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

Abstract

The scalar invariant, I ≡ R µνρσ;δ R µνρσ;δ , 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) Σ × f R, where (3) Σ is the Riemannian hypersurface orthogonal to a timelike Killing vector field with norm given by a positive function, f : (3) Σ −→ 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) 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, R× r(f ) (2) S, for some function r(f ). 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.

A third-order curvature invariant in static spacetimes

“…Thus, in principle, some specific local measurements[2] could be indeed employed by in-falling observers to detect the crossing of the event horizon of spherically symmetric black-holes. Several other aspects and properties of higher order curvature invariants have been also examined [3,4,5,6,7].In [8], the invariant I is considered for static 4-dimensional Einstein spacetimes M = R × Σ, with Σ conformally flat. Several properties of the invariant and some relations to the topology of Σ are discussed.…”

“…In [8], the invariant I is considered for static 4-dimensional Einstein spacetimes M = R × Σ, with Σ conformally flat. Several properties of the invariant and some relations to the topology of Σ are discussed.…”

Tidal indicators in the spacetime of a rotating deformed mass

“…One could also consider other more involved tidal invariants constructed from the covariant derivative of the curvature tensor. Such invariants have received some attention in the recent literature in order to investigate both geometrical and topological properties of certain classes of static as well as stationary spacetimes (see, e.g., [32][33][34]). However, in the context of tidal problems, differential invariants are of interest only when using the Fermi coordinate tidal potential, as discussed in [9].…”

“…Let u = n be the unit tangent vector to the ZAMO family of observer given by equation (3.1) with adapted frame (3.2). The relevant nonvanishing frame components of the electric and magnetic parts of the Riemann tensor are given by E(n) 11 , E(n) 33 and H(n) 12 with…”