Search

The scalar invariant, Iequivalent toR(munurhosigma;delta) R-munurhosigma;delta, constructed from the 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 x (f) R, where ((3))Sigma is the riemannian hypersurface orthogonal to a timelike Killing vector field with norm given by a positive function, f : ((3))Sigma-->R. 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, S-(2), which are totally umbilic spaces of constant curvature, and (b) ((3))Sigma is locally a warp-product space, R x (r(F)) S-(2), for some function r (f). 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.

The scalar invariant, Iequivalent toR(munurhosigma;delta) R-munurhosigma;delta, constructed from the 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 x (f) R, where ((3))Sigma is the riemannian hypersurface orthogonal to a timelike Killing vector field with norm given by a positive function, f : ((3))Sigma-->R. 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, S-(2), which are totally umbilic spaces of constant curvature, and (b) ((3))Sigma is locally a warp-product space, R x (r(F)) S-(2), for some function r (f). 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.