Thermodynamics of black holes from equipartition of energy and holography

A gravitational potential in the relativistic case is introduced as an alternative to Wald's potential used by Verlinde, which reproduces the familiar entropy/area relation $S=A/4$ (in the natural units) when Verlinde's idea is applied to the black hole case. Upon using the equipartition rule, the correct form of the Komar mass (energy) can also be obtained, which leads to the Einstein equations. It is explicitly shown that our entropy formula agrees with Verlinde's entropy variation formula in spherical cases. The stationary space-times, especially the Kerr-Newman black hole, are then discussed, where it is shown that the equipartition rule involves the reduced mass, instead of the Arnowitt-Deser-Misner mass, on the horizon of the black hole.