  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.
