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:
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 [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].
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