Uniqueness theorem for static black hole solutions of  -models in higher dimensions

Abstract

We prove the uniqueness theorem for self-gravitating non-linear σ-models in higher dimensional spacetime. Applying the positive mass theorem we show that Schwarzschild-Tagherlini spacetime is the only maximally extended, static asymptotically flat solution with non-rotating regular event horizon with a constant mapping.Nowadays, much effort is being devoted to the study of mathematical topics related to the black hole equilibrium states. The pioneering investigations in this field were attributed to Israel [1], Müller zum Hagen et al.[2] and Robinson [3]. The alternative approach to the problem of the uniqueness of black hole solutions was proposed by and then strengthened to the Einstein-Maxwell (EM) black holes [5,6]. Heusler [7] comprised the magnetically charged RN solution and static Einstein-σ-model case [8]. Recently the classification of static of vacuum black holes was finished [9]. The condition of non-degeneracy of the event horizon was removed and

Scalar Fields in Multidimensional Gravity. No-Hair and Other No-Go Theorems

“…One can also note that our no-hair theorem is in a complementarity relation with a recent black hole uniqueness theorem [21] (see [22] for a review). In Ddimensional general relativity coupled to the σ -model (3) with V ≡ 0 , it has been proved without assuming spherical symmetry at the outset that "the only black hole solution with a regular, non-rotating event horizon in an asymptotically flat, strictly stationary domain of outer communication is the Schwarzschild-Tangherlini solution with a constant mapping φ" [21].…”

“…In Ddimensional general relativity coupled to the σ -model (3) with V ≡ 0 , it has been proved without assuming spherical symmetry at the outset that "the only black hole solution with a regular, non-rotating event horizon in an asymptotically flat, strictly stationary domain of outer communication is the Schwarzschild-Tangherlini solution with a constant mapping φ" [21]. In contrast to that, our Theorem 3 applies to σ -models with arbitrary V ( ϕ) ≥ 0 but selects the Tangherlini solution among spherically symmetric configurations.…”

Higher dimensional black holes and supersymmetry

“…A uniqueness theorem has been proved for non-degenerate higher dimensional static black holes in Einstein-Maxwell [12] and Einstein-Maxwell-dilaton theory [13], and for Einstein gravity coupled to a σ-model [14]. The uniqueness assumption for static supersymmetric black holes in higher dimensions therefore seems plausible.…”

Mechanics of multidimensional isolated horizons

“…Additionally, in the second case we assume that the metric tensor fields are asymptotically flat 6 .…”

“…Nevertheless, only in the static, non-rotating case do there exist general uniqueness theorems for arbitrary dimension and for σ -model, vacuum and charged black holes [6][7][8].…”