The Einstein Equations on the 3-Brane World

TL;DR

This paper derives effective gravitational equations for a 3-brane in a 5D spacetime, analyzing their implications for brane world models and the nature of gravity, especially in the context of Randall-Sundrum theories and cosmological scenarios.

Contribution

It provides a detailed derivation of the effective Einstein equations on the brane and discusses their physical implications in different brane world models.

Findings

RS1 predicts negative tension branes as anti-gravity worlds

RS2 with positive tension branes correctly reproduces gravity

Effective Einstein equations are invalid during certain cosmological eras

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 $Z_2$ 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.