  We study the effects of general relativistic gravity on the Hill stability,
that is, the stability of a multi-body system against a close approach of one
orbit to another, which has been hitherto studied mainly in Newtonian mechanics
and applied to planetary systems. We focus in this paper on the three-body
problem and extend the Newtonian analyses to the general relativistic regime in
the post-Newtonian approximation. The approximate sufficient condition for the
relativistic Hill stability of three-body systems is derived analytically and
its validity and usefulness are confirmed numerically. In fact, relativity
makes the system more unstable than Newtonian mechanics in the sense of the
Hill stability as expected by our theoretical prediction. The criterion will be
useful to analyze the results of large-scale N-body simulations of dense
environments, in which the stability of three-body sub-systems is important.
