### 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],
(Note that Equation (138) is in fact a solution of the 5D field equations (22) if is any 4D Einstein
vacuum solution, i.e., if , and this can be generalized to the case [7, 15].)
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:

The violation of the perturbative corrections to the potential signals some kind of non- pathology in
the bulk. Indeed, the 5D curvature is unbounded at the Cauchy horizon, as [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 [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].