Linearized Gravity in Isotropic Coordinates in the Brane World

TL;DR

This paper derives an exact first-order solution to Einstein's equations in the Randall-Sundrum brane world model using isotropic coordinates, showing consistency with Schwarzschild metrics and analyzing horizon extensions.

Contribution

It provides a new exact first-order solution in isotropic coordinates that is free of singularities and matches known 4-D Schwarzschild solutions in the brane world context.

Findings

Solution is free from metric singularities away from the source.

The solution matches the 4-D Schwarzschild metric at large distances.

The Schwarzschild horizon extension in the bulk is tubular in isotropic coordinates.

Abstract

We solve the Einstein equations in the Randall-Sundrum framework using an isotropic ansatz for the metric and obtain an exact expression to first order in the gravitational coupling. The solution is free from metric singularities away from the source and it satisfies the Israel matching condition on a straight brane. At distances far away from the source and on the physical brane this solution coincides with the 4-D Schwarzschild metric in isotropic coordinates. Furthermore we show that the extension of the standard Schwarzschild horizon in the bulk is tubular for any diagonal form of the metric while there is no restriction for the extension of the Schwarzschild horizon in isotropic coordinates.