain Weyl components diverge in the free-falling frame although the Kretschmann scalar is finite. The Petrov type is D or O. Explicit solutions for (i) the linear anisotropic equation of state, (ii) Chaplygin gas and (iii) uniform energy density are found.