Gravity and Strings

18.2 General p-brane solutions 513
the definition we gave in Section 18.1.1. This was done in Section 11.3.3, Eqs. (11.149) and
(11.150), and the result can be expressed in this way: we can identify the p-brane tension
as the constant T(p) in the expansion of the gtt component of the metric
gtt ∼ 1 − 16πG(d)
N T(p)
( ˜p + 2)ω( ˜p+2)
1
. (18.60)
r ˜p+1
The coefficient of r −( ˜p+1) is, by definition, the p-brane Schwarzschild radius to the power
˜p + 1.
It is clear from the definition that infinite (uncompactified) p-branes have infinite energy.
Is this Schwarzschild p-brane solution analogous to the Schwarzschild BH solution in
the sense that it exhibits an event horizon? In other words: is it a black p-brane solution?
In the presence of the p translational isometries the only sensible definition of an
event horizon is equivalent to the standard definition of an event horizon in the transverse
space. The event horizon thus defined becomes a (p + 2)-dimensional extended object:
the product of a BH horizon and the p-dimensional Euclidean space spanned by the
p-brane. Clearly, for positive tension, the Schwarzschild p-brane solution has an event horizon
of that form, whereas, for negative tension, the curvature singularity at r = 0 will be
naked.
When the spacetime has n compact dimensions, it is possible to have black p-branes,
p ≤ n, wrapping the compact dimensions with the same (finite) mass and event horizons
of different topologies. Therefore, the uniqueness of four-dimensional BHs is not true in
higher dimensions if some of the dimensions are compact. We know now that it is not
true even in the absence of compact dimensions, as the existence of the asymptotically flat
rotating black ring of [373] shows.
Black p-brane solutions (the Schwarzschild ones and the charged non-extreme ones that
we are going to see next) are classically unstable [477, 478] (the charged, extreme ones
are stable [479], as is usual in supersymmetric solutions) under linear perturbations along
the worldvolume dimensions with wavelengths larger than the Schwarzschild radius. On
the other hand, they are also quantum-mechanically unstable, because there is Hawking
radiation associated with their event horizons, but also because the area of the event horizon
(entropy) of several BHs with the same total mass is in general larger than the area of the
event horizon of a Schwarzschild p-brane. For a Schwarzschild string compactified on
acircle this happens whenever the length of the circle is larger than the Schwarzschild
p-brane radius. The two instabilities seem to be related in the sense that the classical one is
present whenever the thermodynamical one is present [485, 486, 800].
Although the thermodynamical argument seems to indicate that a black p-brane will
break up into several BHs that will eventually merge into one, it has been argued in [551]
that, for the black string, this process cannot take place in a finite time and that, instead,
the black string decays into a new non-translationally invariant (“inhomogeneous”) black
string. Initial data sets for inhomogeneous p-brane solutions have subsequently been proposed
in [552].
As we did in the BH case, we are going to use the Schwarzschild p-brane solution
Eqs. (11.148) as our basic black p-brane solution and we are going to see that the charged