Extrinsically flat static spacetimes

TL;DR

This paper explores static spacetimes with zero extrinsic curvature, revealing two cases with distinct geometric and matter properties, including violations of energy conditions and potential naked horizons, supported by explicit solutions.

Contribution

It classifies extrinsically flat static spacetimes into two cases and provides explicit solutions for various matter models, highlighting unique geometric and physical features.

Findings

Horizon can be naked with diverging Weyl components but finite Kretschmann scalar.

Matter sources can violate null energy condition, indicating phantom-like behavior.

Explicit solutions for linear anisotropic, Chaplygin gas, and uniform density models.

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.