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 [160, 119, 422, 159]. 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., , 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 (41), are at the second post-Newtonian (2PN) level [164, 210]. 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
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