Musings

Almost Hairless

To no one’s great surprise, but to the consternation of some, the blackhole no-hair theorem does not hold in higher dimensions. In four dimensions, blackhole solutions are characterized by their gauge charges: mass, angular momentum, electric charge, … (and perhaps discrete gauge charges, should there be discrete factors in the gauge group).

Not so in five dimensions, where, for given values of the mass, angular momentum and charge, there exist both “blackhole” (whose horizon is $S^3$) and black ring (a rotating black string, whose horizon is $S^2\times S^1$) solutions. There are even supersymmetric black ring solutions.

Such solutions can be said to possess a discrete (in fact, finite) amount of “hair.”

Bena and Warner found a rich class of supersymmetric black ring solutions which seem to comprise an infinite amount of (continuous) “hair.” Their solutions are characterized by seven functions of one variable, the position along a curve, $C$, embedded in $\mathbb{R}^4$. Four of the functions describe the embedding; the other three correspond to certain “charge densities” along the curve. This seems like a rather drastic failure of “no-hair.”

Horowitz and Reall, however, argue that almost all of the BW solutions do not have regular horizons. Only a “round” circle, with uniform charge density on it, has a regular horizon.

Thus is a semblance of hairlessness restored…