Jacobi equations of geodetic brane gravity

TL;DR

This paper derives and analyzes the Jacobi equations for brane gravity in the Regge-Teitelboim model, revealing stability properties of embedded Schwarzschild spacetimes through linearization and Morse index calculations.

Contribution

It introduces the linearization of the equations of motion in brane gravity, deriving the Jacobi equations and applying them to stability analysis of embedded Schwarzschild spacetime.

Findings

Schwarzschild spacetime embedded in six-dimensional Minkowski space is linearly unstable.

Explicit Morse index expression for the Regge-Teitelboim model is obtained.

Jacobi equations exhibit complex features requiring detailed analysis.

Abstract

We consider brane gravity as described by the Regge-Teitelboim geometric model, in any codimension. In brane gravity our spacetime is modeled as the time-like world volume spanned by a space-like brane in its evolution, seen as a manifold embedded in an ambient background Minkowski spacetime of higher dimension. Although the equations of motion of the model are well known, apparently their linearization has not been considered before. Using a direct approach, we linearize the equations of motion about a solution, obtaining the Jacobi equations of the Regge- Teitelboim model. They take a formidable aspect. Some of their features are commented upon. By identifying the Jacobi equations, we derive an explicit expression for the Morse index of the model. To be concrete, we apply the Jacobi equations to the study of the stability of a four-dimensional Schwarzschild spacetime embedded in a six-dimensional Minkowski spacetime. We find that it is unstable under small linear deformations.