Spacetimes characterized by their scalar curvature invariants

Abstract

In this paper we determine the class of four-dimensional Lorentzian manifolds that can be completely characterized by the scalar polynomial curvature invariants constructed from the Riemann tensor and its covariant derivatives. We introduce the notion of an I-non-degenerate spacetime metric, which implies that the spacetime metric is locally determined by its curvature invariants. By determining an appropriate set of projection operators from the Riemann tensor and its covariant derivatives, we are able to prove a number of results (both in the algebraically general and in algebraically special cases) of when a spacetime metric is I-nondegenerate. This enables us to prove our main theorem that a spacetime metric is either I-non-degenerate or a Kundt metric. Therefore, a metric that is not characterized by its curvature invariants must be of Kundt form. We then discuss the inverse question of what properties of the underlying spacetime can be determined from a given a set of scalar polynomial invariants, and some partial results are presented. We also discuss the notions of strong and weak non-degeneracy.

Discriminating the Weyl type in higher dimensions using scalar curvature invariants

“…It was proven that a 4D Lorentzian spacetime metric is either I-non-degenerate or degenerate Kundt [8]. The I-non-degenerate theorem deals with not only zeroth order invariants but also differential scalar polynomial curvature invariants constructed from the Riemann tensor and its covariant derivatives.…”

“…Thus, the necessary conditions in order for a spacetime not to be I-non-degenerate [8] are that the Riemann tensor and all of its covariant derivatives must be of types II or D. By constructing the appropriate curvature operators, these necessary conditions (syzygies) can be obtained using discrimants. In [8] two higher order syzygies were given as sufficient conditions for I-non-degeneracy, which can be expressed in terms of discriminants [14].…”

Local and gauge invariant observables in gravity

“…If A is local and comes from a horizontal density α, there will of course be a relation between these two notions of support. More precisely, we define [6,Eq.5.22] (10) which is always closed. In words, for any point y ∈ M outside supp A, there is a sufficiently small neighborhood V ∋ y so that any perturbation Ψ of the argument of A[Φ] with supp Ψ ⊆ V must leave the numerical value of A unchanged, that is,…”

“…The second inclusion is trivial, because (j k (Φ + Ψ)) * α = (j k Φ) * α whenever supp Ψ is outside of supp M α, since the restriction of both sides of the equality to supp Ψ is simply zero. The rest, namely supp A ⊆ supp M α, follows from the defining Equation (10).…”

“…We can reasonably define supp A| U by replacing C with U in the definition (10). The logical quantifiers are arranged such that supp A| U ⊆ supp A.…”