### 4.3 The “tidal charge” black hole

The equations (143) form a system of constraints on the brane in the stationary case, including the
static spherical case, for which
The nonlocal conservation equations reduce to
where, by symmetry,
for some , with being the unit radial vector. The solution of the brane field equations requires
the input of from the 5D solution. In the absence of a 5D solution, one can make an assumption
about or to close the 4D equations.
If we assume a metric on the brane of Schwarzschild-like form, i.e., in Equation (149), then
the general solution of the brane field equations is [105]

where is a constant. It follows that the KK energy density and anisotropic stress scalar (defined via
Equation (158)) are given by
The solution (159) 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 [104]. If we impose the small-scale perturbative limit () in
Equation (40), we find that

Negative is in accord with the intuitive idea that the tidal charge strengthens the gravitational field,
since it arises from the source mass on the brane. By contrast, in the Reissner–Nordström
solution of general relativity, , where is the electric charge, and this weakens the
gravitational field. Negative tidal charge also preserves the spacelike nature of the singularity,
and it means that there is only one horizon on the brane, outside the Schwarzschild horizon:
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 (159) 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.