The Quantization of a Kerr-AdS Black Hole

Abstract

We apply our model of quantum gravity to a Kerr-AdS space-time of dimension , , where all rotational parameters are equal, resulting in a wave equation in a quantum space-time which has a sequence of solutions that can be expressed as a product of stationary and temporal eigenfunctions. The stationary eigenfunctions can be interpreted as radiation and the temporal ones as gravitational waves. The event horizon corresponds in the quantum model to a Cauchy hypersurface that can be crossed by causal curves in both directions such that the information paradox does not occur. We also prove that the Kerr-AdS space-time can be maximally extended by replacing in a generalized Boyer-Lindquist coordinate system the variable by such that the extended space-time has a timelike curvature singularity in .

1. Introduction

In general relativity the Cauchy development of a Cauchy hypersurface is governed by the Einstein equations, where of course the second fundamental form of has also to be specified.

In the model of quantum gravity we developed in a series of papers [1–7] we pick a Cauchy hypersurface, which is then only considered to be a complete Riemannian manifold of dimension , and define its quantum development to be special solutions of the wave equationdefined in the space-timeThe Laplacian is the Laplacian with respect to , is the scalar curvature of the metric, is the time coordinate defined by the derivation process of the equation, and a cosmological constant. If other physical fields had been present in the Einstein equations then the equation would contain further lower order terms (cf. [5]), but a negative cosmological constant would always have to be present even if the Einstein equations would only describe the vacuum.

Using separation of variables we proved that there is a complete sequence of eigenfunctions (or a complete set of eigendistributions) of a stationary eigenvalue problem and a complete sequence of eigenfunctions of a temporal implicit eigenvalue problem, where plays the role of an eigenvalue, such that the functionsare solutions of the wave equation (cf. [4, Section ] and [5, 6]).

In a recent paper [8] we applied this model to quantize a Schwarzschild-AdS space-time and we shall prove in the present paper that similar arguments can also be used to quantize a Kerr-AdS space-time with a rotating black hole.

We consider an odd-dimensional Kerr-AdS space-time , , where all rotational parameters are equal:and where we also setReplacing the coordinate in a generalized Boyer-Lindquist coordinate system bywe shall prove that in the new coordinate system the metric is smooth in the intervaland that the extended space-time has a timelike curvature singularity in (cf. Lemma 9).

For the quantization we first assume that there is a nonempty interior black hole region which is bounded by two horizons:where the outer horizon is the event horizon. Picking a Cauchy hypersurface in of the formwe shall prove that the induced metric of the Cauchy hypersurface can be expressed in the formwhere, and is a smooth Riemannian metric on depending on , and the cosmological constant . The metric in (10) is free of any coordinate singularity; hence we can let tend to such that the Cauchy hypersurfaces converge to a Riemannian manifold which represents the event horizon at least topologically. Furthermore, the Laplacian of the metric in (10) comprises a harmonic oscillator with respect to which enables us to write the stationary eigenfunctions in the formwhere is an eigenfunction of the elliptic operatorwhere, and is an eigenfunction of the harmonic oscillator the frequency of which is still to be determined.

Due to the presence of the harmonic oscillator we can now consider an explicit temporal eigenvalue problem; that is, we consider the eigenvalue problemwith a fixed , where we choose to be the cosmological constant of the Kerr-AdS space-time.

The eigenvalue problem (15) has a complete sequence of eigenfunctions with finite energies such thatand by choosing the frequencies of appropriately we can arrange that the stationary eigenvalues of agree with the temporal eigenvalues . If this is the case then the eigenfunctionswill be a solution of the wave equation. More precisely we proved the following.

Theorem 1. Let (resp., ) be eigenfunctions of the elliptic operator in (13), respectively, the temporal eigenfunctions, and, for a given index , let be the smallest eigenvalue of the with the propertythen, for any , there existsand corresponding satisfysuch thatThe functionsare then solutions of the wave equation with bounded energies satisfying for someMoreover, one hasIfthen one defines

Remark 2. (i) The event horizon corresponds to the Cauchy hypersurface in and the open set in , whereto the regionwhile the partis represented byThe timelike black hole singularity corresponds to which is a spacelike curvature singularity in the quantum space-time provided we equip with a metric such that the hyperbolic operator is normally hyperbolic (cf. [5, Lemma ]). Moreover, in the quantum space-time the Cauchy hypersurface can be crossed by causal curves in both directions; that is, the information paradox does not occur.
(ii) The stationary eigenfunctions can be looked at as being radiation because they comprise the harmonic oscillator, while we consider the temporal eigenfunctions to be gravitational waves.

As it is well-known the Schwarzschild black hole or more specifically the extended Schwarzschild space has already been analyzed by Hawking [9] and Hartle and Hawking [10]; see also the book by Wald [11], using quantum field theory, but not quantum gravity, to prove that the black hole emits radiation.

The metric describing a rotating black hole in a four-dimensional vacuum space-time was first discovered by Kerr [12]. Carter [13] generalized the Kerr solution by describing a rotating black hole in a four-dimensional de Sitter or anti-de Sitter background. Higher dimensional solutions for a rotating black hole were given by Myers and Perry [14] in even-dimensional Ricci flat space-time and by Hawking et al. [15] in five-dimensional space-time satisfying the Einstein equations with cosmological constant.

A general solution in all dimension was given in [16] by Gibbons et al. and we shall use their metric in odd dimensions, with all rotational parameters supposed to be equal, to define our space-time , though we shall maximally extend it.

Notations 3. We apply the summation convention and label coordinates with contravariant indices, for example, . However, for better readability we shall usually writeinstead of

2. Preparations

We consider odd-dimensional Kerr-AdS space-time , , , assuming that all rotational parameters are equal:The Kerr-Schild form of the metric can then be expressed aswhereand is the cosmological constant such that the Einstein equationsare satisfied in and is the mass of the black hole:which is the standard metric of , where are the azimuthal coordinates, the values of which have to be identified modulo , and are the latitudinal coordinates subject to the side-condition: also satisfyThe coordinates are defined inrespectively, (cf. [16, Section and Appendix  B]).

The horizons are hypersurfaces , where satisfies the equationLetbe the polynomial on the left-hand side of (43); then is strictly convex in and we havefrom which we deduce that (43) is satisfied if and only if and in casewe have exactly two solutions, otherwise only one. If there are two solutions ,   , such thatthen the outer horizon is called event horizon and the black hole has an interior regionin which the variable is a time coordinate. If there is only one solution , then is empty and the black hole is called extremal.

We shall first quantize a black hole with ; the quantization of an extremal black hole is then achieved by approximation.

Thus, let us consider a nonextremal black hole and let be a spacelike coordinate slice:where also denotes the constant value.

In view of (36), the induced metric can be expressed asfrom which we deduceThe other components of the metric are either or are represented by the line elementnote constraint (40).

To eliminate we shall introduce new coordinates. First, let us make the simple change by defining throughwhere is a constant which will be specified later, and dropping the prime in the sequel, resulting in a replacement of the components in (52) and (53) byrespectively,

Next, we define new coordinates bywhere are nonvanishing constants to specified later, such thatIn the new coordinates the only interesting new components areWe therefore deduce, in view of (54), (57), and (58),Choosing nowwe concludeCombining then (64) and (62) by setting we obtainDefiningthensince the function in (44) is negative in . Writingwe inferThe term in the brackets vanishes on the event horizon and is strictly positive in , in view of (68) and the identityHence, for any satisfyingwe can choose such thatWriting instead of we can then state the following.

Lemma 4. For any hypersurfacethe induced metric can be expressed in the formwhereis a smooth Riemannian metric on and ranges in , while, in casethe induced metric is Lorentzian of the formIf tends to , then the hypersurfaces S(r) converge topologically to the event horizon and the induced metrics to the Riemannian metric

Proof. We only have to prove case (77). However, the proof of this case is identical to the proof when (74) is valid by observing that then the term in (70) is strictly positive.

3. The Quantization

We are now in a position to have an argument very similar to that in our former paper [8, Section ]. For the convenience of the reader we shall repeat some of the arguments so that the results can be understood directly without having to look up the details in the reference.

The interior of the black hole is a globally hyperbolic space-time and the slices , withare Cauchy hypersurfaces. Let tend to and let be the resulting limit Riemannian manifold; that is, topologically it is the event horizon but equipped with the metric in (79) which we shall write in the form as in (75). By a slight abuse of language we shall also call to be a Cauchy hypersurface though it is only the geometric limit of Cauchy hypersurfaces. However, is a genuine Cauchy hypersurface in the quantum model which is defined by (1).

Let us now look at the stationary eigenvalue equation, where we recall that , in , whereand is the Laplacian in the Riemannian manifoldmoreover the scalar curvature is also the scalar curvature with respect to in view of (81). Using separation of variables let us writeto conclude that the left-hand side of (82) can be expressed in the form Hence, eigenvalue problem (82) can be solved by settingwhere , , is an eigenfunction of the elliptic operatorsuch that and is an eigenfunction of the harmonic oscillator. The eigenvalue of the harmonic oscillator can be arbitrarily positive or zero. We define it at the moment aswhere will be determined later. For we shall consider the real eigenfunction which represents the ground state in the intervalwith vanishing boundary values. is a solution of the variational problemin the Sobolev space .

Multiplying by a constant we may assumeObviously,and though is defined in and is even an eigenfunction it has infinite norm in . However, when we consider a finite disjoint union of open intervals ,wherethenis a unit eigenfunction in with vanishing boundary values having the same energy as in . Hence, it suffices to consider only in since this configuration can immediately be generalized to arbitrary large bounded open intervalsWe then can state the following.

Lemma 5. There exists a complete sequence of unit eigenfunctions of the operator in (88) with eigenvalues such that the functionswhere is a constant multiple of the function in (91) with unit norm, are solutions of the eigenvalue problem (82) with eigenvalueThe eigenfunctions form an orthogonal basis for .

To solve wave equation (1) let us first consider the following eigenvalue problem:in the Sobolev spaceHere,can in principle be an arbitrary negative parameter but in the case of a Kerr-AdS black hole it seems reasonable to choose the cosmological constant of the Kerr-AdS space-time.

Eigenvalue problem (102) can be solved by considering the generalized eigenvalue problem for the bilinear forms in the Sobolev space which is the completion ofin the norm defined by the first bilinear form.

We then look at the generalized eigenvalue problemwhich is equivalent to (102).

Theorem 6. Eigenvalue problem (107) has countably many solutions such that are complete in as well as in .

Proof. The quadratic form is compact with respect to the quadratic form as one can easily prove (cf. [17, Lemma ]), and hence a proof of the result, except for the strict inequalities in (108), can be found in [18, Theorem , p. 37]. Each eigenvalue has multiplicity one since we have a linear ODE of order two and all solutions satisfy the boundary conditionThe kernel is two-dimensional and condition (111) defines a one-dimensional subspace. Note that we considered only real valued solutions to apply this argument.

We are now ready to define the solutions of wave equation (1).

Theorem 7. Let (resp., ) be eigenfunctions of the elliptic operator in (88) (resp., the temporal eigenfunctions) and let be the smallest eigenvalue of with the propertythen, for any , there existsand corresponding satisfysuch thatThe functionsare then solutions of the wave equation with bounded energies satisfyingfor someMoreover, one hasIfthen one defines

Proof. The proof is obvious.

Remark 8. (i) By construction the temporal and spatial energies of the solutions of the wave equation have to be equal.
(ii) The stationary solutions comprising a harmonic oscillator can be looked at as being radiation while we consider the temporal solutions to be gravitational waves.
(iii) If one wants to replace the bounded Interval by then the eigenfunctions have to be replaced by eigendistributions. An appropriate choice would bealso see [6] for a more general setting.

The hyperbolic operator defined by wave equation (1) can be defined in the space-time which can be equipped with the Lorentzian metrics as well as with the metricwhere is the metric defined on and the indices now have the range . In both metrics is globally hyperbolic provided is complete, which is the case for the metric defined in (81). The hyperbolic operator is symmetric in the first metric but not normally hyperbolic while it is normally hyperbolic but not symmetric in the second metric. Normally hyperbolic means that the main part of the operator is identical to the Laplacian of the space-time metric.

Hence, if we want to describe quantum gravity not only by an equation but also by the metric of a space-time then the metric in (125) has to be chosen. In this metric has a curvature singularity in (cf. [5, Remark ]). The Cauchy hypersurface then corresponds to the hypersurfacewhich also follows from the derivation of the quantum model where we consider a fiber bundle with base space and the elements of the fibers were Riemann metrics of the formwhere were metrics defined in and is the time coordinate that we use in ; that is,In the present situation we used the symbol to denote the metric on since is supposed to be a metric on .

Thus the event horizon is characterized by the Cauchy hypersurfaceIf , that is, in case we consider a Schwarzschild-AdS black hole as in [8], then we shall obviously assume that the black hole singularity,corresponds to the curvature singularityof ; that is, the open black hole region is described in the quantum model byand the open exterior region by

If , then there is no curvature singularity in , only a coordinate singularity in our present coordinate system. Indeed, if we choose generalized Boyer-Lindquist coordinates (cf. [16, equation ]), the metric has the formThen, definingsuch thatwe obtain new coordinates in which the metric is smooth up to ; indeed the metric is even smooth in the intervalIn there is curvature singularity.

Lemma 9. The extended space-time has a timelike curvature singularity in .

Proof. The fact that the curvature singularity is timelike follows immediately from (78), where we proved that outside the black hole region the hypersurfacesare timelike.
To prove the existence of a curvature singularity we first consider the case . Looking at the metric in (134) we observe that the components with respect to the coordinates form a diagonal matrix without any cross terms with the other coordinates; namely,where are subject to the side-condition:that is, (139) represents the metric of a sphere of radiusembedded in and the corresponding sectional curvatures in are defined independently of the other components of the metric in and they obviously become unbounded when tends to , since the sectional curvature in a point of a plane spanned by two linearly independent vectors inis equal toSecondly, in case , we used the package GREAT [19] in Mathematica to compute the squared Riemannian curvature tensor in dimension and obtainedcompleting the proof of the lemma.

Since the curvature singularity is timelike and not spacelike as the singularity of a Schwarzschild-AdS space-time or the singularity in our quantum space-time, equipped with the metric in (125), it is easily avoidable. Despite this difference, we stipulate that the region in (133) corresponds to whereand the region in (132) corresponds to

Remark 10. The time coordinate in a generalized Boyer-Lindquist coordinate system is a time function in where is the extended Kerr-AdS space-time. We proved it directly with the help of Mathematica, if , by proving For a proof in any odd dimension it will be sufficient to prove that the slices are spacelike in the region specified in (148). Looking at the metric (134) we immediately see, by setting , that the induced metric is Riemannian.

Remark 11. When we have an extremal black hole with mass and corresponding radius for the event horizon; then the function in (44), where satisfies and hence From the definition of we then conclude that any black hole with mass while the other parameters remain equal, will have an interior region. Hence, our previous arguments could then be applied to yield a quantum model depending on the Riemannian metric in (79). Letting tend to the corresponding radii of the event horizons will then converge to leading to a quantum model for an extremal black hole.

Remark 12. In the quantum model of the black hole the event horizon is a regular Cauchy hypersurface and can be crossed in both directions by causal curves; hence no information paradox can occur.

Conflicts of Interest

The author declares that he has no conflicts of interest.