Black Hole Instabilities and Exponential Growth

Recently, a general analysis has been given of the stability with respect to axisymmetric perturbations of stationary-axisymmetric black holes and black branes in vacuum general relativity in arbitrary dimensions. It was shown that positivity of canonical energy on an appropriate space of perturbations is necessary and sufficient for stability. However, the notions of both "stability" and "instability" in this result are significantly weaker than one would like to obtain. In this paper, we prove that if a perturbation of the form $\pounds_t \delta g$---with $\delta g$ a solution to the linearized Einstein equation---has negative canonical energy, then that perturbation must, in fact, grow exponentially in time. The key idea is to make use of the $t$- or ($t$-$\phi$)-reflection isometry, $i$, of the background spacetime and decompose the initial data for perturbations into their odd and even parts under $i$. We then write the canonical energy as $\mathscr E\ = \mathscr K + \mathscr U$, where $\mathscr K$ and $\mathscr U$, respectively, denote the canonical energy of the odd part (kinetic energy) and even part (potential energy). One of the main results of this paper is the proof that $\mathscr K$ is positive definite for any black hole background. We use $\mathscr K$ to construct a Hilbert space $\mathscr H$ on which time evolution is given in terms of a self-adjoint operator $\tilde {\mathcal A}$, whose spectrum includes negative values if and only if $\mathscr U$ fails to be positive. Negative spectrum of $\tilde{\mathcal A}$ implies exponential growth of the perturbations in $\mathscr H$ that have nontrivial projection into the negative spectral subspace. This includes all perturbations of the form $\pounds_t \delta g$ with negative canonical energy. A "Rayleigh-Ritz" type of variational principle is derived, which can be used to obtain lower bounds on the rate of exponential growth.