  As is well known from the work of R. Glassey} and J. Schaeffer, the main
energy estimates which are used in global existence results for the
gravitational Vlasov-Poisson system do not apply to the relativistic version of
this system, and smooth solutions to the initial value problem with spherically
symmetric initial data of negative energy blow up in finite time. For similar
reasons the variational techniques by which Y. Guo and G. Rein obtained
nonlinear stability results for the Vlasov-Poisson system do not apply in the
relativistic situation. In the present paper a direct, non-variational approach
is used to prove nonlinear stability of certain steady states of the
relativistic Vlasov-Poisson system against spherically symmetric, dynamically
accessible perturbations. The resulting stability estimates imply that smooth
solutions with spherically symmetric initial data which are sufficiently close
to the stable steady states exist globally in time.
