  A self-interacting dark matter halo can experience gravothermal collapse,
resulting in a central core with an ultrahigh density. It can further contract
and collapse into a black hole, a mechanism proposed to explain the origin of
supermassive black holes. We study dynamical instability of the core in general
relativity. We use a truncated Maxwell-Boltzmann distribution to model the dark
matter distribution and solve the Tolman-Oppenheimer-Volkoff equation. For
given model parameters, we obtain a series of equilibrium configurations and
examine their dynamical instability based on considerations of total energy,
binding energy, fractional binding energy, and adiabatic index. Our numerical
results indicate that the core can collapse into a black hole when the
fractional binding energy reaches $0.035$ with a central gravitational redshift
of $0.5$. We further show for the instability to occur in the classical regime,
the boundary temperature of the core should be at least $10\%$ of the mass of
dark matter particles; for a $10^9~{\rm M_\odot}$ seed black hole, the particle
mass needs to be larger than a few keV. These results can be used to constrain
different collapse models, in particular, those with dissipative dark matter
interactions.
