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 determinin...

A classical result in Lorentzian geometry states that a strongly causal spacetime is globally hyperbolic if and only if the Lorentzian distance is finite valued for every metric choice in the conformal class. It is proven here that a non-total imprisoning spacetime is causally simple if and only if for every metric choice in the conformal class the Lorentzian distance is continuous wherever it ...

The canonical form of the Alcubierre warp drive metric is considered to gain insight into the mathematical mechanism triggering the effect. A parallel with the Chung-Freese spacetime metric is drawn to demonstrate that the spacetime expansion boost can be considered a 3 + 1 on-brane simplification for higher dimensional geometric effects. The implications for baryonic matter of higher dimension...

‎The object of the present paper is to study spacetimes admitting‎ ‎quasi-conformal curvature tensor‎. ‎At first we prove that a quasi-conformally flat spacetime is Einstein‎ ‎and hence it is of constant curvature and the energy momentum tensor of such a spacetime satisfying‎ ‎Einstein's field equation with cosmological constant is covariant constant‎. ‎Next‎, ‎we prove that if the perfect flui...

In this paper we first study some global properties of the energy functional on a nonreversible Finsler manifold. In particular we present a fully detailed proof of the Palais–Smale condition under the completeness of the Finsler metric. Moreover we define a Finsler metric of Randers type, which we call Fermat metric, associated to a conformally standard stationary spacetime. We shall study the...

We give a set of local geometric conditions on a spacetime metric which are necessary and sufficient for it to be a null electrovacuum, that is, the metric is part of a solution to the Einstein-Maxwell equations with a null electromagnetic field. These conditions are restrictions on a null congruence canonically constructed from the spacetime metric, and can involve up to five derivatives of th...

The theory of massive gravity possesses ambiguities when the spacetime metric evolves far from the non-dynamical fiducial metric used to define it. We explicitly construct a spherically symmetric example case where the metric evolves to a coordinate-independent determinant singularity which does not exist in the initial conditions. Both the metric and the vierbein formulation of the theory are ...

The causal structure of spacetime defines a partial order on the events of spacetime. In an earlier paper, using techniques from domain theory, we showed that for globally hyperbolic spacetimes one could reconstruct the topology from the causal structure. However, the causal structure determines the metric only up to a local rescaling (a conformal transformation); in a four dimensional spacetim...

Spacetime geometry is treated as a fluctuating, stochastic quantum system allowing an effective quantum gravity solution to the cosmological constant problem. A Fokker-Planck equation for the probability density of spacetime metric fluctuations is solved for an FRW universe and the most probable value for the metric scale factor R is for a very small cosmological constant. e-mail: [email protected]...