Contents
Abstract

Motivated by the Kerr-CFT conjecture, we investigate perturbations of the near-horizon extreme Kerr spacetime. The Teukolsky equation for a massless field of arbitrary spin is solved. Solutions fall into two classes: normal modes and traveling waves. Imposing suitable (outgoing) boundary conditions, we find that there are no unstable modes. The explicit form of metric perturbations is obtained using the Hertz potential formalism, and compared with the Kerr-CFT boundary conditions. The energy and angular momentum associated with scalar field and gravitational normal modes are calculated. The energy is positive in all cases. The behaviour of second order perturbations is discussed.

Kerr-CFT and gravitational perturbations



Óscar J.C. Dias, Harvey S. Reall, Jorge E. Santos


DAMTP, Centre for Mathematical Sciences, University of Cambridge,

Wilberforce Road, Cambridge CB3 0WA, United Kingdom

[.3em]


O.Dias@damtp.cam.ac.uk, hsr1000@cam.ac.uk, jss55@cam.ac.uk

June 12, 2009

1 Introduction

Some time ago, Bardeen and Horowitz (BH) showed that one can take a near-horizon limit of the extreme Kerr geometry to obtain a spacetime similar to [1]. This near-horizon extreme Kerr (NHEK) geometry has an isometry group, where the is inherited from the axisymmetry of the Kerr solution and the extends the Kerr time-translation symmetry. Recently, Guica, Hartman, Song and Strominger (GHSS) have conjectured that quantum gravity in the NHEK geometry with certain boundary conditions is equivalent to a chiral conformal field theory (CFT) in 1+1 dimensions [2]. Using this, they gave a statistical calculation of the entropy of an extreme Kerr black hole.

More precisely, GHSS showed that there exist boundary conditions on the asymptotic behaviour of the metric such that the asymptotic symmetry group is generated by time translations plus a single copy of the Virasoro algebra, the latter extending the symmetry of the background. Hence, if a consistent theory of quantum gravity can be defined in NHEK with these boundary conditions then it must be a chiral CFT. There has been considerable interest in extending the Kerr-CFT conjecture, and entropy calculation, to other extremal black holes [3].

The GHSS boundary conditions are unusual in two respects. First, they specify the rate at which components of (the deviation of the metric from the NHEK geometry) should behave asymptotically. We shall refer to these as the “fall-off” conditions. Most components decay relative to the background but some are allowed to be relative to the background. Secondly, GHSS impose a supplementary boundary condition, namely that the energy (the conserved charge associated with the generator of ) should vanish.

One motivation for this paper is that the GHSS fall-off conditions are motivated entirely by considerations of the asymptotic symmetry group. However, boundary conditions are also required for classical physics to be predictable from initial data in a non-globally hyperbolic spacetime such as NHEK (or anti-de Sitter). It is not clear whether these boundary conditions will be compatible with the unusual GHSS boundary conditions. Indeed, it is not even clear whether the GHSS boundary conditions allow propagating gravitational degrees of freedom, or whether they lead to physics similar to Einstein gravity in , where non-trivial physics is associated with large gauge transformations (i.e., non-trivial elements of the asymptotic symmetry group) and black holes that are locally, but not globally, gauge [4]. We shall investigate these issues by studying linearized gravitational perturbations of NHEK.

Another motivation for studying perturbations of NHEK is associated with positivity of the energy. The GHSS “zero energy” condition arises from the desire to consider only the ground states corresponding to an extreme Kerr black hole, rather non-extremal excitations. However, this presupposes that the energy must be non-negative. The NHEK geometry possesses an ergoregion, inherited from the ergoregion of the Kerr black hole. It is well-known that, in the presence of an ergoregion, one can construct initial data for test matter fields for which the energy of these fields is negative [5]. For a Kerr black hole, this is not a problem because the positive energy theorem  [6] ensures that the total energy of the spacetime (black hole plus matter) is non-negative. This is a non-trivial result, which may not extend to NHEK.111 If one wanted to prove such a theorem using spinorial methods then NHEK would have to admit a spinor field covariantly constant with respect to some connection. As far as we know, no such spinor field has been constructed. Furthermore, in a spacetime with an ergoregion but no event horizon, e.g. NHEK (adopting the global perspective), if one imposes boundary conditions such that there is no energy in the matter fields entering from infinity, then the total energy of these fields can only decrease. If it is initially negative then it will become more negative, suggesting an instability [5].

It should be noted that the issue of NHEK stability is subtle: BH pointed out that the singularity theorems imply that there exist small perturbations of NHEK that will lead to the formation of a singularity. In this sense, NHEK is unstable. However, as BH also observed, such a singularity might be hidden inside a tiny black hole.222The same might be true in for : is like a confining box, and a small gravitational perturbation in a box might be expected to evolve ergodically. If so, eventually sufficient energy will be concentrated into a small enough region to produce a tiny black hole. We thank G. Horowitz for discussion of this point. If this has positive mass then there would not be a problem. However, if the energy is negative, or the singularity is naked, then it would be difficult to make sense of NHEK.

The NHEK geometry shares many similarities with : indeed, it is foliated by warped submanifolds, which have been discussed extensively in recent work on topologically massive gravity (TMG) [7]. In TMG, there are propagating gravitational degrees of freedom but some of these turn out to have negative energy, signaling a potential instability of [8]. In the chiral limit, the propagating modes are eliminated by boundary conditions at infinity [8, 9], leaving only pure gauge modes and BTZ black holes, just as in Einstein gravity . Away from the chiral limit, is unstable but there exist warped solutions that might provide an alternative ground state [10]. The stability of some of these has been investigated recently [11]. Again, there are propagating modes with negative energy but these are excluded by boundary conditions.

We now describe the approach we shall take. NHEK is a type D vacuum spacetime so one can obtain decoupled equations describing gravitational perturbations using Teukolsky’s method [12, 13]. The Teukolsky equation turns out to be very similar to the equation governing a massless scalar field in NHEK, which was discussed by BH, and the qualitative features of our solutions closely resemble theirs.

By expanding in (spin-weighted, spheroidal) harmonics on the of the NHEK geometry, we reduce the Teukolsky equation to the equation of a charged massive scalar in with a homogeneous electric field. This equation can be solved in terms of hypergeometric functions. Depending on the labels of the spheroidal harmonics, the solutions either grow or decay as powers of the radial coordinate, or they are oscillating at infinity. In the former case, the natural “normalizable” boundary conditions lead to quantized frequencies: we shall refer to these as normal modes. These modes fill out highest-weight representations of a Virasoro algebra which extends the isometry group of , indeed such modes have been obtained previously in the context of a charged scalar in with electric field [14]. A particularly important set of normalizable modes are those arising from axisymmetric () perturbations of NHEK.

The other set of modes are those that oscillate at infinity. Following BH, we refer to these as traveling waves. These modes typically have large for given : . From the perspective, these correspond to modes that have complex weight with respect to the generator of and so would not normally be considered. However, in NHEK it would be very restrictive to discard these modes since that would correspond to a restriction on the allowed values of . Even if such a restriction were imposed at the linearized level, it would be violated at the nonlinear level through interactions between modes.

The traveling waves carry energy and angular momentum to infinity. BH showed that such modes are associated with superradiant scattering in the NHEK geometry. However, rather than considering scattering, we are interested in the question of what happens to localized initial data. We therefore impose purely outgoing boundary conditions at infinity. We find that the modes corresponding to traveling waves become exponentially damped, i.e., they are quasinormal modes of NHEK, describing the decay of a small perturbation via radiation to infinity. Therefore NHEK is stable against linearized gravitational perturbations. The reason that the above argument for instability based on the energy in matter (or linearized gravitational) fields fails is that some outgoing waves carry negative energy to infinity. Hence the energy flux through infinity need not be positive and so the energy need not decrease with time.

So far, our discussion of gravitational perturbations has been based entirely on the Teukolsky equation. However, in order to calculate the energy, or discuss fall-off conditions on the metric, we need to know the perturbed metric tensor rather than just the Teukolsky scalars. Fortunately, there exists a method for determining the metric perturbation in terms of a scalar potential, called the Hertz potential [15]-[19]. This satisfies an equation closely related to the Teukolsky equation. Using this, we obtain explicit results for the form of the metric perturbation.

We find that most (but not quite all) normal modes satisfy the GHSS fall-off conditions but traveling waves violate these conditions. Although one can construct localized wavepackets involving the latter, they will eventually propagate to infinity and violate the fall-off conditions. Therefore, at the linearized level, they should be excluded, leaving just the normal modes.

Next, we consider the energy of the normal modes. To warm-up, we start by considering a massless scalar field. We are able to show that an arbitrary superposition of normal modes has positive energy. Then we turn to gravitational perturbations. We define the energy of the latter in the usual way using the Landau-Lifshitz “pseudotensor”. Since the metric perturbation involves second derivatives of the Hertz potential, the energy involves an integral of a complicated quantity sixth order in derivatives. Nevertheless, using a combination of analytical and numerical methods, we find that the energy of gravitational normal modes is positive, thus supporting the validity of the GHSS zero-energy condition.

This positive energy result is satisfying but the exclusion of the traveling waves is worrying. First, it is worrying that we can construct initial data that satisfy the fall-off conditions, but violate these conditions when evolved. It suggests that the initial value problem, at least for linearized fields, may not be well-posed. Furthermore, if one goes beyond linearized theory then interactions between modes will excite traveling waves even if they are not present initially.333 The only way to escape this conclusion is to consider only axisymmetric () modes, which form a consistent truncation of the full set of modes. So one might worry about well-posedness of the nonlinear theory too. It is possible that these problems are cured by backreaction, i.e, going beyond the linearized approximation. We shall discuss this further at the end of the paper.

This paper is organized as follows. In section 2, we derive and solve the Teukolsky equation in the NHEK background, obtaining the spectrum of normal, and quasinormal modes. In section 3 we introduce the Hertz potential and use it to obtain the explicit form of linearized perturbations. We compare the asymptotic behaviour of these with the GHSS boundary conditions. We then calculate the energy of scalar field and gravitational normal modes. Finally, section 4 discusses how going beyond the linearized approximation may solve some of the problems just discussed.

Note added. As this work was nearing completion, we learned that another group is exploring similar issues [20].

2 Massless fields of arbitrary spin in NHEK

2.1 NHEK and its Newman-Penrose tetrad

In global coordinates the NHEK metric is [1] (we use the notation of Ref. [2] and, because we shall employ the Newman-Penrose formalism, a negative signature metric)

(2.1)

with

(2.2)

Surfaces of constant are warped geometries, i.e., a circle fibred over with warping parameter . The isometry group is . BH showed that the solution is geodesically complete, with timelike infinities at . There is an ergoregion (where is spacelike) which extends to .

In the next subsection we study perturbations in the NHEK using the Teukolsky formulation. For that we need the Newman-Penrose (NP) tetrad, spin coefficients and directional derivatives. In Appendix A we obtain the shear-free null geodesics of this background and use them to construct the associated NP null tetrad [21], , , , , where (coordinates are listed in the order )

(2.3)

and with non-vanishing symmetric given by . This NP tetrad satisfies the normalization and orthogonality conditions (A.16), and the null vector is tangent to affinely parametrized geodesics: .

The unperturbed Weyl scalars in the NHEK geometry are computed using (A.20), yielding

(2.4)

The first line confirms that this solution is indeed Petrov type D.

2.2 Teukolsky master equation

Teukolsky has shown how, for type D spacetimes, one can use the NP formalism to derive a system of decoupled equations, that furthermore separate into an angular and radial part, for the perturbations of several NP scalars [12, 13]. For gravitational perturbations, the relevant quantities are the perturbed Weyl scalars (spin ) and (); the complex NP scalars for spin Maxwell perturbations; the Weyl fermionic scalars for massless spin perturbations; and the scalar field for massless spin perturbations. Teukolsky’s master equation encompasses all of these cases [13].

Using the NP quantities listed in appendix A, we find that the Teukolsky master equation for spin field perturbations in the NHEK geometry is

(2.5)

We have allowed for the possibility of a source term on the RHS (see Appendix C). The relation between the nomenclature used here and the original notation of Teukolsky [13] is and for positive spin. For negative spin the map is . Here, the powers of the unperturbed Weyl scalar are those that allow for the separation of the master equation, when we further assume an ansatz for the perturbation that is a radial function times the spin-weighted spheroidal harmonic; see (2.6). For the source term one has the map . These relations are summarized in Table 1.

Table 1: Teukolsky fields , spin and source terms for the master equation (2.5).

2.3 Separation of variables

We shall solve the Teukolsky equation in the NHEK geometry by separation of variables. Assuming

(2.6)

equation (2.5) separates into an angular and radial equations. The angular equation is

(2.7)

for and where is the separation constant. Its eigenfunctions are the spin-weighted spheroidal harmonics (the nomenclature usually includes an appropriate normalization factor; see e.g., [22]), with positive integer specifying the number of zeros, , of the eigenfunction. The associated eigenvalues can be computed numerically with very good accuracy and are specified by subject to the regularity constraints that must be an integer and . The transformation can be used to show that

(2.8)

We also note that, to leading order in , . This is useful when .

Equation (2.7) represents the most standard way to write the spin-weighted spheroidal harmonic equation. However, it will be convenient here to work with shifted eigenvalues defined by

(2.9)

The advantage of using these quantities is that they have the symmetry

(2.10)

Notice that in the Kerr background with mass and angular velocity the angular equation for spin perturbations is also (2.7) but with , where is Kerr’s rotation parameter and the wave’s frequency in this geometry. As observed in [1], in the near-horizon limit of extreme Kerr, all finite frequencies in the NHEK throat correspond to the single frequency in the extreme Kerr geometry. This corresponds precisely to the marginally unstable superradiant frequency, and in the NH limit one finds .

Writing for any spin,

(2.11)

we find that the radial equation associated with (2.5) can be written also in a unified way as

(2.12)

with

(2.13)

                 

Figure 1: Values of , defined in (2.20), for , and a) and b) .

                 

Figure 2: Values of , defined in (2.20), for , and a) and b) .

This is exactly the equation for a charged massive scalar field in with a homogeneous electric field: take the metric in global coordinates,

(2.14)

and the electric field to arise from the potential

(2.15)

Define the covariant derivative for a field of charge as

(2.16)

where is the Levi-Civita connection in . The equation for a charged scalar field with mass is then

(2.17)

Assuming

(2.18)

the equation of motion reduces to (2.12). Therefore, a general spin perturbation with angular momentum in NHEK obeys the wave equation for a massive charged scalar field in with a homogeneous electric field. However, note that the charge is complex, as is the squared mass , although is real. The problem of a massive charge scalar field in with homogeneous electric field was studied in Ref. [14], where solutions corresponding to highest weight representations of a Virasoro algebra extending were obtained. We shall recover the same solutions in the next section.

2.4 Solving the radial equation

Asymptotically, the solutions of (2.12) behave as

(2.19)

where

(2.20)

Note that . We can now see that the modes can exhibit qualitatively different behaviour, depending on the value of , as first noticed by BH (for ). Some modes have real and others have imaginary . For example, axisymmetric modes (), have, for general ,

(2.21)

i.e., such modes exhibit power-law behaviour at infinity. However, for certain other modes, specifically those with , is imaginary and hence the solutions oscillate at infinity. In Figs. 1 and 2 we show how depends on for gravitational perturbations with some different values of .

It is interesting to ask which modes have the smallest real value for since these will give the normal modes that decay most slowly at infinity. For gravitational perturbations () we have calculated for all with and find that the mode with the smallest real value for occurs for , , which gives .

Equation (2.12) can be solved exactly. This is not a surprise since in the Kerr geometry, Teukolsky and Press [23] found that the corresponding Teukolsky radial equation can also be analytically solved in the particular case where we have extreme Kerr and a wave frequency that saturates the superradiant bound, . As discussed after (2.10), all frequencies in NHEK correspond to the single superradiant threshold frequency in the extreme Kerr. So we indeed expect this property for the radial equation in NHEK.

Introducing the new radial coordinate,

(2.22)

and redefining the radial wavefunction as

(2.23)

the radial equation (2.12) can be rewritten as

(2.24)

This wave equation is a standard hypergeometric equation [24], , with

(2.25)

and hence the most general solution in the neighborhood of is [24]

(2.26)

We render this function single valued in the complex plane by taking branch cuts to run from to and from to , corresponding to taking , . Note that the branch cuts do not intersect the line , which corresponds to real .

2.5 Boundary conditions

The above solution of the radial equation is regular for all finite . Using standard properties of the hypergeometric function, we find that it exhibits the following behaviour as :

(2.27)

where

(2.28)

The boundary conditions now depend on whether is real or imaginary.

2.5.1 Normal modes

Assume that is real. In this case, we impose normalizable boundary conditions, corresponding to demanding that , a pair of simultaneous equations for , . Non-zero solutions exist only if the determinant of this system vanishes. Using , this gives

(2.29)

This imposes a quantization condition on , corresponding to the two solutions and where .444At first sight, condition (2.29) could also be satisfied if we imposed , i.e., . However, a more careful analysis rules out this possibility because for , (2.26) is not a solution of the problem: one must allow for a logarithmic dependence in the second part. Redoing the analysis with the appropriate regular radial solution for this special case [24], we conclude that nothing physically special occurs for . The former solution gives

(2.30)

and the latter gives

(2.31)

We can summarize the normal mode spectrum as

(2.32)

This is precisely the spectrum of normal modes found for a massive charged scalar in with a homogeneous electric field in Ref. [14].

Note that we have allowed to be positive or negative. This is because the Teukolsky equation for is not invariant under complex conjugation, so negative frequency solutions are not simply related to positive frequency solutions by complex conjugation, they have to be considered separately. The two possible signs correspond to the two different helicities of the field. The radial equation is invariant under , hence .

For , the positive frequency solution of the radial equation is

(2.33)

and the negative frequency solution is obtained by , i.e., . The solutions with positive are related to these solutions by multiplication by a polynomial of degree in .

2.5.2 Traveling waves

Now consider the case of imaginary . Define by . The radial function oscillates at infinity, corresponding to incoming or outgoing waves (see Appendix B for details). Rather than considering scattering in NHEK, we shall impose boundary conditions corresponding to purely outgoing waves at infinity, which will discretize the frequency and render it complex. A solution with positive imaginary part corresponds to an instability, and a solution with negative imaginary part is a quasinormal mode.

As discussed by BH, there are two inequivalent notions of “outgoing” that one can use in NHEK because the phase velocity and group velocity of wavepackets need not have the same sign, e.g. for positive and , the group and phase velocities have the same sign at but opposite sign at (see Table 3). Physical boundary conditions correspond to the notion of “outgoing” defined using the group velocity. However, it is easier to analyze the case of outgoing phase, so we shall consider this case first.

Assume that . Then the solutions with outgoing phase at are the solutions with . This leads to the quantization condition

(2.34)

with solution , ( is inconsistent with ), which gives . Repeating the exercise for requires , and leads to . We can summarize the result as

(2.35)

The imaginary part is negative, hence these are quasinormal modes. This is a little surprising. BH pointed out that the energy flux (for positive frequency modes) has the same sign as the phase velocity. Hence outgoing phase should correspond to outgoing energy at infinity. As discussed in the introduction, this is precisely the situation in which one expects an instability associated with the negative energy in matter fields within the ergoregion becoming increasingly negative. We have found that outgoing phase leads to stable quasinormal modes rather than an instability. However, these boundary conditions are unphysical: we are arranging that an initial wavepacket (at finite ) composed of modes with positive does not propagate to by sending in an appropriate (finely tuned) wavepacket from to scatter with it in such a way as to produce only a wavepacket propagating to . This is analogous to boundary conditions for a Kerr black hole in which one arranges that initial data leads to no waves crossing the future horizon by sending in appropriate waves from the past horizon. Presumably, the fine-tuning is the reason that we do not see an instability here.

Now consider the physical boundary conditions corresponding to “outgoing” defined with respect to the group velocity. Assume that and . BH showed that, under these conditions, the phase and group velocities have the same sign for but opposite sign for . Hence the boundary conditions that we need are . In fact, the same holds for and (see Appendix B). Using the identity , we find that the quantization condition is

(2.36)

which gives555A solution corresponding to taking integer values is ruled out for the reason discussed in footnote 4.

(2.37)

where we have specialized to scalar field () or gravitational () perturbations for simplicity. Repeating the analysis for requires . The general result is

(2.38)

We see that hence these are stable quasinormal modes. So NHEK is stable against linearized gravitational (and scalar field) perturbations.

3 Metric perturbations

3.1 Hertz potentials

The NP scalar perturbations are useful because they are invariant under infinitesimal diffeomorphisms and under rotations of the NP tetrad. Many physically interesting quantities can be computed directly from the knowledge of these NP fields [12, 13, 21]. However, in some problems as is our case, we really need to know the perturbations of the metric itself, , or the perturbations of the Maxwell or Weyl fermionic vector fields, respectively and . Cohen and Kegeles [15, 17], and Chrzanowski [16] have proposed a unique map that provides the , or perturbations given the so-called Hertz potential (see a good discussion also in [19]). Wald proved Cohen-KegelesChrzanowski’s results [31]. See Appendix C for a detailed discussion of these works. The main conclusion is that the Hertz potential also obeys a pair of decoupled equations, again one for positive and the other for negative . These are written in equations (C.18) and (C.19).

For gravitational perturbations, this method yields the metric perturbation in a particular gauge: the ingoing (outgoing) radiation gauge IRG (ORG), specified by the conditions

(3.39)

At first sight, these gauge conditions appear overdetermined but it has been shown that, for perturbations of a type II vacuum spacetime, there is a residual gauge freedom that allows one to impose the IRG provided that where is the repeated principal null direction and the stress-tensor of any matter perturbation present [25]. Similarly, for type D one can impose either the IRG or the ORG (if ). The spin of the Hertz potential corresponds to these two different gauges: the metric perturbation in the IRG (ORG) is obtained from the Hertz potential with (). The two Hertz potentials contain exactly the same physical information, so one need only work with one of them.

For vacuum type D spacetimes, the Hertz potential itself satisfies a master equation. For the Kerr solution, this master equation turns out to be exactly the same as for the original NP scalars , equation (2.5), with no source term on the RHS [26]. We have checked that the same is true for NHEK. More concretely, are the Hertz potentials conjugate to the positive spin Teukolsky perturbations but satisfy exactly the same master equation (2.5) as for negative spin. Similarly, are the Hertz potentials conjugate to the negative spin Teukolsky perturbations but the positive spin Hertz potential obeys the same master equation as for positive spin. In short, the Hertz potential obeys the same master equation as its conjugated Teukolsky field but with spin sign traded. This relation is better clarified if we use Tables 1 and 2.

Table 2: Spin Hertz fields that satisfy the master equation (2.5) with no source term.

Assuming perturbations for the Hertz potentials of the form

(3.40)

where further satisfies (2.11), equation (2.5) separates into an angular and radial equations. The angular equation is (2.7) with , and the radial equation is (2.12). Its solution is given by (2.26).

As stated above, given the Hertz potential for the gravitational field there is a unique map between it and the metric perturbations [15, 16, 17, 31]. A similar map exists between the spin Hertz potentials and the Maxwell and Weyl fermionic vector perturbations, but we leave the discussion of these cases to Appendix C. In the ingoing radiation gauge the metric perturbation in NP notation is given by (see Appendix C)

and a similar correspondence exists between the Hertz potential and the metric perturbations in the outgoing radiation gauge. (See the second relation of (LABEL:MapHertzHuv).) One can check that (LABEL:huvHertz) indeed satisfies the linearized Einstein’s equations for a traceless metric perturbation:

(3.42)

3.2 Behaviour of solutions

The basis vector fields and are globally well-defined. However, the vector field is singular at . Nevertheless, one can check that angular dependence of the Hertz potential contains a sufficiently high power of to ensure that the above metric perturbation is smooth at .

The asymptotic behaviour of the Hertz potential can be obtained using (2.11) and (2.19). Use of (LABEL:huvHertz) yields then for the asymptotic behaviour (rows and columns follow the order: )

(3.43)

where is given by (2.20). Exactly the same result is obtained in the outgoing radiation gauge. In (3.43) we have not imposed any boundary condition. These were discussed in subsection 2.5; e.g., for , the lower sign would correspond to normal modes.

We shall now compare the above asymptotic behaviour of metric perturbations with the GHSS fall-off conditions. The and components are the most restrictive. For these to satisfy the fall-off conditions, must be real, so traveling waves are excluded, we must use normalizable boundary conditions (i.e. the lower sign choice) and we need . Recall that there are normal modes with , so it appears that the GHSS fall-off conditions exclude some of the normal modes.666It is conceivable that a gauge transformation could be used to bring a mode violating the fall-off conditions to one that satisfies these conditions but this seems unlikely, especially for traveling waves.

As emphasized in the introduction, at the nonlinear level, we expect that interactions will lead to modes corresponding to traveling waves () being excited, which would lead to a violation of the GHSS fall-off conditions. The only modes that escape this conclusion are the axisymmetric ones (which have with ), which always obey the GHSS boundary conditions. Axisymmetric modes form a consistent truncation of the full set of modes in the sense that linearized axisymmetric modes will not excite non-axisymmetric modes at next order in perturbation theory.

3.3 The energy

3.3.1 Massless scalar field

We want to compute the energy associated with the gravitational perturbations that we found in the previous subsection. Since this will involve a rather lengthy calculation, we shall start with the conceptually simpler case of a massless complex scalar field:

(3.44)

The canonical energy momentum tensor is given by

(3.45)

Let be a spacelike hypersurface with future-directed unit normal . Then, given any Killing vector , we can define the associated conserved charge

(3.46)

where is the induced metric on . We shall choose to be a surface of constant in the NHEK geometry. The conserved charges of interest are the energy , for , and the angular momentum , for . (The latter is the charge of GHSS.) Written out explicitly, these are

(3.47)

In the energy integrand, all the terms except the last are manifestly positive. The last is proportional to and thus is positive only outside the ergosphere where . The energy can thus be negative (for a rigorous proof of this, see Ref. [5]).

Consider a general superposition of normalizable modes (recall that is defined by the frequency quantization (2.32)):

(3.48)

First, we shall show that the conserved charges associated with such a solution can be decomposed into a sum of conserved charges of the individual modes.

A charge integral can be regarded as defining a (typically indefinite) norm on the space of solutions of the wave equation. Given a norm , it can be “polarized” to obtain a Hermitian scalar product : the real and imaginary parts of are given by and respectively. In our case, polarizing the charge integral defines a scalar product , antilinear in and linear in . Since the norm is conserved, so will be the scalar product. Note that .

We shall now argue that modes with different are orthogonal with respect to this scalar product. The scalar product has the form

(3.49)

where is preserved by any Killing vector field that commutes with . Now let be such a Killing field. We can then write

(3.50)

If the RHS vanishes then this shows that is self-adjoint with respect to the this scalar product. For NHEK, we take or . Taking , the RHS vanishes because the scalar product is conserved, and hence independent of . Taking , the RHS vanishes because it is a total derivative on . It follows that modes with different or diffferent will be orthogonal with respect to this scalar product. Hence, in calculating the charge associated with (3.48), there are no cross-terms in the charge arising from modes with different or (in particular, there are no cross-terms between the positive and negative frequency parts of (3.48)).

Now consider the -dependence. Since is not associated with a Killing symmetry of the background, we cannot use the above argument. Instead, for separable solutions, the angular dependence will be given by (2.7) with . This equation is self-adjoint, so two solutions with different values of will be orthogonal with respect to the measure , i.e.,

(3.51)

Fortunately, it turns out that is precisely the measure that arises in the scalar products associated with the energy and angular momentum.

From these results, we see that no cross-terms between modes with different contribute to the energy and angular momentum. Substituting (3.48) into (3.46) for gives the energy as a sum over contributions from individual modes:

(3.52)

where

(3.53)

Note that is manifestly positive only when . However, we have evaluated the radial integral above for many cases, namely for , and . In all these cases, it is positive. Hence, for a massless complex scalar field in the NHEK geometry, the energy of an arbitrary superposition of normalizable modes is positive.

The angular momentum can be similarly decomposed:

(3.54)

where we find the simple result

(3.55)

3.3.2 Gravitational perturbations

The energy of gravitational perturbations is calculated from the Landau-Lifshitz “pseudotensor” defined as follows. Consider metric perturbations around NHEK up to second order in the amplitude,

(3.56)

The linearized Einstein equation is777Written out, this takes the standard Lichnerowicz form. If we assume that is traceless then this equation reduces to (3.42).