The Einstein Tensor and Its Generalizations

Abstract

The Einstein tensor Gij is symmetric, divergence free, and a concomitant of the metric tensor gab together with its first two derivatives. In this paper all tensors of valency two with these properties are displayed explicitly. The number of independent tensors of this type depends crucially on the dimension of the space, and, in the four dimensional case, the only tensors with these properties are the metric and the Einstein tensors.

Brane cosmology with curvature corrections

“…Varying eq. (2.1) with respect to the bulk metric (5) g AB , we obtain the field equations: 6) where (4) g AB = (5) g AB − n A n B is the induced metric on the hypersurfaces {y = constant}, with n A the normal vector. The localized energy-momentum tensor of the brane is…”

A quasi-thick codimension 2 braneworld

“…where κ 2 is the 6-dimensional gravitational constant, g (6)µν and g µν are the 6-dimensional bulk and the our 4-dimensional brane metrics, respectively. Here, L matter is the Lagrangian density of the matter on the brane, and σ is the brane tension.…”

“…For this case, two kinds of singular structures exist in Eqs. (8) and (9) under the boundary condition (6). One is a 2-dimensional delta function sin-…”

“…Because a natural extension of Einstein gravity to higher dimensions is the Lovelock gravity and it reduces to the Einstein GaussBonnet gravity in 6-dimensions [6]. Moreover, the GaussBonnet term appears in the low energy effective action in string theory.…”

Some considerations on the Mac Dowell-Mansouri action

“…Up to now the results appear to be very dissimilar in odd and even dimensions. While odd dimensional Lovelock theories [1] can be used to construct gauge theories of gravity, that, moreover, have a topological interpretation using Chern-Simons (CS) forms [2,3]. Even dimensional Lovelock theories appear to not be embeddable in topological structures.…”