Black holes in higher dimensional space-times

Gravitational Field of a Spinning Mass as an Example of Algebraically Special Metrics

Separable wave equations with source terms are presented for electromagnetic and gravitational perturbations of an uncharged, rotating black hole. These equations describe the radiative field completely, and also part of the nonradiative field. Nontrivial, source-free, stationary perturbations are shown not to exist. The barrier integral governing synchrotron radiation from particles in circular orbits is shown to be the same as for scalar radiation. Future applications (stability of rotating black holes, "spin-down," superradiant scattering, floating orbits) are outlined.

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Rotating black holes: separable wave equations for gravitational and electromagnetic perturbations.

It is a deceptively simple question to ask how acoustic disturbances propagate in a non-homogeneous flowing fluid. Subject to suitable restrictions, this question can be answered by invoking the language of Lorentzian differential geometry. This paper begins with a pedagogical derivation of the following result: if the fluid is barotropic and inviscid, and the flow is irrotational (though possibly time dependent), then the equation of motion for the velocity potential describing a sound wave is identical to that for a minimally coupled massless scalar field propagating in a (3 + 1)-dimensional Lorentzian geometry The acoustic metric governing the propagation of sound depends algebraically on the density, flow velocity, and local speed of sound. Even though the underlying fluid dynamics is Newtonian, non-relativistic, and takes place in flat space plus time, the fluctuations (sound waves) are governed by an effective (3 + 1)-dimensional Lorentzian spacetime geometry. This rather simple physical system exhibits a remarkable connection between classical Newtonian physics and the differential geometry of curved (3 + 1)-dimensional Lorentzian spacetimes, and is the basis underlying a deep and fruitful analogy between the black holes of Einstein gravity and supersonic fluid flows. Many results and definitions can be carried over directly from one system to another. For example, it will be shown how to define the ergosphere, trapped regions, acoustic apparent horizon, and acoustic event horizon for a supersonic fluid flow, and the close relationship between the acoustic metric for the fluid flow surrounding a point sink and the Painleve-Gullstrand form of the Schwarzschild metric for a black hole will be exhibited. This analysis can be used either to provide a concrete non-relativistic analogy for black-hole physics, or to provide a framework for attacking acoustics problems with the full power of Lorentzian differential geometry.

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Acoustic black holes: horizons, ergospheres and Hawking radiation

The following theorem is established. Among all static, asymptotically flat electrovac fields with closed, simply-connected equipotential surfacesg0 0=const.. the only ones which have regular event horizonsg0 0=0 are the Reissner-Nordstrom family of spherisymmetric solutions withm≧G1/2|e|/c. In the special case where the gravitational coupling of the electromagnetic energy density is neglected (G=0) all solutions are computed explicitly, thus extending an earlier result ofGinzburg for a magnetic dipole inSchwarzschild's space-time. Possible implications for gravitational collapse are briefly discussed.

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Event horizons in static electrovac space-times

Kruskal's transformation of the Schwarzschild metric is generalized to apply to the stationary, axially symmetric vacuum solution of Kerr, and is used to construct a maximal analytic extension of the latter. In the low angular momentum case, a2 < m2, this extension consists of an infinite sequence Einstein‐Rosen bridges joined in time by successive pairs of horizons. The number of distinct asymptotically flat sheets in the extended space can be reduced to four by making suitable identifications. Several properties of the Kerr metric, including the behavior of geodesics lying in the equatorial plane, are examined in some detail. Completeness is demonstrated explicitly for a special class of geodesics, and inferred for all those that do not strike the ring singularity.

Maximal analytic extension of the Kerr metric

A class of solutions of the time‐symmetric initial‐value equations for gravitation and electromagnetism is obtained on a two‐sheeted manifold containing N Einstein‐Rosen bridges. The initial metric tensor and electric field are expressed in terms of a pair of harmonic functions, called ``metric potentials,'' which are required to be analytic and asymptotically flat and to satisfy certain match‐up conditions; these potentials are then determined by applying the method of inversion images. The particular case of two identical Einstein‐Rosen bridges is examined in detail, and is shown to correspond to a wormhole in an asymptotically flat (one‐sheeted) universe. A charge and ``renormalized mass'' are defined for each Einstein‐Rosen bridge, as well as an interaction energy (gravitational plus electrostatic potential energy) for the system as a whole. Expressions for these and other physical characteristics are evaluated explicitly in the two‐body case.

Richard W. Lindquist

Papers

Maximal analytic extension of the Kerr metric

Initial‐Value Problem on Einstein‐Rosen Manifolds