Soft Hair on Black Holes

It has recently been shown that Bondi–van der Burg–Metzner–Sachs supertranslation symmetries imply an infinite number of conservation laws for all gravitational theories in asymptotically Minkowskian spacetimes. These laws require black holes to carry a large amount of soft (i.e., zero-energy) supertranslation hair. The presence of a Maxwell field similarly implies soft electric hair. This Letter gives an explicit description of soft hair in terms of soft gravitons or photons on the black hole horizon, and shows that complete information about their quantum state is stored on a holographic plate at the future boundary of the horizon. Charge conservation is used to give an infinite number of exact relations between the evaporation products of black holes which have different soft hair but are otherwise identical. It is further argued that soft hair which is spatially localized to much less than a Planck length cannot be excited in a physically realizable process, giving an effective number of soft degrees of freedom proportional to the horizon area in Planck units.

Figure 1

Penrose diagram for a black hole formed by gravitational collapse. The blue line is the event horizon. The red line indicates a null spherical shell of collapsing matter. Spacetime is flat prior to collapse and Schwarzschild after. The event horizon $\mathcal{H}$ and the $S^2$ boundaries ${\mathcal{I}}_{\pm }^{\pm }$ of ${\mathcal{I}}^{\pm }$ are indicated. ${\mathcal{I}}^{-}$ and ${\mathcal{I}}^{+}\cup \mathcal{H}$ are Cauchy surfaces for massless fields.

Figure 2

Penrose diagram for a semiclassical evaporating black hole, outlined in blue. The red arrows denote the classical collapsing matter and the orange arrows the outgoing quantum Hawking radiation. The horizontal dashed line divides the spacetime into two regions long after classical black hole formation is complete and long before the onset of Hawking evaporation. The conserved charges $Q_{ϵ}^{+}$ defined at ${\mathcal{I}}_{-}^{+}$ can be evaluated as a volume integral either over the green slice comprising ${\mathcal{I}}^{+}$ or the yellow slice involving the classical part of the horizon ${\mathcal{H}}_{<}$. The equality of these two expressions yields infinitely many deterministic constraints on the evaporation process.