Spectral Geometry of Cosmological and Event Horizons for Kerr-Newman de Sitter metrics

We study the Laplace spectra of the intrinsic instantaneous metrics on the event and cosmological horizons of a Kerr-Newman de Sitter space-time and prove that the spectral data from these horizons uniquely determine the space-time. This is accomplished by exhibiting formulae relating the parameters of the space-time metric to the traces of invariant and equivariant Green's operators associated with these Laplacians. In particular, an interesting explicit formula for the cosmological constant is found.


Introduction
The Kerr-Newman de Sitter metric exhibits four horizons. Three of these have physical interpretations as: a Cauchy (inner) horizon, a black-hole event horizon, and a cosmological horizon. In this paper, we study the two largest: the event and cosmological horizons.
The space-time metric induces on each of these horizons an S 1 -invariant Riemannian metric which we call intrinsic instantaneous metrics. Each of these metrics' Green's operators yield S 1 -invariant and k-equivariant trace formulae. In [3] we proved, for the case of a single horizon in the Kerr metric, that the two trace formulae uniquely determine the two parameters of this space-time. In the present paper there are four parameters which must be determined: the mass, angular momentum, charge, and cosmological constant. It is, therefore, fortunate that there are two horizons each providing two distinct types of trace formulae which produce a non-linear system of four equations in the parameters which can be shown to have a unique solution. Each of the parameters is given semi-explicitly in terms of the spectrum but in the case of the cosmological constant we are, in fact, able to find an explicit formula.
In general, of course, the equation ∆ r = 0 has four roots. Here we are only interested in two positive real roots, denoted by r e and r c , which correspond to the event and cosmological horizons respectively. We assume, of course, that r e < r c .
Let r 0 denote either r e or r c . To obtain the intrinsic instantaneous metrics on these surfaces we pull back the space-time metric (1) to the surface defined by r = r 0 (so that dr = 0) and dt = 0 to obtain two dimensional Riemannian metrics on each of the event and cosmological horizons. Both take the form: Following the notation of [10],we define the scale parameter by η = r 2 0 + a 2 and the distortion parameter by β = a √ r 2 0 +a 2 . We also define a new parameter with the change of variable x = − cos θ one finds that the horizon metric is The area of this metric is A = 4πη 2 (1 − ξ) (see [2]). It is well known that the Gauss curvature of such a metric takes the form K(x) = −f ′′ (x)/(2η 2 (1 − ξ)) so that in this case from (5) the curvature is: The special case β 2 = ξ gives a constant curvature metric on a horizon, but yields only one horizon corresponding to a positive solution of ∆ r = 0.

Spectrum of S 1 invariant metrics
For any Riemannian manifold with metric g ij the Laplacian, in local coordinates, is given by This is the Riemannian version of the Klein-Gordon, or D'Alembertian, or wave operator usually denoted by .
In this section we outline some our previous work on the spectrum of the Laplacian on S 1 invariant metrics on S 2 . The interested reader may consult [4], [5], [6], and [16] for further details.
To simplify the discussion the area of the metric is normalized to A = 4π for this section only. The metrics we study have the form: where . In this form, it is easy to see that the Gauss curvature of this metric is given by K(x) = (−1/2)f ′′ (x). The canonical (i.e. constant curvature) metric is obtained by taking f (x) = 1 − x 2 and the metric (4) is a homothety (scaling) of a particular example of the general form (7).
The Laplacian for the metric (7) is Let λ be any eigenvalue of −∆. We will use the symbols E λ and dim E λ to denote the eigenspace for λ and it's multiplicity (degeneracy) respectively. In this paper the symbol λ m will always mean the mth distinct eigenvalue. We adopt the convention λ 0 = 0. Since S 1 (parametrized here by 0 ≤ φ < 2π) acts on (M, g) by isometries we can separate variables and because dim E λm ≤ 2m + 1 (see [6] for the proof), the orthogonal decomposition of E λm has the special form is the "eigenspace" (it might contain only 0) of the ordinary differential operator f (x) with suitable boundary conditions. It should be observed that dim W k ≤ 1, a value of zero for this dimension occuring when λ m is not in the spectrum of L k .
The set of positive eigenvalues is given by Spec(dl 2 ) = k∈Z SpecL k and consequently the nonzero part of the spectrum of −∆ can be studied via the spectra SpecL k = {0 < λ 1 k < λ 2 k < · · · < λ j k < · · · }∀k ∈ Z. The eigenvalues λ j 0 in the case k = 0 above are called the S 1 invariant eigenvalues since their eigenfunctions are invariant under the action of the S 1 isometry group. If k = 0 the eigenvalues are called k equivariant or simply of type k = 0. Each L k has a Green's operator, Γ k : (H 0 (M )) ⊥ → L 2 (M ), whose spectrum is {1/λ j k } ∞ j=1 , and whose trace is defined by The formulas of present interest were derived in [4] and [5] and are given by and Remark. One must be careful with the definition of γ 0 since λ 0 0 = 0 is an S 1 invariant eigenvalue of −∆. To avoid this difficulty we studied the S 1 invariant spectrum of the Laplacian on 1-forms in [5] and then observed that the nonzero eigenvalues are the same for functions and 1-forms.

Spectral Determination of Horizons
In case f (x) is given by (5) the metric (4) is related to (7) via the homothety ds 2 r0 = η 2 (1 − ξ)dl 2 , and it is well known that so that, after an elementary integration and some algebra, the trace formulae ( (9) and (10)), for either horizon, take the form and An immediate consequence of (12) is that the area of the metric has a representation for each k ∈ N given by A = 4πkγ k .
We now define: Then equation (11) has the form: One now has, for each of the two physical horizons r = r e and r = r c , corresponding parameters and traces denoted by η e , β e , γ e k and η c , β c , γ c k respectively. Using the definitions of the parameters and the system of equations consisting of (12) and (15) (for k = 1) for both horizons, one easily obtains the following formula for the cosmological constant. Theorem 1. If a = 0, Λ > 0, and r c = r e then After defining the pairs of equations (12) and (15) for each horizon yield And once it is verified that (17) is invertible, we obtain From the definition (3), the angular momentum parameter is given by: From (12) (k = 1), (19), and (20) one can solve for r e and r c respectively. The resulting equations are: and we have Λ, a 2 , r e and r c in terms of the traces. Finally, after substituting these (distinct) values of r into the equation ∆ r = 0, a nonsingular linear system in the variables m and Q 2 is obtained and, therefore, m and Q 2 are uniquely determined. We have thus proved: Theorem 2. The Kerr-Newman de Sitter space-time is uniquely determined by the union of the spectra of the cosmological and event horizons.

Discussion
The spectra we study in this paper should not be confused with the quasinormal mode frequencies arising from the angular part of the Teukolsky master equation (see [1], [7], [13], [9], [12] and references therein, among many others). On the other hand, in certain limiting cases for the parameters, they do coincide but we will not pursue this comparison here.
The reader may have noticed that Theorem 2 is consistent with the holographic principle ( [11], [14]) in as much as the structure of the (3 + 1 dimensional) Kerr-Newman de-Sitter space-time is encoded in the intrinsic spectral data of the two (two-dimensional) horizon surfaces.
Many of the calculations in this paper are independent of which pair of horizons are being used. This leads to the conjecture that one can use the spectra of the inner Cauchy horizon, together with that of the event horizon to obtain the uniqueness result and the formula for Λ.

Acknowledgements
A special thanks goes out to María del Rio for her support, especially for the second author, during the writing of this paper. This work was partially supported by the NSF Grants: Model Institutes for Excellence and AGMUS Institute of Mathematics at UMET.