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4.1 The black string

The projected Weyl term vanishes in the simplest candidate for a black hole solution. This is obtained by assuming the exact Schwarzschild form for the induced brane metric and “stacking” it into the extra dimension [52Jump To The Next Citation Point],
(5) 2 − 2|y|∕ℓ μ ν 2 ds = e &tidle;gμνdx dx + dy , (138 ) 2|y|∕ℓ 2 ----dr2----- 2 2 g&tidle;μν = e gμν = − (1 − 2GM ∕r )dt + 1 − 2GM ∕r + r dΩ . (139 )
(Note that Equation (138View Equation) is in fact a solution of the 5D field equations (22View Equation) if &tidle;gμν is any 4D Einstein vacuum solution, i.e., if &tidle;R μν = 0, and this can be generalized to the case R&tidle;μν = −Λ&tidle;&tidle;gμν [7, 15Jump To The Next Citation Point].)

Each {y = const.} surface is a 4D Schwarzschild spacetime, and there is a line singularity along r = 0 for all y. This solution is known as the Schwarzschild black string, which is clearly not localized on the brane y = 0. Although (5) CABCD ⁄= 0, the projection of the bulk Weyl tensor along the brane is zero, since there is no correction to the 4D gravitational potential:

GM V (r) = ----- ⇒ ℰμν = 0. (140 ) r
The violation of the perturbative corrections to the potential signals some kind of non-AdS5 pathology in the bulk. Indeed, the 5D curvature is unbounded at the Cauchy horizon, as y → ∞ [52]:
40 48G2M 2 (5)RABCD (5)RABCD = -4-+ ----6----e4|y|∕ℓ. (141 ) ℓ r
Furthermore, the black string is unstable to large-scale perturbations [127].

Thus the “obvious” approach to finding a brane black hole fails. An alternative approach is to seek solutions of the brane field equations with nonzero ℰμν [72Jump To The Next Citation Point]. Brane solutions of static black hole exteriors with 5D corrections to the Schwarzschild metric have been found [49, 72Jump To The Next Citation Point, 78, 111, 159, 160, 314], but the bulk metric for these solutions has not been found. Numerical integration into the bulk, starting from static black hole solutions on the brane, is plagued with difficulties [54, 292].


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