4 Gravitational Collapse and Black Holes on the Brane

The physics of brane-world compact objects and gravitational collapse is complicated by a number of factors, especially the confinement of matter to the brane, while the gravitational field can access the extra dimension, and the nonlocal (from the brane viewpoint) gravitational interaction between the brane and the bulk. Extra-dimensional effects mean that the 4D matching conditions on the brane, i.e., continuity of the induced metric and extrinsic curvature across the 2-surface boundary, are much more complicated to implement [160Jump To The Next Citation Point, 119Jump To The Next Citation Point, 422Jump To The Next Citation Point, 159Jump To The Next Citation Point]. High-energy corrections increase the effective density and pressure of stellar and collapsing matter. In particular this means that the effective pressure does not in general vanish at the boundary 2-surface, changing the nature of the 4D matching conditions on the brane. The nonlocal KK effects further complicate the matching problem on the brane, since they in general contribute to the effective radial pressure at the boundary 2-surface. Gravitational collapse inevitably produces energies high enough, i.e., ρ ≫ λ, to make these corrections significant.

We expect that extra-dimensional effects will be negligible outside the high-energy, short-range regime. The corrections to the weak-field potential, Equation (41View Equation), are at the second post-Newtonian (2PN) level [164, 210Jump To The Next Citation Point]. However, modifications to Hawking radiation may bring significant corrections even for solar-sized black holes, as discussed below.

A vacuum on the brane, outside a star or black hole, satisfies the brane field equations

μ μ ν R μν = − ℰμν, R μ = 0 = ℰ μ, ∇ ℰμν = 0. (143 )
The Weyl term ℰ μν will carry an imprint of high-energy effects that source KK modes (as discussed above). This means that high-energy stars and the process of gravitational collapse will in general lead to deviations from the 4D general relativity problem. The weak-field limit for a static spherical source, Equation (41View Equation), shows that ℰμν must be nonzero, since this is the term responsible for the corrections to the Newtonian potential.

 4.1 The black string
 4.2 Taylor expansion into the bulk
 4.3 The “tidal charge” black hole
 4.4 Realistic black holes
 4.5 Oppenheimer–Snyder collapse gives a non-static black hole
 4.6 AdS/CFT and black holes on 1-brane RS-type models

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