

4.4 Realistic black
holes
Thus a simple brane-based approach, while giving useful insights,
does not lead to a realistic black hole solution. There is no known
solution representing a realistic black hole localized on the
brane, which is stable and without naked singularity. This remains
a key open question of nonlinear brane-world gravity. (Note that an
exact solution is known for a black hole on a
-brane in a 4D bulk [96], but this is a very special case.)
Given the nonlocal nature of
, it is possible that the
process of gravitational collapse itself leaves a signature in the
black hole end-state, in contrast with general relativity and its
no-hair theorems. There are contradictory indications about the
nature of the realistic black hole solution on the brane:
- Numerical simulations of highly relativistic
static stars on the brane [319] indicate that general relativity remains
a good approximation.
- Exact analysis of Oppenheimer-Snyder collapse
on the brane shows that the exterior is non-static [109
], and this is
extended to general collapse by arguments based on a generalized
AdS/CFT correspondence [303
, 94
].
The first result suggests that static black holes
could exist as limits of increasingly compact static stars, but the
second result and conjecture suggest otherwise. This remains an
open question. More recent numerical evidence is also not
conclusive, and it introduces further possible subtleties to do
with the size of the black hole [183
].
On very small scales relative to the
curvature scale,
, the
gravitational potential becomes 5D, as shown in Equation (40),
In this regime, the black hole is so small that it does not “see”
the brane, so that it is approximately a 5D Schwarzschild (static)
solution. However, this is always an approximation because of the
self-gravity of the brane (the situation is different in ADD-type
brane-worlds where there is no brane tension). As the black hole
size increases, the approximation breaks down. Nevertheless, one
might expect that static solutions exist on sufficiently small
scales. Numerical investigations appear to confirm this [183]: Static metrics satisfying
the asymptotic
boundary conditions are found if the
horizon is small compared to
, but no numerical convergence
can be achieved close to
. The numerical instability
that sets in may mask the fact that even the very small black holes
are not strictly static. Or it may be that there is a transition
from static to non-static behaviour. Or it may be that static black
holes do exist on all scales.
The 4D Schwarzschild metric cannot describe the
final state of collapse, since it cannot incorporate the 5D
behaviour of the gravitational potential in the strong-field regime
(the metric is incompatible with massive KK modes). A
non-perturbative exterior solution should have nonzero
in order to be compatible with massive KK modes in
the strong-field regime. In the end-state of collapse, we expect an
which goes to zero at large distances, recovering
the Schwarzschild weak-field limit, but which grows at short range.
Furthermore,
may carry a Weyl “fossil record” of the
collapse process.

