Gravity on branes

TL;DR

This paper investigates gravity on a four-dimensional brane embedded in a five-dimensional space, deriving conditions under which Einstein's equations are recovered or approximated, highlighting the importance of bulk geometry.

Contribution

It provides a simplified derivation of brane world gravity equations and clarifies when Einstein's equations hold on the brane based on bulk geometry assumptions.

Findings

Einstein's equations are approximately recovered if the bulk is an Einstein space near the brane.

Strict anti-de Sitter bulk space prevents Einstein equations from holding on a quasi-Minkowskian brane.

Matter must obey specific equations of state for Einstein equations to hold in certain bulk conditions.

Abstract

We consider the four dimensional discontinuity generated by two identical pieces of a five dimensional space pasted along their edge (that is a "brane" in a "$Z_2$-symmetric" "bulk"). Using a four plus one decomposition of the Riemann tensor we write the equations for gravity on the brane and recover in a simple manner a number of known "brane world" scenarios. We study under which conditions these equations reduce, exactly or approximately, to the four dimensional Einstein equations. We conclude that if the bulk is imposed to be only an Einstein space near the brane, Einstein's equations can be recovered approximately on the brane, but if it is imposed to be strictly anti-de Sitter space then the Einstein equations cannot hold, even approximately, on a quasi-Minkowskian brane, unless matter obeys a very contrived equation of state.