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 [105
]. 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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