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

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, $~^{(3)}Σ\times_{f} \reals$, where $^{(3)}Σ$ is the Riemannian hypersurface orthogonal to a timelike Killing vector field with norm given by a positive function, $f$ on $^{(3)}Σ$. 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 $^{(3)}Σ$ is conformally flat; also, the vanishing of this term implies (a) $~^{(3)}Σ$ is locally foliated by level surfaces of $f$, $^{(2)}S$, which are totally umbilic spaces of constant curvature, and (b) $^{(3)}Σ$ is locally a warp-product space. Futhermore, if $^{(3)}Σ$ 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.