Each surface is a 4D Schwarzschild spacetime, and there is a line singularity along for all . This solution is known as the Schwarzschild black string, which is clearly not localized on the brane . Although , the projection of the bulk Weyl tensor along the brane is zero, since there is no correction to the 4D gravitational potential:5 pathology in the bulk. Indeed, the 5D curvature is unbounded at the Cauchy horizon, as : .
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 . Brane solutions of static black hole exteriors with 5D corrections to the Schwarzschild metric have been found [105, 160, 119, 422, 224, 73, 223], 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 [389, 78].
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