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4.6 AdS/CFT and black holes on 1-brane RS-type models

Oppenheimer–Snyder collapse is very special; in particular, it is homogeneous. One could argue that the non-static exterior arises because of the special nature of this model. However, the underlying reasons for non-static behaviour are not special to this model; on the contrary, the role of high-energy corrections and KK stresses will if anything be enhanced in a general, inhomogeneous collapse. There is in fact independent heuristic support for this possibility, arising from the AdS/CFT correspondence.

The basic idea of the correspondence is that the classical dynamics of the AdS5 gravitational field correspond to the quantum dynamics of a 4D conformal field theory on the brane. This correspondence holds at linear perturbative order [87], so that the RS 1-brane infinite AdS5 brane-world (without matter fields on the brane) is equivalently described by 4D general relativity coupled to conformal fields,

(cft) G μν = 8πGT μν . (172 )
According to a conjecture [303Jump To The Next Citation Point], the correspondence holds also in the case where there is strong gravity on the brane, so that the classical dynamics of the bulk gravitational field of the brane black hole are equivalent to the dynamics of a quantum-corrected 4D black hole (in the dual CFT-plus-gravity description). In other words [94, 303Jump To The Next Citation Point]:

A further remarkable consequence of this conjecture is that Hawking evaporation is dramatically enhanced, due to the very large number of CFT modes of order (ℓ∕ℓ )2 p. The energy loss rate due to evaporation is

( ) M ˙ 1 --- ∼ N -2---3 , (173 ) M G M
where N is the number of light degrees of freedom. Using N ∼ ℓ2∕G, this gives an evaporation timescale [303]
( )3 ( )2 t ∼ -M-- 1-mm-- yr. (174 ) evap M ⊙ ℓ
A more detailed analysis [95] shows that this expression should be multiplied by a factor ≈ 100. Then the existence of stellar-mass black holes on long time scales places limits on the AdS5 curvature scale that are more stringent than the table-top limit, Equation (6View Equation). The existence of black hole X-ray binaries implies
ℓ ≲ 10− 2 mm, (175 )
already an order of magnitude improvement on the table-top limit.

One can also relate the Oppenheimer–Snyder result to these considerations. In the AdS/CFT picture, the non-vanishing of the Ricci scalar, Equation (170View Equation), arises from the trace of the Hawking CFT energy-momentum tensor, as in Equation (172View Equation). If we evaluate the Ricci scalar at the black hole horizon, R ∼ 2GM, using λ = 6M 56∕M 2p, we find

μ M512-ℓ6- R μ ∼ M 4 . (176 )
The CFT trace on the other hand is given by (cft) 4 2 T ∼ N T h∕M p, so that
12 6 8πGT (cft) ∼ M5---ℓ-. (177 ) M 4
Thus the Oppenheimer–Snyder result is qualitatively consistent with the AdS/CFT picture.

Clearly the black hole solution, and the collapse process that leads to it, have a far richer structure in the brane-world than in general relativity, and deserve further attention. In particular, two further topics are of interest:


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