Goldstone origin of black hole hair from supertranslations and criticality

Abstract

Degrees of freedom that carry black hole entropy and hair can be described in the language of Goldstone phenomenon. They represent the pseudo-Goldstone bosons of certain supertranslations, called [Formula: see text]-transformations, that are spontaneously broken by the black hole metric. This breaking gives rise to a tower of Goldstone bosons created by the spontaneously-broken generators that can be labeled by spherical harmonics. Classically, the number of charges is infinite, they have vanishing vacuum expectation values (VEVs) and the corresponding Goldstone modes are gapless. The resulting hair and entropy are infinite, but unresolvable. In quantum theory, the two things happen. The number of legitimate Goldstone modes restricted by requirement of weak-coupling, becomes finite and scales as black hole area in Planck units. The Goldstones generate a tiny gap, controlled by their gravitational coupling. The gap turns out to be equal to the inverse of black hole half-life, [Formula: see text]. Correspondingly, in quantum theory the charges are neither conserved nor vanish, but non-conservation time is set by [Formula: see text]. This picture nicely matches with the idea of a black hole as critical system is composed of many soft gravitons. The [Formula: see text]-Goldstones of geometric picture represent the near-gapless Bogoliubov–Goldstone modes of critical soft-graviton system.