Towards uniqueness of degenerate axially symmetric Killing horizon

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 (non-symmetric) linear perturbation of near horizon geometry. The result is also applicable for other situations like Kundt’s class spacetimes or isolated horizons.


Introduction
Let us consider the following basic equation on a two-dimensional compact manifold 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 thatS AB vanishes. See also [4] or [9].
Some geometric consequences of the basic equation 1 (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][3][4][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 A||B is controlled by the equation but f := 1 2 ω A||B ε AB is an unknown function on a sphere. We have The integrability condition: implies that there exists non-empty open subset, where 12ω A ω A > R > 0. In this paper we analyze a linear perturbation of extremal Kerr solution. More precisely, in Sect. 2 we perform linearization of Eq. (1) around extremal Kerr solution (20). Axial symmetry of the background solution gives possibility to decompose linearized solution into Fourier series. Each Fourier mode v k fulfills ordinary differential Eq. (42). Using functional analysis methods we prove (in Appendix C) that there are no regular solutions for |k| > 8. We hope to check numerically the nonexistence of low modes for |k| ≤ 8.
Moreover, in this Section below we give some new results like equivalent formulations of the full nonlinear problem (cf. Theorem 4), equivalence between (2) and (18) or some properties contained in formulae at the end of Sect. 1.
Finally, some nontrivial calculations are shifted to the Appendix which also contains some useful formulae.

Transformation to linear problem
Let us denote For any domain, where ω B ω B > 0, equation (1) implies which simply means that the one-form A dx A is closed, and locally there exists a coordinate such that Moreover, from (1) we get hence the potential is a solution of the Poisson's equation: Remark If we choose one isolated point, where ω vanishes, then for a given metric g we have unique solution of the above Laplace-Beltrami equation (Green's function in the enlarged sense cf. [12]). For more isolated points we can take linear combination of such solutions. More precisely, let G x 0 be a unique solution (for a given metric g) of the Eq.

Two zeros of ω
Suppose ω A vanishes at two distinct points in a generic way (i.e. ω A||B is nondegenerate at those points). Then the Eqs. (6) and (5) extend (in the sense of distributions) as follows: Integration of the above equations on S 2 implies d 1 = d 2 = d and c 1 + c 2 = λ =(total volume of S 2 ). Hence, for A = ∂ A + ε A B ∂ B˜ the potentials ,˜ fulfill Laplace equations: and their solutions may be expressed in terms of generalized Green's functions on S 2 which are well defined as the distributions (they are integrable functions, smooth outside poles with log divergence at poles). Moreover, the trace of (1) may be expressed in terms of A as follows: 1.3 Equivalent form of the basic equation in terms of the covector A and its conformal rescaling Equations (5) and (6) together with (13) written as follows: for the conformally equivalent metric h AB = exp(−2u)g AB (cf. Eq. (22)) are almost the same Moreover, we have the following is not vanishing.
Proof Let us represent tensor ω A||B as a sum of three parts: skewsymmetric ( f ), traceless symmetric (τ AB ) and trace (τ ): We have to show that τ AB and τ are determined by Eq. (14-16). It is easy to check that (16) implies 2τ = K − ω 2 = ω A ||A . Moreover, (14) gives and similarly (15) implies Let us observe that any two-dimensional traceless symmetric tensor has only two independent components, hence the last two conditions determine τ AB uniquely in the following form: Finally, the above formula together with τ = 1 2 (K − ω 2 ) give the Eq. (2).
One can also check the following formula: which is equivalent to (2) but in terms of . Let us observe that B ||B A = 0 hence the symmetry of the tensor A||B implies and we obtain the following nice formulae: Moreover,

Linearization of basic equation around extremal Kerr
After introducing a new coordinate x := cos θ the (two-dimensional) extremal Kerr (see [6]) takes the following form: where a 2 := 2 1−x 2 1+x 2 and λ := The components of various objects for Kerr are the following: The nearby metric g we describe by conformal factor: and we get Let us denote by u B := h B A ∂ A u the gradient of u with respect to the metric h. We have Moreover, the Gaussian curvatures K h and K g for the conformally related metrics h and g respectively are related as follows This gives the following transformation for the right-hand side of (1): Using (27) and (29) we rewrite basic Eq. (1) as follows: Let us denote the linear part of the covector ω by Now we are ready to linearize basic equation.
Finally, for covector w A and conformal factor u in (26) the linearization of (1) takes the following form: where now ω and ∇ are background objects (corresponding to the Kerr solution (22)), and denotes the traceless symmetric part of the tensor t AB .
We show in Appendix B that after elimination of u A we get: where Remark The Eqs. (34-35) are conformally covariant with respect to the rescaling of the two-metric h. More precisely, the form of these equations is the same for two conformally related metrics provided that , * are vector fields and w and ω are covector fields. One can easily verify this observation multiplying the above equations by scalar density λ. One can also introduce another pair of variables: where a 2 := 2 1−x 2 1+x 2 . The formula (36) takes a simple form: Moreover, the inverse transformation implies the following form of the Eqs. (34-35) in terms of variables α, β: where , then the Eqs.

Boundary data
A small perturbation of Kerr data (20-21) does not destroy the number of two zeros for covector ω A . This is a simple consequence of the "inverse function theorem". More precisely, the non-vanishing curvature in the neighborhood of "spherical pole" (zero of where ω A vanishes. The freedom of global conformal transformations enables one to introduce "new conformal coordinates" in such a way that the spherical poles are always at the points where ω A vanishes. Hence, we can always assume that the perturbed ω A vanishes at spherical poles which implies zero (homogeneous) boundary data for linear perturbation w A or equivalently for v = (α, β) 3 . One can also show that respectively chosen conformal vector field X enables one to change w A → w A + L X ω A in such a way that it will vanish at a given point (see Appendix D).

Hypothesis The Eq. (41) has no regular solutions for homogeneous boundary data
Proof attempt Let us consider Fourier series for v: 5 and it vanishes only on the equator x = 0. 3 It is not obvious that w A = 0 corresponds to v = 0 and it is not true for k = 0.
It leads to ODE for v k (x): We check that v k vanishes at poles for |k| ≥ 1, because w A vanishes there. For k = 0 we have axial symmetry, hence we already have uniqueness in full nonlinear case, however it would be nice to check this fact independently.
For |k| > 8 we show in Appendix C that there are no regular solutions. There are some initial numerical results which confirm nonexistence hypothesis for |k| ≤ 8. We are going to check numerically the existence or nonexistence of low modes. The results will be published in a separate paper.
The above Hypothesis implies stability of the extremal Kerr horizon. It is true for |k| > 8 and adding this assumption we get Theorem 5. More precisely, Eq. (42) has no regular solutions for |k| > 8. The analytical proof is given in Appendix C.

Acknowledgments This research was supported by Polish Ministry of Science and Higher Education grant Nr N N201 372736.
Open Access This article is distributed under the terms of the Creative Commons Attribution License which permits any use, distribution, and reproduction in any medium, provided the original author(s) and the source are credited.

Appendix A: Kerr in conformal coordinates
The background metric (22) can be conformally related to unit sphere metric as follows: is the usual unit sphere metric and

B.1 Elimination of u A
We start from traceless part (33): The two independent components (AB) = (x x) and (AB) = (xφ) can be written as follows. Component (x x): or in an equivalent form (dividing by h x x ): Component (xφ): Finally we have (in matrix form) Let Hence Multiplying by A −1 we get or in simpler form Similarly, component φ: Equations (48) and (49) we can rewrite in covariant form:

B.2 Equations for w A
The trace and curl of u A gives: Using formula and Eq. (54) we obtain where by R A BC D we denote Riemann curvature tensor. We have where by R AB we denote Ricci tensor. The symmetry of Ricci Hence Using formula (56) and Eq. (55) we get Using identity ε AB ε AC = −δ B C , we get and finally we obtain (34)

Appendix C: Proof for large k
Stability for the extremal Kerr leads to the following equation: where • a 2 = 2 1−x 2 1+x 2 .

Theorem 5 Equation (57) has no solutions for |k| > 8.
Proof For functions f, g : [−1, 1] → C 2 let us define a standard scalar product: Let us consider an operator X := a d dx and its hermitian conjugate X * = − d dx a. The Eq. (57) takes the form: The left-hand side we denote by Lv k , where L is a linear operator and v k ∈ ker L, i.e. Lv k = 0. For (v k |Lv k ) we have: Introducing real numbers x := X v k v k , y := 1 a v k v k we obtain: and absolute value one can estimate as follows: From Cauchy-Schwarz inequality and from Av ≤ A v we get: Hence x 2 + k 2 y 2 ≤ x a B + |k|y aC + D or in an equivalent form:

The CVF equation
∇ A X B + ∇ B X A = ∇ C X C g AB applied to our field X reduces to They can be written in an equivalent form: 2 1 − x 2 (A + B) cos φ, and finally we obtain system of ODE's:

4B
1 which leads to the second order ODE for the function B: B (x). We get the following solution: If we assume that the field X vanishes at one "pole" (x = ±1) we obtain the relation for constants C i : C 2 = ±C 1 . For C 2 = −C 1 we have: Finally the CVF X takes the form: