  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.
