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

Entropic force in black hole binaries and its Newtonian limits