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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 m n 2 ds = e ~gmndx dx + dy , (138) dr2 g~mn = e2|y| /lgmn = -(1 - 2GM/r)dt2 + ------------+ r2d_O_2. (139) 1 - 2GM/r
(Note that Equation (138View Equation) is in fact a solution of the 5D field equations (22View Equation) if ~gmn is any 4D Einstein vacuum solution, i.e., if ~R = 0 mn, and this can be generalized to the case R~ = - ~/\~g mn mn [715Jump 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) = ----- ==> Emn = 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 --> oo [52]:
(5) (5) ABCD 40- 48G2M---2 4|y|/l RABCD R = l4 + r6 e . (141)
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 Emn [73Jump To The Next Citation Point]. Brane solutions of static black hole exteriors with 5D corrections to the Schwarzschild metric have been found [73Jump To The Next Citation Point1107831416049159], 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 [29254].



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