A scalar invariant and the local geometry of a class of static spacetimes

TL;DR

This paper investigates a scalar invariant derived from the curvature tensor's derivative to analyze the local geometry of static Einstein spacetimes, revealing conditions for conformal flatness and classifying certain vacuum solutions.

Contribution

It provides an explicit form of the scalar invariant for static Einstein spacetimes and links its vanishing to geometric properties and classification of solutions.

Findings

Scalar invariant I detects conformal flatness of spatial slices.

Vanishing of I implies spatial slices are conformally flat and warped-product spaces.

Classifies static vacuum solutions as Nariai-type or Kottler-type under certain conditions.

Abstract

The scalar invariant, I, constructed from the "square" of the first covariant derivative of the curvature tensor is used to probe the local geometry of static spacetimes which are also Einstein spaces. We obtain an explicit form of this invariant, exploiting the local warp-product structure of a 4-dimensional static spacetime, $~^{(3)}\Sigma \times_{f} \reals$, where $^{(3)}\Sigma $ is the Riemannian hypersurface orthogonal to a timelike Killing vector field with norm given by a positive function, $f$ on $^{(3)}\Sigma $. For a static spacetime which is an Einstein space, it is shown that the locally measurable scalar, I, contains a term which vanishes if and only if $^{(3)}\Sigma$ is conformally flat; also, the vanishing of this term implies (a) $~^{(3)}\Sigma$ is locally foliated by level surfaces of $f$, $^{(2)}S$, which are totally umbilic spaces of constant curvature, and (b) $^{(3)}\Sigma$ is locally a warp-product space. Futhermore, if $^{(3)}\Sigma$ is conformally flat it follows that every non-trivial static solution of the vacuum Einstein equation with a cosmological constant, is either Nariai-type or Kottler-type - the classes of spacetimes relevant to quantum aspects of gravity.