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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 1+2-brane in a 4D bulk [96], but this is a very special case.) Given the nonlocal nature of ℰ μν, 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:

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 [183Jump To The Next Citation Point].

On very small scales relative to the AdS5 curvature scale, r ≪ ℓ, the gravitational potential becomes 5D, as shown in Equation (40View Equation),

GℓM G5M V (r ) ≈ -r2---= --r2--. (158 )
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 AdS5 boundary conditions are found if the horizon is small compared to ℓ, but no numerical convergence can be achieved close to ℓ. 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 ℰμν in order to be compatible with massive KK modes in the strong-field regime. In the end-state of collapse, we expect an ℰμν which goes to zero at large distances, recovering the Schwarzschild weak-field limit, but which grows at short range. Furthermore, ℰ μν may carry a Weyl “fossil record” of the collapse process.

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