Abstract
In this article we review the present status of the numerical construction of black holes in the Randall–Sundrum II braneworld model. After reviewing the new numerical methods to solve the elliptic Einstein equations, we numerically construct a black hole solution in five-dimensional anti-de Sitter (AdS5) space whose boundary geometry is conformal to the four-dimensional Schwarzschild solution. We argue that such a solution can be viewed as the infinite radius limit of a braneworld black hole, and we provide convincing evidence for its existence. By deforming this solution in AdS we can then construct braneworld black holes of various sizes. We find that standard 4d gravity on the brane is recovered when the radius of the black hole on the brane is much larger than the radius of the bulk AdS space.
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Notes
- 1.
In this discussion we are implicitly assuming that \({\xi }^{a}\) is not a Killing vector.
- 2.
In Riemannian manifolds with boundaries, Anderson [29] has shown that imposing ξ a = 0 and Dirichlet or Neumann conditions for the induced metric on an given boundary gives rise to an ill posed problem.
- 3.
The Schwarzschild radial coordinate R is related to the compact radial coordinate r as \(R = \frac{R_{0}} {1-{r}^{2}}\).
- 4.
For a well-posed elliptic problem, as is our case, one should expect that such a solution exists and it is close to the solution in Sect. 3.
- 5.
Recall that a classical solution to the Einstein equations need only be C 2.
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Acknowledgements
It is a great pleasure to acknowledge the contributions of my collaborators J. Lucietti and especially T. Wiseman, without whom this work would not have been possible. I would also like to thank the organisers of the ERE2012 meeting in Guimarães (Portugal) for the invitation and for such a successful and enjoyable event. I am supported by an EPSRC postdoctoral fellowship [EP/H027106/1].
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Figueras, P. (2014). Braneworld Black Holes. In: GarcÃa-Parrado, A., Mena, F., Moura, F., Vaz, E. (eds) Progress in Mathematical Relativity, Gravitation and Cosmology. Springer Proceedings in Mathematics & Statistics, vol 60. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-40157-2_3
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