  We numerically study the stability of collisionless equilibria in the context
of general relativity. More precisely, we consider the spherically symmetric,
asymptotically flat Einstein-Vlasov system in Schwarzschild and in maximal
areal coordinates. Our results provide strong evidence against the well-known
binding energy hypothesis which states that the first local maximum of the
binding energy along a sequence of isotropic steady states signals the onset of
instability. We do however confirm the conjecture that steady states are stable
at least up to the first local maximum of the binding energy. For the first
time, we observe multiple stability changes for certain models. The equations
of state used are piecewise linear functions of the particle energy and provide
a rich variety of different equilibria.
