Riemann curvature tensor

About: Riemann curvature tensor is a research topic. Over the lifetime, 6248 publications have been published within this topic receiving 138871 citations. The topic is also known as: Riemann–Christoffel tensor.

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 $\mathcal{I}$-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 $\mathcal{I}$-non-degenerate. This enables us to prove our main theorem that a spacetime metric is either $\mathcal{I}$-non-degenerate or a Kundt metric. Therefore, a metric that is not characterized by its curvature invariants must be of degenerate 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 \emph{strong} and \emph{weak} non-degeneracy.

Effective action approach to cosmological perturbations in dark energy and modified gravity

Abstract: In light of upcoming observations modelling perturbations in dark energy and modified gravity models has become an important topic of research. We develop an effective action to construct the components of the perturbed dark energy momentum tensor which appears in the perturbed generalized gravitational field equations, ?G?? = 8?G?T??+?U?? for linearized perturbations. Our method does not require knowledge of the Lagrangian density of the dark sector to be provided, only its field content. The method is based on the fact that it is only necessary to specify the perturbed Lagrangian to quadratic order and couples this with the assumption of global statistical isotropy of spatial sections to show that the model can be specified completely in terms of a finite number of background dependent functions. We present our formalism in a coordinate independent fashion and provide explicit formulae for the perturbed conservation equation and the components of ?U?? for two explicit generic examples: (i) the dark sector does not contain extra fields, = (g??) and (ii) the dark sector contains a scalar field and its first derivative = (g??,,??). We discuss how the formalism can be applied to modified gravity models containing derivatives of the metric, curvature tensors, higher derivatives of the scalar fields and vector fields.