Towards uniqueness of degenerate axially symmetric Killing horizon
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Towards uniqueness of degenerate axially symmetric Killing horizon Jacek Jezierski · Bartek Kaminski ´
Received: 13 October 2012 / Accepted: 3 February 2013 / Published online: 22 February 2013 © The Author(s) 2013. This article is published with open access at Springerlink.com
Abstract For near horizon geometry we examine the linearized equations around extremal Kerr horizon (which is a unique axially symmetric near horizon geometry) and give some arguments towards stability of this horizon with respect to generic (nonsymmetric) linear perturbation of near horizon geometry. The result is also applicable for other situations like Kundt’s class spacetimes or isolated horizons. Keywords Near horizon geometry · Extremal Kerr horizon · Kundt’s class spacetimes
1 Introduction Let us consider the following basic equation on a two-dimensional compact manifold ω A||B + ω B||A + 2ω A ω B = R AB ,
(1)
where ω A dx A is a covector field, || denotes covariant derivative compatible with the metric g AB , and R AB is its Ricci tensor. The Eq. (1) is a starting point of our considerations and it is a special case of (3.7) in [1], if we assume that S˜ AB vanishes. See also [4] or [9].
J. Jezierski (B) · B. Kami´nski Department of Mathematical Methods in Physics, Faculty of Physics, University of Warsaw, ul. Ho˙za 74, 00-682 Warsaw, Poland e-mail: [email protected] B. Kami´nski e-mail: [email protected]
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J. Jezierski, B. Kami´nski
Some geometric consequences of the basic equation1 (resulting from Einstein equations) were discussed in [6]. This is a non-linear PDE for unknown covector field and unknown Riemannian structure on the two-dimensional manifold. It appears in the context of Kundt’s class metrics (cf. [7,8]), degenerate Killing horizons [4,9], or vacuum degenerate isolated horizons [1,10,11]. Several important results are already proved, like topological rigidity of the horizon and integrability conditions (cf. [6]). Moreover, when the one-form ω B dx B is closed (e.g. static degenerate horizon [4]) there are no solutions of (1). The transformation (4) of a covector ω A leads to (partially) linear problem (invented in [6]) and simplifies the proof of the uniqueness of extremal Kerr for axially symmetric horizon. However, the problem of the existence of non-symmetric solutions to the basic equation remains open. The solutions of this equation enables one to construct near horizon metric (cf. [2–5,9]), Kundt’s class spacetime or isolated horizon neighborhood. In [6] the following results were proved: Theorem 1 For any Riemannian metric g AB on a two-dimensional, compact, connected manifold B with no boundary and genus g ≥ 2 there are no solutions of basic equation. Theorem 2 For any Riemannian metric g AB on a two-dimensional torus Eq. (1) possesses only trivial solutions ω A ≡ 0 ≡ K and the metric g AB is flat. Theorem 3 There are no solutions of Eq. (1) with the following properties: • ω A = 0 only at finite set of points, • B is a sphere with non-negative Gaussian curvature. The symmetric part of ω
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