6 Stationary and Axisymmetric Space-Times5 Applications of the Coset 5.2 Mass Formulae

5.3 The Israel-Wilson Class 

A particular class of solutions to the stationary EM equations is obtained by requiring that the Riemannian manifold tex2html_wrap_inline3827 is flat [102Jump To The Next Citation Point In The Article]. For tex2html_wrap_inline4421, the three-dimensional Einstein equations obtained from variations of the effective action (28Popup Equation) with respect to tex2html_wrap_inline3837 become Popup Footnote

  equation1117

Israel and Wildon [102Jump To The Next Citation Point In The Article] have shown that all solutions of this equation fulfill tex2html_wrap_inline4429 . In fact, it is not hard to verify that this ansatz solves Eq. (43Popup Equation), provided that the complex constants tex2html_wrap_inline4431 and tex2html_wrap_inline4433 are subject to tex2html_wrap_inline4435 . Using asymptotic flatness, and adopting a gauge where the electro-magnetic potentials and the twist potential vanish in the asymptotic regime, one has tex2html_wrap_inline4437 and tex2html_wrap_inline4439, and thus

  equation1144

It is crucial that this ansatz solves both the equation for tex2html_wrap_inline3933 and the one for tex2html_wrap_inline4199 : One easily verifies that Eqs. (29Popup Equation) reduce to the single equation

  equation1152

where tex2html_wrap_inline3947 is the three-dimensional flat Laplacian.

For static, purely electric configurations the twist potential Y and the magnetic potential tex2html_wrap_inline4151 vanish. The ansatz (44Popup Equation), together with the definitions of the Ernst potentials, tex2html_wrap_inline4453 and tex2html_wrap_inline4455 (see Sect. 4.5), yields

  equation1161

Since tex2html_wrap_inline4457, the linear relation between tex2html_wrap_inline4045 and the gravitational potential tex2html_wrap_inline4461 implies tex2html_wrap_inline4463 . By virtue of this, the total mass and the total charge of every asymptotically flat, static, purely electric Israel-Wilson solution are equal:

  equation1171

where the integral extends over an asymptotic two-sphere. Popup Footnote The simplest nontrivial solution of the flat Poisson equation (45Popup Equation), tex2html_wrap_inline4471, corresponds to a linear combination of n monopole sources tex2html_wrap_inline4475 located at arbitrary points tex2html_wrap_inline4477,

  equation1182

This is the Papapetrou-Majumdar (PM) solution [143], [128], with spacetime metric tex2html_wrap_inline4479 and electric potential tex2html_wrap_inline4481 . The PM metric describes a regular black hole spacetime, where the horizon comprises n disconnected components. Popup Footnote In Newtonian terms, the configuration corresponds to n arbitrarily located charged mass points with tex2html_wrap_inline4487 . The PM solution escapes the uniqueness theorem for the Reissner-Nordström metric, since the latter applies exclusively to space-times with M > | Q |.

Non-static members of the Israel-Wilson class were constructed as well [102], [145]. However, these generalizations of the Papapetrou-Majumdar multi black hole solutions share certain unpleasant properties with NUT spacetime [140] (see also [15], [136]). In fact, the work of Hartle and Hawking [81], and Chrusciel and Nadirashvili [42] strongly suggests that - except the PM solutions - all configurations obtained by the Israel-Wilson technique are either not asymptotically Euclidean or have naked singularities. In order to complete the uniqueness theorem for the PM metric among the static black hole solutions with degenerate horizon, it basically remains to establish the equality M = Q under the assumption that the horizon has some degenerate components. Until now, this has been achieved only by requiring that all components of the horizon have vanishing surface gravity and that all ``horizon charges'' have the same sign [90].



6 Stationary and Axisymmetric Space-Times5 Applications of the Coset 5.2 Mass Formulae

image Stationary Black Holes: Uniqueness and Beyond
Markus Heusler
http://www.livingreviews.org/lrr-1998-6
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