  We give an exact solution for the static force between two black holes at the
turning points in their binary motion. The results are derived by Gibbs'
principle and the Bekenstein-Hawking entropy applied to the apparent horizon
surfaces in time-symmetric initial data. New power laws are derived for the
entropy jump in mergers, while Newton's law is shown to derive from a new
adiabatic variational principle for the Hilbert action in the presence of
apparent horizon surfaces. In this approach, entropy is strictly monotonic such
that gravity is attractive for all separations including mergers, and the
Bekenstein entropy bound is satisfied also at arbitrarily large separations,
where gravity reduces to Newton's law. The latter is generalized to point
particles in the Newtonian limit by application of Gibbs' principle to
world-lines crossing light cones.
