Uniqueness of Smooth Stationary Black Holes in Vacuum: Small Perturbations of the Kerr Spaces

Abstract

Following the program started in [24], we attempt to remove the analyticity assumption in the the well known Hawking-Carter-Robinson uniqueness result for regular stationary vacuum black holes. Unlike [24], which was based on a tensorial characterization of the Kerr solutions, due to Mars [29], we rely here on Hawking's original strategy, which is to reduce the case of general stationary space-times to that of stationary and axi-symmetric spacetimes for which the Carter-Robinson uniqueness result holds. In this reduction Hawking had to appeal to analyticity. Using a variant of the geometric Carleman estimates developed in [24], in this paper we show how to bypass analyticity in the case when the stationary vacuum space-time is a small perturbation of a given Kerr solution. Our perturbation assumption is expressed as a uniform smallness condition on the Mars-Simon tensor. The starting point of our proof is the new local rigidity theorem established in [2].

On the classification of stationary electro-vacuum black holes

“…We further note that a new approach to Hawking's rigidity without analyticity [1,2] as yield significant breakthroughs in the vacuum case. According to A.D. Ionescu (private communication) the generalization of the results to electro-vacuum should follow by similar techniques.…”

“…Also, no previous work known to us establishes the asymptotic behavior, as needed for the proof of uniqueness, of the relevant harmonic maps. More specifically: the necessity to control the behavior at points where the horizon meets the rotation axis, prior to [13], seems to have been neglected; at infinity, which requires special attention in the electro-vacuum case, part of the necessary estimates were imposed as extra conditions, beyond asymptotic flatness; (2) also, an apparent disregard for the singular character, at the axis, of the hyperbolic distance (4.14) between the maps, even at large distance, appears to be the norm. A detailed asymptotic analysis is carried out in Section 5.…”

Bulk supertranslation memories: A concept reshaping the vacua and black holes of general relativity

“…Assuming zero angular momentum, the final black hole is constrained by uniqueness theorems to be diffeomorphic to the Schwarzschild black hole [27][28][29][30][31]. However, the diffeomorphism might be singular inside the Killing horizon and act as a source for the supertranslation field.…”

“…Since all physical processes involved in merging, accretion and collapse are highly non-spherically symmetric, the ≥ 2 spherical harmonics in the right-hand side of (2) will be of order of the mass. Assuming C| initial = ∆C| i 0 = 0 (which would have to be assessed) and after solving for the differential operator, the supertranslation field will be O(M ).Assuming zero angular momentum, the final black hole is constrained by uniqueness theorems to be diffeomorphic to the Schwarzschild black hole [27][28][29][30][31]. However, the diffeomorphism might be singular inside the Killing horizon and act as a source for the supertranslation field.…”

On the Construction of a Geometric Invariant Measuring the Deviation from Kerr Data

“…A peculiarity of this procedure is that if the spacetime is not stationary, the approximate Killing vector associated with a time translation does not have the same asymptotic behaviour as a time translation. 1 The Killing spinor initial data equations consist of three conditions: one of them differential (the spatial Killing spinor equation) 2 and two algebraic conditions. Following the spirit of [15] we construct a generalisation of the spatial Killing spinor equation-the approximate Killing A precise formulation will be given in the main text.…”