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 [73
]. Brane solutions of
static black hole exteriors with 5D corrections to the
Schwarzschild metric have been found [73
, 110, 78, 314, 160, 49, 159], 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 [292, 54].