Instability of the negative mass Schwarzschild naked singularity

TL;DR

This paper demonstrates that the negative mass Schwarzschild spacetime with a naked singularity is perturbatively unstable by constructing explicit exponentially growing solutions to the linearized Einstein equations.

Contribution

It introduces a modified perturbation approach to handle the kinematic singularity for negative masses and provides explicit unstable solutions, contrasting previous stability analyses.

Findings

Negative mass Schwarzschild spacetime is perturbatively unstable.

Explicit exponentially growing solutions are constructed.

Perturbations are smooth and well-behaved at the singularity and infinity.

Abstract

We study the negative mass Schwarzschild spacetime, which has a naked singularity, and show that it is perturbatively unstable. This is achieved by first introducing a modification of the well known Regge - Wheeler - Zerilli approach to black hole perturbations to allow for the presence of a ``kinematic'' singularity that arises for negative masses, and then exhibiting exact exponentially growing solutions to the linearized Einstein's equations. The perturbations are smooth everywhere and behave nicely around the singularity and at infinity. In particular, the first order variation of the scalar invariants can be made everywhere arbitrarily small as compared to the zeroth order terms. Our approach is also compared to a recent analysis that leads to a different conclusion regarding the stability of the negative mass Schwarzschild spacetime. We also comment on the relevance of our results to the stability of more general negative mass, nakedly singular spacetimes.