Article
Ida, D.: A boundary value problem for the fivedimensional stationary rotating black holes. Phys. Rev. D 69, 124005
Kyoto University, Kioto, Kyōto, Japan
Physical Review D (Impact Factor: 4.64). 01/2004; 69(12). DOI: 10.1103/PhysRevD.69.124005 Source: arXiv
ABSTRACT
We study the boundary value problem for the stationary rotating black hole solutions to the fivedimensional vacuum Einstein equation. Assuming the two commuting rotational symmetry and the sphericity of the horizon topology, we show that the black hole is uniquely characterized by the mass, and a pair of the angular momenta. Comment: 16 pages, no figures
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 "In the analytic context recent uniqueness theorems for vacuum stationary black holes in higher dimensions [9] [10] also phrase the problem as an elliptic system, and so far have restricted attention to D dimensional spacetimes with D2 commuting Killing vectors, where again the WeylPapapetrou form is used (see also [11]). "
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ABSTRACT: Introduction Whilst black holes in four dimensions are well mannered, being spherically symmetric or having special algebraic properties which enable them to be found analytically, moving beyond four dimensions many solutions of interest appear to have no manners whatsoever. The problem of finding these unruly black holes becomes that of solving a nonlinear coupled set of partial differential equations (PDEs) for the metric components given by the Einstein equations. In general it is unlikely that closedform analytic solutions will be found for many of the exotic black holes discussed earlier in this book. If we are to understand their properties then we must turn to numerical techniques to tackle the PDEs that describe them. It is the purpose of this chapter to develop general numerical methods to address the problem of finding static and stationary black holes. Surely the phrase “the devil is in the detail” could not have a truer application than to numerics. The emphasis of this chapter will be to provide a road map in which we formulate the problem in as unified, elegant and geometric a way as possible. We will also discuss concrete algorithms for solving the resulting formulation, but the extensive details of implementation will not be addressed, probably much to the reader's relief. Such details can be found in the various articles cited in this chapter. 
 "In addition at axes of symmetry associated to the fixed action of a compact R a , det G AB will vanish. Following the uniqueness theorem treatment of 4D stationary black holes as an elliptic problem on the two dimensional Riemannian orbit space bounded by the horizon and axes of symmetry (as for example discussed in [6]) and its generalization to D dimensional metrics with (D − 2) commuting Killing vectors [7] [8] which is treated in the same manner, we make the follow key assumption; • We assume that the orbit space base manifold (M, h) is a smooth Riemannian manifold with boundaries given by the horizons and axes of symmetry of the R a that generate rotational isometries. "
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ABSTRACT: The Harmonic Einstein equation is the vacuum Einstein equation supplemented by a gauge fixing term which we take to be that of DeTurck. For static black holes analytically continued to Riemannian manifolds without boundary at the horizon this equation has previously been shown to be elliptic, and Ricci flow and Newton's method provide good numerical algorithms to solve it. Here we extend these techniques to the arbitrary cohomogeneity stationary case which must be treated in Lorentzian signature. For stationary spacetimes with globally timelike Killing vector the Harmonic Einstein equation is elliptic. In the presence of horizons and ergoregions it is less obviously so. Motivated by the Rigidity theorem we study a class of stationary black hole spacetimes, considered previously by Harmark, general enough to include the asymptotically flat case in higher dimensions. We argue the Harmonic Einstein equation consistently truncates to this class of spacetimes giving an elliptic problem. The Killing horizons and axes of rotational symmetry are boundaries for this problem and we determine boundary conditions there. As a simple example we numerically construct 4D rotating black holes in a cavity using Anderson's boundary conditions. We demonstrate both Newton's method and Ricci flow to find these Lorentzian solutions. 
 "The proof relies heavily on the analysis in [7], as well as on the results in [9] which are reviewed in our context in Section 3. Next, inspection of the uniqueness arguments in [22] [24] [27] [28] shows that serious difficulties arise there if the orbits of the isometry group cease to be timelike on A . The second main result of our work is Theorem 6.1 below, that this does not occur. "
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ABSTRACT: We consider (n+1)dimensional, stationary, asymptotically flat, or KaluzaKlein asymptotically flat black holes, with an abelian $s$dimensional subgroup of the isometry group satisfying an orthogonal integrability condition. Under suitable regularity conditions we prove that the area of the group orbits is positive on the domain of outer communications, vanishing only on its boundary and on the "symmetry axis". We further show that the orbits of the connected component of the isometry group are timelike throughout the domain of outer communications. Those results provide a starting point for the classification of such black holes. Finally, we show nonexistence of zeros of static Killing vectors on degenerate Killing horizons, as needed for the generalisation of the static nohair theorem to higher dimensions.