The black string

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 [52

(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 (138

) is in fact a solution of the 5D field equations (22

) 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 , 15

].)

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:

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 ]:

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 $ℰμν$ [72 ]. Brane solutions of static black hole exteriors with 5D corrections to the Schwarzschild metric have been found [49 , 72 , 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 ].