4.6 Classification theorems for KK-black holes

As usual, the static case requires separate consideration. The first classification results addressed static five-dimensional solutions with KK asymptotics and with a ℝ × U (1) factor in the group of isometries. In such a setting, the Kaluza–Klein reduction leads to gravity coupled with a Maxwell field F and a “dilaton” field ϕ, with a Lagrangean
L = R − 2 (∇ ϕ )2 − e−2αϕ|F |2,
where √ -- α = 3. In the literature one also considers more general theories where α does not necessarily take the Kaluza–Klein value. All current uniqueness proofs require that the mass, the Maxwell charges, and the dilaton charge satisfy a certain genericity condition, and that all horizon components have non-vanishing surface gravity. When α = 1, Mars and Simon [224] show that the generic static solutions belong to the family found by Gibbons and Maeda [131Jump To The Next Citation Point, 126Jump To The Next Citation Point, 122Jump To The Next Citation Point]. For other values of α, in particular for the KK value, a purely electric or purely magnetic configuration is assumed, and then the same conclusion is reached. The result is an improvement on the original uniqueness theorems of Simon [294] and Masood-ul-Alam [226], and has been generalized to higher dimensions in [129]. The analyticity assumption, which is implicit in all the above proofs, can be removed using [72Jump To The Next Citation Point].

The remaining classfication results assume cohomogeneity-two isometry actions [169Jump To The Next Citation Point]:

Theorem 4.1 Let (Mi, gi), i = 1,2, be two + I-regular, (n + 1)-dimensional, n ≥ 3, stationary toroidal Kaluza–Klein spacetimes, with five asymptotically-large dimensions (N = 4). Assume, moreover, that the event horizon is connected and mean non-degenerate. If the interval structure and the set of angular momenta coincide, then the domains of outer communications are isometric.

This theorem generalizes previous results by the same authors [167, 168] as well as a uniqueness result for a connected spherical black hole of [240Jump To The Next Citation Point].

The proof of Theorem 4.1 can be outlined as follows: After establishing the, mainly topological, results of Sections 4.3, 4.4 and 4.5, the proof follows closely the arguments for uniqueness of 4-dimensional stationary and axisymmetric electrovacuum black holes. First, a generalized Mazur identity is valid in higher dimensions (see [218Jump To The Next Citation Point, 31Jump To The Next Citation Point] and Section 7.1). From this Hollands and Yazadjiev show that (compare the discussion in Sections 8.4.1 and 8.4.2)

Δ ψ ≥ 0, (4.2 ) δ
where Δ δ is the flat Laplacian on ℝ3, the function
ψ : ℝ3 ∖ {z = 0} → ℝ,
is defined as
ψ = Trace(Φ2 Φ−1 1− 𝟙),
the Φi’s, i = 1,2, are the Mazur matrices [169Jump To The Next Citation Point, Eq. (78)] associated with the two black-hole spacetimes that are being compared. In terms of twist potentials ((i)χ ) i and metric components of the axisymmetric Killing vectors (generators of the toroidal symmetry)
(i) ((i) ) fmn = g(km, kn) ,i = 1,2,
we have the following explicit formula (see also [218, 240])
( )( ) (1)f (1)fij (1)χi − (2) χi (1)χj − (2) χj ( ) ψ = − 1 + (2)--+ ---------------(2)---------------- + (1)f ij (2)fij − (1)fij , (4.3 ) f f
where ( ) (i)f = det (i)fmn , and where (1)fij is the matrix inverse to (1)fij.

It should be noted that this provides a variation on Mazur’s and the harmonic map methods (see Sections 3.2.4 and 3.2.5), which avoids some of their intrinsic difficulties. Indeed, the integration by parts argument based on the Mazur identity requires detailed knowledge of the maps under consideration at the singular set { ρ = 0}, while the harmonic map approach requires finding, and controlling, the distance function for the target manifold. (In some simple cases ψ is the desired distance function, but whether this is so in general is unclear.) The result then follows by a careful analysis of the asymptotic behavior of the relevant fields; such analysis was also carried out in [169].

In this context, the degenerate horizons suffer from the supplementary difficulty of controlling the behavior of the fields near the horizon. One expects that an exhaustive analysis of near-horizon geometries would allow one to settle the question; some partial results towards this can be found in [204, 203, 202Jump To The Next Citation Point, 164].


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