If we assume a metric on the brane of
Schwarzschild-like form, i.e., in
Equation (143
), then the general
solution of the brane field equations is [73]
The solution (153) has the form of the
general relativity Reissner-Nordström solution, but there is no electric field on the brane. Instead, the
nonlocal Coulomb effects imprinted by the bulk Weyl tensor have
induced a “tidal” charge parameter
, where
, since
is the source of the bulk Weyl field.
We can think of the gravitational field of
being “reflected back” on the brane by the negative
bulk cosmological constant [71]. If we impose the small-scale
perturbative limit (
) in Equation (40
), we find that
The tidal-charge black hole metric does not
satisfy the far-field correction to the
gravitational potential, as in Equation (41
), and therefore cannot
describe the end-state of collapse. However, Equation (153
) shows the correct 5D
behaviour of the potential (
) at short
distances, so that the tidal-charge metric could be a good
approximation in the strong-field regime for small black holes.