4.3 The “tidal charge” black hole

The equations (143View Equation) form a system of constraints on the brane in the stationary case, including the static spherical case, for which
Θ = 0 = ω μ = σμν, ρ˙ℰ = 0 = qℰ = ˙πℰ . (156 ) μ μν
The nonlocal conservation equations ∇ νℰμν = 0 reduce to
1 4 -⃗∇ μρℰ + --ρℰA μ + ∇⃗νπ ℰμν + A νπℰμν = 0, (157 ) 3 3
where, by symmetry,
( ) ℰ 1 πμν = Π ℰ 3-hμν − rμrν , (158 )
for some Π ℰ(r), with rμ being the unit radial vector. The solution of the brane field equations requires the input of ℰμν from the 5D solution. In the absence of a 5D solution, one can make an assumption about ℰ μν or g μν to close the 4D equations.

If we assume a metric on the brane of Schwarzschild-like form, i.e., H = F in Equation (149View Equation), then the general solution of the brane field equations is [105]

2GM--- 2G-ℓQ-- F = 1 − r + r2 , (159 ) 2G ℓQ ℰμν = − ---4--[uμuν − 2rμrν + hμν], (160 ) r
where Q is a constant. It follows that the KK energy density and anisotropic stress scalar (defined via Equation (158View Equation)) are given by
ℓQ 1 ρℰ = ----4 = --Πℰ. (161 ) 4 πr 2

The solution (159View Equation) 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 Q, where Q = Q (M ), since M is the source of the bulk Weyl field. We can think of the gravitational field of M being “reflected back” on the brane by the negative bulk cosmological constant [104]. If we impose the small-scale perturbative limit (r ≪ ℓ) in Equation (40View Equation), we find that

Q = − 2M. (162 )
Negative Q is in accord with the intuitive idea that the tidal charge strengthens the gravitational field, since it arises from the source mass M on the brane. By contrast, in the Reissner–Nordström solution of general relativity, 2 Q ∝ +q, where q 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:
[ ∘ ----------] [ ∘ ---------] 2ℓQ 4ℓ rh = GM 1 + 1 − -----2 = GM 1 + 1 + ----- . (163 ) GM GM

The tidal-charge black hole metric does not satisfy the far-field −3 r correction to the gravitational potential, as in Equation (41View Equation), and therefore cannot describe the end-state of collapse. However, Equation (159View Equation) shows the correct 5D behaviour of the potential (∝ r−2) at short distances, so that the tidal-charge metric could be a good approximation in the strong-field regime for small black holes.

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