  We improve and extend Shapiro's model of a relativistic, compact object which
is stable in isolation but is driven dynamically unstable by the tidal field of
a binary companion. Our compact object consists of a dense swarm of test
particles moving in randomly-oriented, initially circular, relativistic orbits
about a nonrotating black hole. The binary companion is a distant, slowly
inspiraling point mass. The tidal field of the companion is treated as a small
perturbation on the background Schwarzschild geometry near the hole; the
resulting metric is determined by solving the perturbation equations of Regge
and Wheeler and Zerilli in the quasi-static limit. The perturbed spacetime
supports Bekenstein's conjecture that the horizon area of a near-equilibrium
black hole is an adiabatic invariant. We follow the evolution of the system and
confirm that gravitational collapse can be induced in a compact collisionless
cluster by the tidal field of a binary companion.
