Horizon entropy does not by itself determine a force between black holes

Abstract

The Bekenstein-Hawking law fixes the total entropy of a horizon. Membrane, holographic, and entropic-force pictures go further and follow the entropy of individual horizon elements as a configuration changes. We show that this step needs structure that the horizon geometry does not supply. Along a family of horizons labeled by $λ$, a relabeling of horizon points by a tangential field $v^A$ shifts the temperature-weighted entropy variation $\int T\partial_λ(dS)$ exactly by $-\int v^AD_AT dS$, so the horizon data determine it only when the temperature is uniform. We test this on two black holes in exact Brill-Lindquist data. At fixed physical masses each horizon area is independent of the separation through five relative orders; the low-order equilibrium force reported earlier comes from holding the bare parameters fixed instead. Weighting the redistribution of area by the nonuniform Noether-charge temperature gives a candidate repulsion equal to $(ν/8)(GM/r_{12})^3$ times the Newtonian attraction, for total mass $M$, symmetric mass ratio $ν$, and coordinate separation $r_{12}$. As a static interaction it would shift the third post-Newtonian coefficient of the circular-orbit binding energy by $5ν/48$, in conflict with general relativity. The identity resolves the conflict. Through fourth post-Newtonian order the candidate lies entirely in the dipole of the area distribution, which the uniform part of the companion's field generates as a Möbius map of the labels, and which vanishes in the area-balanced conformal frame, where the functional begins at fifth order with a coefficient that a further relabeling can still change. The marginally trapped tube of the data and the tidally deformed horizon of a post-Newtonian binary, both of which do supply an identification, transport no dipole at the order the candidate requires.