Mode Stability For Massless Scalars In Five-Dimensional Black Hole Backgrounds

The mode stability of the Kerr black hole in four dimensions was demonstrated by Whiting in 1989, by separating the Teukolsky equation that describes gravitational perturbations and then transforming the radial and angular equations in such a way that the problem can be reformulated as a wave equation in an auxiliary spacetime in which the proof of stability is greatly simplified, owing to the absence of an ergoregion. As a preliminary step towards extending these ideas to higher-dimensional black holes, we study the mode stability of the massless scalar wave equation in the five-dimensional black hole solutions of Einstein gravity and supergravity. We show how the wave equation can again be mapped into one in an auxiliary spacetime in which there is no ergoregion, allowing us to give a proof of the mode stability of the solutions of the scalar wave equation.


Introduction
Establishing results on the stability of black hole solutions has been a major activity in the general relativity community for many years. This is especially subtle, and also important, in the case of stationary rotating black holes. Many different approaches have been followed, but one of the most fruitful has involved finding an integral transformation that maps the difficult analysis in the original variables into a considerably simpler analysis in the transformed variables. The idea was first developed in [1], where it was employed to establish the mode stability of the Kerr black hole [1]. The technique has been developed further in recent years, and found application in studies such as the global stability of black holes (see, for example, [2,3]), the stability of extremal black holes [4], and mode stability on the real frequency axis [5,6].
Establishing the mode stability of the Kerr black hole involved studying the properties of the mode functions in the separation of variables for the Teukolsky equation, which provides a gauge-invariant description of the perturbations around the Kerr background. In [1] the generalised Teukolsky equation with a spin parameter s was studied, with s = ±2 corresponding to the actual case of interest in which the equation describes the gravitational perturbations themselves. The case s = 0 corresponds to the massless scalar wave equation (the massless Klein-Gordon equation), while the s = ±1 case governs gauge-invariant components of the Maxwell field.
The techniques for analysing the Teukolsky equation for spin s that were developed in [1] were broadly similar for all s, and in fact the essential features associated with the stability of the solutions could already be seen in the s = 0 case. This is a useful observation because if one looks at more complicated situations than black holes in pure Einstein gravity, such as black holes in Einstein-Maxwell theory or in supergravity, the analogue of the Teukolsky analysis has not been implemented. In the absence of a gauge-invariant treatment of the perturbations of the black holes in these more complicated situations, one can at least study the analogue of s = 0 Teukolsky equation, that is, the massless scalar wave equation in the black hole background. One may hope that this can provide a "proxy" for the full analysis, and that establishing stability results for the solutions of the massless wave equation may be indicative of what one might find in a more elaborate and complete perturbative analysis.
This approach was adopted in [7], where the techniques of [1] were applied to the study of the mode stability for solutions of the massless scalar wave equation in the background of a class of four-dimensional supergravity black holes carrying four independent electric charges [8,9].
In this paper, we extend some of the four-dimensional techniques for mode-stability analysis that were developed in [1] to the case of five dimensions. Even in the pure Einstein case, the analogue of the four-dimensional Teukolsky analysis is unknown. The stumbling block is that the four-dimensional analysis depended heavily upon the use of the Newman-Penrose formalism, and no particularly useful extension of this to five or higher dimensions has been constructed. 1 Thus for now, our approach will mirror the one that can be followed for more complicated theories in four dimensions, namely, we shall focus attention on establishing stability results for solutions of the five-dimensional massless scalar wave equation.
This already allows us to develop a rather non-trivial generalisation of the techniques that were employed in four dimensions, and it reveals ways in which the integral transformation that allows us to establish mode-stability results is substantially different from the one in four dimensions.
We shall describe in the subsequent sections how one can establish mode stability results 1 However, see [10].
for solutions of the massless wave equation in the background of the five-dimensional Myers-Perry rotating black hole [11], and also in the background of general 3-charge rotating black holes in the five-dimensional STU supergravity theory [12].

Massless Scalar Wave Equation in Five-Dimensional Black
Hole Background

Five-dimensional Myers-Perry black hole
The natural generalisation of the four-dimensional rotating Kerr [13] black hole to higher spacetime dimensions is provided by the Myers-Perry black hole solutions [11]. These vacuum solutions of the D-dimensional Einstein equations are characterised by their mass M and by [(D − 1)/2] independent angular momenta, reflecting the fact that independent rotations can occur in each orthogonal spatial 2-plane. In this paper we shall be concerned specifically with the example of the five-dimensional rotating black hole. Its metric is given by [11] Here a and b are the two independent rotation parameters, with φ and ψ being the two associated azimuthal angles (each with period 2π). The latitude coordinate θ ranges over

Massless scalar wave equation
Our focus will be the investigation of solutions of the massless scalar wave equation Ψ = 0 in the Myers-Perry background, with the goal of establishing that modes with time dependence e −i ωt that are ingoing on the future horizon and outgoing at future null infinity cannot have a frequency ω with a positive imaginary part. In other words, we seek to show that there cannot exist spatially regular modes that would give rise to instabilities growing exponentially in time.
One can define new radial and latitude coordinatesx and y by writing Using these, we have and defining the quantity G µν ≡ √ −g g µν , we have It can be shown [14] that the massless scalar wave equation Ψ = 0, which may be written as ∂ µ (G µν ∂ ν Ψ) = 0, is separable.
We can obtain the separatedx and y equations in Schrödinger form by writing 2 Ψ = e −i ωt+i mφ+i nψ R(x) S(y) (2.7) and then defining This gives the separated equations It is convenient now to make a further change of the radial variable. Defining ǫ + and a further coordinate transformation fromx to x, given bỹ implies that D, defined in eqn (2.6), becomes The outer horizon r = r + is located at x = 1, and the inner horizon r = r − at x = 0. The region where r goes to infinity corresponds to x goes to infinity.
In terms of the variable x, we can now write the radial equation in the form In the angular direction, the equation for Y in (2.9) is of the form

Transforming the radial equation
We start with the radial equation given by eqn (2.13), with the constants given in (2.14).
Next, introduce a new function f (x), related to X(x) by Next, we make an integral transform to a new radial variable h(z), by defining Now, multiplying eqn (2.18) by e 2α x z and integrating gives, after integration by parts, where B is the boundary term Note that eqn (2.21) can be written as It can then be seen that provided the boundary term B vanishes, a question to which we shall return later, then h(z) defined in eqn (2.20) satisfies O z eα z e −ν/z z −1−γ h(z) = 0, and thus where the constants are chosen so that The constantα is arbitrary at this stage (it really just sets the scale of the new radial variable z that is introduced in eqn (2.20)). We shall find it convenient to define it to bẽ

Behaviour of unstable modes
If they existed, unstable modes would be solutions than were purely outgoing at I and purely ingoing at the horizon H, that is, asymptotically, they would have support only at I + and H + . We are interested in establishing the non-existence of unstable modes arising from frequencies in the upper half of the complex ω-plane, that is for ω = ω 0 + iω 1 , with ω 1 > 0, since these would grow exponentially in the future. Thus, near I + , we must have: and, near H + , we must have: From the definition of the r * coordinate in eqn (A.1), it can be seen that in terms of the radial coordinate x introduced in eqn (2.11), we shall have asymptotically Near I : Near H : From the radial equation (2.13), it can be seen that near the horizon x = 1, the solutions will have the form where the functions F (1) ± (0) a non-zero constant. As noted above, an unstable mode would have ω dependence of the form e −i ωr * near H, and hence from (2.31) and (2.14) it would correspond to the minus-sign choice in eqn (2.32).
Thus for an unstable mode X um , we must have the near-horizon behaviour In the asymptotic region near x = ∞, it can be seen from the radial equation (2.13) that the asymptotic form of the solutions will be ± (0) a non-zero constant. Now, as we saw, an unstable mode would correspond to a having an ω dependence of the form e i ωr * near infinity, and so from (2.30) and (2.14) it would correspond to the plus-sign choice in eqn (2.34). Thus, for an unstable mode X um (x), we must have the asymptotic behaviour Near I : Before going on, we point out that infinitely many modes with these analytic properties do exist with ω 1 , the imaginary part of ω, being < 0. These modes decay exponentially in time, have been extensively studied [15], and are known as the quasi-normal modes which, taken together, uniquely characterise a black hole spacetime to which they correspond.
We turn now to the behaviour of the transformed radial function h(z), defined by eqn (2.20). In particular, we shall be concerned with the behaviour in the coordinate range 0 ≤ z ≤ ∞.
Near z = 0, it can be seen from eqn (2.25) that the leading-order behaviour of the solutions will be where G (0) ± (z) are analytic functions with G (0) ± (0) being non-zero constants. It is evident from (2.20) that at z = 0 the integrand for an unstable mode is just an analytic function of x, and so the leading-order behaviour of h(z) near z = 0 will be given by the pre-factor functions eν /z z 1+γ . In other words, the unstable mode corresponds to the plus-sign choice in eqn (2.36): As a check, we see from eqns (2.14), (2.26) and (2.27) that (2.40) Substituting v = −2α(x − 1)z, we then have: Combining with the pre-factor we then see, provided that: Thus for an unstable mode, the plus-sign choice in eqn (2.39) is selected: As a check, we note that if ω has a positive imaginary part,α, given in eqn (2.27), will have a negative real part, and so h um (z) will be finite as z goes to infinity.  Together with the terms involving the radial and angular derivatives these are (see eqns (2.10)), and we always assume a and b are non-negative, it follows that ǫ − ≥ ǫ + .
Consequently, all the ω 2 terms in eqn (3.3) will be non-negative, provided that we choose the sign of ǫ 2 to be Note that the necessity for this choice of sign was already seen in the previous section, in eqn (2.42). The choice of sign for ǫ 1 is undetermined by these considerations. We shall, for definiteness, make the choice For the remaining ω 2 terms, namely with coefficient we note that the range of the angular coordinate is 0 ≤ y ≤ 1, and so −1 ≤ 1 − 2y ≤ 1. We also have that In summary, we have seen that the overall coefficient of ω 2 in eqn (3.3) is always positive, implying that in the transformed metricg µν obtained by the process of unseparating variables, ∂ ∂t is always timelike outside the horizon. We now present the complete result for the combination of the radial and angular equations. With ǫ 1 = ǫ 2 = −1 as discussed above we obtain Using the replacements we can read off the componentsg µν of an inverse metric in a transformed spacetime, such that eqn (3.9) can be written as withg zz = z 2 ,g yy = y(1 − y) ,g φφ = 1 4(1 − y) ,g ψψ = 1 4y , We may find a suitable conformal factor Ω 2 and a redefined wave function Φ such that the D'Alembertian of Φ in a rescaled metricĝ µν = Ω 2g µν gives rise to eqn (3.11). We define Φ(t, z, y, φ, ψ) = 1 z y(1 − y) Ψ(t, z, y, φ, ψ) , (3.13) and, noting from (3.12) that we have it can be seen that if we choose Ω so that then the transformed equation (3.11) is equivalent to the following equation for Φ in thê g µν metric: This can be derived from the Lagrangian From the resulting energy-momentum tensor we may construct a conserved current J µ = −K ν T µ ν , where K = ∂ ∂t is the time-translation Killing vector. This gives rise to a conserved energy with The integrand in the energy integral (3.19) is therefore given by where The four-dimensional integration in eqn (3.19) is over the coordinates of the 3-sphere (with the ranges 0 ≤ y ≤ 1, 0 ≤ φ < 2π, 0 ≤ ψ < 2π), and over the transformed radial variable z. This ranges over 0 ≤ z ≤ ∞, and as we discussed in section 2.4 the transformed radial function h(z) for any putative unstable mode goes rapidly to zero at z = 0 (see eqn (2.37)), and it goes rapidly to zero at z = ∞ (see eqn (2.44)), ensuring the convergence of the integrals of all the terms in eqn (3.19).
From eqns (3.4) we have |a 2 − b 2 | ≤ 2M , and since y lies in the interval 0 ≤ y ≤ 1, it follows that the quantity P satisfies P ≥ 0. Since every term in the energy integral (3.19) is integrable for any putative unstable mode, and each contribution is non-negative, it follows in particular that the integral of the 1 2z 2 P |∂ t Φ| 2 term is bounded from above by the conserved energy E. Thus Φ cannot grow exponentially in time, and therefore there cannot in fact exist any exponentially unstable modes.

3-Charge Five-Dimensional STU Supergravity Black Holes
The 3-charge rotating black-hole solution in five-dimensional STU supergravity was obtained in [12], by using a solution-generating procedure. A convenient form of the solution was given in [16]. With minor change of notation, to achieve consistency with our present conventions, the metric is given by where with The various functions above are given by 3 Here s i = sinh δ i and c i = cosh δ i , where δ i are the boost parameters that correspond to turning on the three electric charges. When δ i = 0, the metric reduces to the fivedimensional Myers-Perry black hole.
The coordinates σ and χ are related to the standard azimuthal angular coordinates φ and ψ (each with period 2π) by as can be seen from eqn (15) in [16] after turning off the gauge coupling constant g. The standard radial and angular coordinates r and θ are related tox andỹ bỹ x = r 2 ,ỹ = a 2 cos 2 θ + b 2 sin 2 θ . The coefficients κ, Λ, β and γ in the potential for the radial equation, generalising those in (2.14) for Myers-Perry, are now given by Crucial properties that held previously in the uncharged case continue to hold here. In together with the usual inequalities ǫ + ≤ ǫ − ≤ 1, it follows that when ω has a positive imaginary part, the real parts of β and γ will be negative.
For the angular equation, the hatted quantitiesκ,Λ,β andγ are given bŷ unchanged from the results (2.16) for the uncharged black holes.
Following the same steps as we did previously for the uncharged Myers-Perry black hole, we find that after implementing the same integral transformation of the radial equation as before, we again arrive at an "unseparated" equation of the form (3.9), with the only difference being in the coefficient of ω 2 : This correspondingly implies that the components of the tilded inverse metricg µν are un-changed except forg tt , which becomes Calculating the determinant, we now find that instead of eqn (3.14) we have Following the remaining steps of the previous discussion for the uncharged case, we find that the conserved energy is given by integrating √ −ĝ J 0 as in equation (3.21), with the function P now given not by eqn (3.22) but instead (4.14) The same arguments that established that P was non-negative in the uncharged case show that P ≥ 0 here also, and hence again there cannot exist any unstable exponentially-growing modes.

Discussion
It has become apparent that many of the equations governing massless fields on black hole spacetimes are of Heun type (in cases where the cosmological constant is non-zero), or one of its many confluent variants, such as for non-extreme Kerr [1], or the extreme case [4].
The differential equations we find for massless scalar fields in the five-dimensional Myers-Perry black hole spacetime (see (2.13) and (2.15)) are of yet another confluent Heun type.
That observation has allowed us to extend to this (and the related STU) case the analysis originally applied to massless fields of all spin in the Kerr spacetime [1]. Before our present work, that analysis had also been extended: i) by using a different integration contour, to rule out unstable modes on the real axis for the Kerr spacetime [6], ii) by considering a modified integral transform, to deal with the extreme (|a| = M ) Kerr black hole [4], and iii), by looking carefully at more complicated examples, to establish the absence of unstable modes for massless scalar fields in STU spacetimes and all more specialized sub-cases [7].
Remarkably, the integral transform we have used here is, effectively, an inverse of that developed for the extreme Kerr spacetime [4]. In this context, it is also worth noting that quite different techniques, stemming from Seiberg-Witten theory (see, for example [17]), and based on the spectral properties of the operators involved, have been used to discuss both Kerr quasi-normal modes [18] and Kerr-de Sitter stability [19]. The relevance of such an approach to the spacetimes we consider here is yet to be determined.

Conclusion
We have shown that a massless scalar field has no exponentially unstable modes in the fivedimensional Myers-Perry black hole spacetime. We have also shown that the same is true in the five-dimensional supergravity-motivated STU spacetimes and, previously [7], that this holds, too, for the four-dimensional STU spacetimes. Together, these encompass a number of other special cases which arise from restricting the parameters in these more general examples. Although these results may serve as suggestive for the behaviour for fields of higher spin -in particular, Maxwell fields and gravitational perturbations -it would be useful to have some more direct indication, perhaps by writing down (at least) the analogue of the Teukolsky equation in these more general cases. That task currently remains for future work.
This form of the metric is regular in the neighbourhood of future null infinity.
It can be seen from this form of the metric that it is regular as one crosses the future horizon.