Black holes on the brane
Introduction
Recent developments in string theory have shown that if matter fields are localized on a 3-brane in 1+3+d dimensions, while gravity can propagate in the d extra dimensions, then the extra dimensions can be large (see, e.g., [1]). The extra dimensions need not even be compact, as in the 5-dimensional warped space models of Randall and Sundrum [2]. (See also [3] for earlier work.) In particular, they showed that it is possible to localize gravity on a 3-brane when there is one infinite extra dimension.
If matter on a 3-brane collapses under gravity, without rotating, to form a black hole, then the metric on the brane-world should be close to the Schwarzschild metric at astrophysical scales in order to preserve the observationally tested predictions of general relativity. Collapse to a black hole in the Randall-Sundrum brane-world scenario was studied by Chamblin et al. [4] (see also [5], [6], [7], [8]). They gave a `black string' solution which intersects the brane in a Schwarzschild solution.
Here we give an exact localized black hole solution, which remarkably has the mathematical form of the Reissner-Nördstrom solution, but without electric charge being present. Instead the Reissner-Nördstrom-type correction to the Schwarzschild potential can be thought of as a `tidal charge', arising from the projection onto the brane of free gravitational field effects in the bulk. These effects are transmitted via the bulk Weyl tensor (see below). The Schwarzschild potential Φ=−M/(Mp2r), where Mp is the effective Planck mass on the brane, is modified towhere the constant Q is a `tidal charge' parameter, which may be positive or negative.
A geometric approach to the Randall-Sundrum scenario has been developed by Shiromizu et al. [9] (see also [10]), and proves to be a useful starting point for formulating the problem and seeing clear lines of approach. The field equations in the bulk are (modifying the notation of [9])where the tildes denote bulk quantities. The fundamental 5-dimensional Planck mass enters via . The brane tension is λ, and is the bulk cosmological constant. The brane is located at χ=0 (so that x4=χ is a natural choice for the fifth dimension coordinate), and is the induced metric on the brane, with nA the spacelike unit normal to the brane. The brane energy-momentum tensor is TAB, and TABnB=0. The brane is a fixed point of the Z2 symmetry.
Section snippets
Field equations on the brane
The field equations induced on the brane arise from Eq. (2), the Gauss-Codazzi equations and the matching conditions with Z2-symmetry, and they may be written as a modification of the standard Einstein equations, with the new terms carrying bulk effects onto the brane [9]:where κ2=8π/Mp2. The energy scales are related to each other and to the cosmological constants viaTypically, the fundamental Planck scale is much lower than the
Solutions with tidal charge
Algebraic symmetry properties imply that in general we can decompose irreducibly with respect to a chosen 4-velocity field uμ as [11]where hμν=gμν+uμuν projects orthogonal to uμ. Hereis an effective energy density on the brane arising from the free gravitational field in the bulk—but note that this energy density need not be positive. Indeed, as we argue below, is the natural case. The effective anisotropic stress from the free gravitational
Properties of the black hole
The 4-dimensional horizon structure of the brane-world black hole depends on the sign of q. For q≥0, there is a direct analogy to the Reissner-Nördstrom solution, with two horizons:As in general relativity, both horizons lie inside the Schwarzschild horizon: , and there is an upper limit on q:The intriguing new possibility that q<0, which is impossible in the general relativity Reissner-Nördstrom case, leads to only one horizon, lying
Conclusion
We have not investigated fully the effect of the brane-world black hole on the bulk geometry, and in particular the nature of the off-brane horizon structure. This has been done for solutions which reduce to the Schwarzschild black hole on the brane [4]. In these solutions, the bulk metric iswhere gμν is the Schwarzschild metric. We have adopted a different approach: instead of starting from an induced Schwarzschild metric, we have solved the effective field
Acknowledgments
V.R. was supported by a Royal Society grant while at Portsmouth, and thanks the Relativity and Cosmology Group for hospitality. R.M. thanks IUCAA, Pune for hospitality during a visit, which was partially supported by the Royal Society. We thank Bruce Bassett, Marco Bruni, Roberto Casadio, Roberto Emparan, David Langlois, Jose Senovilla, Carlo Ungarelli and David Wands for useful discussions and comments, and especially Tetsuya Shiromizu.
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