  A massive spin-2 field can grow unstably around a black hole, giving rise to
a potential probe of the existence of such fields. In this work, we use
time-domain evolutions to study such instabilities. Considering the linear
regime by solving the equations generically governing a massive tensor field on
the background of a Kerr black hole, we find that black hole spin increases the
growth rate and, most significantly, the mass range of the axisymmetric
(azimuthal number $m=0$) instability, which takes the form of the
Gregory-Laflamme black string instability for zero spin. We also consider the
superradiant unstable modes with $1 \leq m \leq 3$, extending previous results
to higher spin-2 masses, black hole spins, and azimuthal numbers. We find that
the superradiant modes grow slower than the $m=0$ modes, except for a narrow
range of high spins and masses, with $m=1$ and 2 requiring a dimensionless
black hole spin of $a_{\rm BH}\gtrsim 0.95$ to be dominant. Thus, in most of
the parameter space, the backreaction of the $m=0$ instability must be taken
into account when using black holes to constrain massive spin-2 fields. As a
simple model of this, we consider nonlinear evolutions in quadratic gravity, in
particular Einstein-Weyl gravity. We find that, depending on the initial
perturbation, the black hole may approach zero mass with the curvature blowing
up in finite time, or can saturate at a larger mass with a surrounding cloud of
the ghost spin-2 field.
