Recovery of the Schwarzschild Metric in Theories with Localized Gravity Beyond Linear Order

TL;DR

This paper demonstrates that in the Randall-Sundrum model with localized matter, the gravitational field outside a spherical source matches the Schwarzschild metric up to second order, confirming consistency with classical tests of General Relativity.

Contribution

It provides an approximate second-order solution to Einstein's equations in the Randall-Sundrum framework, showing the recovery of the Schwarzschild metric on the physical brane.

Findings

The Schwarzschild metric is recovered to second order in the Randall-Sundrum model.

The model is consistent with Mercury's perihelion precession.

Second-order corrections do not violate classical gravitational tests.

Abstract

We solve the Einstein equations in the Randall-Sundrum framework with a static, spherically symmetric matter distribution on the {\it physical brane} and obtain an approximate expression for the gravitational field outside the source to second order in the gravitational coupling. This expression when confined on the {\it physical brane} coincides with the standard form of the Schwarzschild metric. Therefore, the Randall-Sundrum scenario is consistent with the Mercury precession test of General Relativity.