The Einstein equations on the 3-brane world

Abstract

We carefully investigate the gravitational equations of the brane world, in which all the matter forces except gravity are confined on the 3-brane in a 5-dimensional spacetime with Z2 symmetry. We derive the effective gravitational equations on the brane, which reduce to the conventional Einstein equations in the low energy limit. From our general argument we conclude that the first Randall & Sundrum-type theory (RS1) [hep-ph/9905221] predicts that the brane with the negative tension is an anti-gravity world and hence should be excluded from the physical point of view. Their second-type theory (RS2) [hep-th/9906064] where the brane has the positive tension provides the correct signature of gravity. In this latter case, if the bulk spacetime is exactly anti-de Sitter, generically the matter on the brane is required to be spatially homogeneous because of the Bianchi identities. By allowing deviations from anti-de Sitter in the bulk, the situation will be relaxed and the Bianchi identities give just the relation between the Weyl tensor and the energy momentum tensor. In the present brane world scenario, the effective Einstein equations cease to be valid during an era when the cosmological constant on the brane is not well-defined, such as in the case of the matter dominated by the potential energy of the scalar field.

Brane cosmology with curvature corrections

“…On the contrary, in the pure Gauss-Bonnet equation (3.27), the brane tension is essential, since λ = 0 implies G eff = 0. This is like the pure Randall-Sundrum case, where positive brane tension is necessary in order to recover the standard Friedmann equation [2,20].…”

“…To determine the Friedmann equation on the brane, one can project the five-dimensional Einstein equations to four dimensions, following the approach of [11,20]. This is a difficult task because of the presence of the Lovelock tensor.…”

“…(2.13) in the simplest possible form. We define the dimensionless parameter 20) and the dimensionless variables…”

Oxidised cosmic acceleration

“…• Braneworld cosmologies, with a single warped extra dimension, satisfy the conditions of the theorems for curvature-free M [13,14,62,63,64]. When the extra dimension is compact, a negative-tension brane, which violates the NEC, must be present even in reductions to Minkowski space.…”

Brane cosmology solutions with bulk scalar fields

“…By projecting the higher dimensional curvature tensors onto the brane, and applying the relevant boundary conditions it is possible to obtain four-dimensional Einstein equations [3]. In the case of brane cosmologies with constant bulk energy densities, this approach allows the evolution of the fields on the brane to be completely determined.…”