# The Superradiant Instability in AdS

###### Abstract

We consider the intermediate and end state behavior of the superradiantly perturbed Kerr black hole. Superradiant scattering in an asymptotically flat background is considered first. The case of a Kerr black hole in an Anti de-Sitter background is then discussed. Specifically we review what is known about the superradiant instability arising in AdS and its possible end state behavior.

## Acknowledgements

I would like to dedicate this essay to the memory of Professor Steve Detweiler. It was during our many conversations during my time as an undergraduate that he first spurred my interest in the physics of black holes. I am just one of many students who has benefited immensely from his guidance and insight.

I would like to thank Dr. Jorge Santos for shepparding me toward a better understanding of the problems at the heart of this essay, for taking the time to thoroughly review & critique my work and for the introduction to AdS. I also owe Dr. Mike Blake a debt of gratitude for the mathematical clarifications he provided. While I’m at it, I should probably also thank my mother.

## 1 Introduction

This essay is concerned with the phenomena of superradiance in the setting of the Kerr black hole (Kerr BH). Superradiance is a wave phenomena in which an ingoing wave scatters off an object and in the process extracts some energy; the scattered wave is more energetic than the incident one. This is a particularly interesting process in the context of black hole physics because it provides an outlet for energy dissipation by a BH. Much of our discussion will be concerned with the results of [1] and [2]. In [2] superradiant scattering was studied in the space-time of a Kerr BH in Minkowski background. Frequency dependent conditions for superradiance and calculations of the extent of amplification were obtained; we will review these.

We also consider superradiance in a spacetime comprised of a Kerr BH in an Anti de-Sitter (AdS) background. While superradiant scattering in a Minkowski background is interesting in its own right, we will see that the corresponding problem in AdS is much richer and more complex. This is due in large part to the box-like nature of AdS. A scattered wave can now reflect off the boundary at infinity and return, in finite time, to the Kerr BH to extract more energy. This process can repeat many times suggesting that the Kerr-AdS BH is susceptible to a superradiant instability. This motivates a plethora of fascinating questions; can we characterize this instability, is there a relationship between general instabilities and these superradaint ones, what is the end state of the superradaintly perturbed Kerr-AdS BH…etc? In an effort to answer these questions we will draw heavily from the results of [1].

Rather than dive right into a discussion of superradiance in the two Kerr BH spacetimes, we first provide an introduction to some of the objects and concepts fundamental to the problem. To begin we give a treatment of the Kerr BH. Next we discuss the ergoregion of the Kerr BH and the associated phenomena of the Penrose Process to motivate the idea of Kerr BHs being susceptible to energy extraction. To aid our discussion of the superradiant instability we give an overview of some of the important properties of AdS. We next introduce the Teukolsky formalism and comment on its importance to Kerr BH perturbation theory. Finally, with the Teukolsky formalism in our tool box, we give a generic analysis of BH superradiance.

### 1.1 The Kerr Black Hole

A generic uncharged rotating BH in Minkowski background belongs to the Kerr family. Remarkably, this is just a two parameter family of solutions characterized by and which describe the mass and angular momentum of the BH. In natural units, with and , the metric [3] is given by:

(1.1) |

with

where the mass of the spinning BH is given by and the angular momentum is given by . We see that the metric has a coordinate singularity at the roots of with the larger root determining the event horizon and the smaller root corresponding to a Cauchy horizon.

### 1.2 Ergoregion and Energy Extraction

Note that the spacetime has the Killing vector fields (KVFs)

Observe that which, while monotone with respect to and negative for sufficiently large , is not strictly negative in the region . Thus finding the roots of we see is timelike in the region , null at and actually spacelike in the region . This latter region defines the ergoregion of the Kerr BH.

A static observer, i.e person with 4-velocity parallel , is not allowed in the ergoregion as curves with tangent vector are spacelike when . Interpreting this physically, an observer cannot simply sit still in the ergoregion but is forced to rotate with the BH.

On the other hand we can consider a stationary observer at constant with 4-velocity

Such an observer can exist provided he/she travels on a timelike curve or equivalently . This provides a condition for the existence of a stationary observer:

The zeros of the above expression are given by

Note that for hence there cannot be a stationary observer in this region. The permissible values are . At , meaning there is only one possibly stationary observer at the event horizon.

To motivate superradiance we now give an example of how a rotating BHs allow energy extraction. Suppose we have a particle with 4-momentum which approaches the Kerr BH along a geodesic. The energy of the particle as measured by a static observer at infinity is conserved along the geodesic: . Now suppose that inside the ergosphere the particle decays into two other particles with momenta . But since is spacelike in the ergoregion it is possible that which implies that . It can be shown that must fall into the BH while can escape to infinity greater energy than the incident particle. Hence the BH will actually decrease in mass and energy will be extracted. . Momentum must be conserved so

There are limits to the amount of energy which can extracted in this way. A particle crossing must have as both are future directed causal curves. Defining , one has . So the particle carries energy and angular mom into the BH. Hence and . Our inequality gives

Defining we clearly see that

so there is a bound to how much energy can be extracted from the BH. One can show that .

### 1.3 Anti de-Sitter Space

Because one of our interests is superradiance in Kerr-AdS we give here a quick introduction into Anti de-Sitter space and its properties.

The simplest vacuum solution of Einstein’s equation with cosmological constant,

are spacetimes of constant curvature. They are locally characterized by the condition

where d is the dimension of spacetime. Making use of these expressions we see that

hence the Ricci scalar is the constant . We are in the domain of AdS when .

In , which is our primary dimension on interest, the metric can be written as

(1.2) |

The quantity in eq 1.2 is the radius of curvature of the spacetime and is related to by

This is a maximally symmetric solution to Einstein’s equation.

One of the most famous properties of AdS, and one which is central to the discussion of the superradiant instability later on, is that it acts like a “box”. To demonstrate this consider the norm of the tangent vector of a radially outward null geodesic:

from this we have

We see that it takes a finite time for a radial null geodesic to reach the boundary and so we can think of AdS in some sense as an enclosed space. Note that it follows from this that AdS is not globally hyperbolic. For any hypersurface one can always construct a timelike curve which reaches the boundary before it is able to intersect the surface. Hence when evolving AdS initial data, boundary conditions (BCs) become very important.

### 1.4 Perturbations of Kerr Black Holes

Black Hole perturbation theory is an incredibly complex and rich subject. Here we will simply introduce what is relevant to the Kerr BH. For convenience we have left out a satisfactory discussion of Newman-Penrose (NP) formalism. For those unfamiliar, we highly recommend that the reader consult chapters 2,6 and 7 of [4].

Newman-Penrose formalism is a tetrad formalism in which the basis vectors are selected so as to emphasize the lightcone structure of the space-time. We pick an ”isotropic tetrad”

with real valued and complex valued, such that the only non-zero inner products are

Given a tensor we can project onto the tetrad frame and express the object in tetrad coordinates:

We can pass freely in either direction, considering the problem in which ever frame provides the most simplification. The original metric of the spacetime can be recovered via

For electromagnetic and gravitational perturbations we can consider the relevant tensors and in the tetrad frame. Because are complex valued we can actually express the and independent components of the above tensors as a set of and complex valued scalars respectively. The information contained in the Maxwell tensor is encoded in the following

(1.3) |

and the Weyl tensor is distilled into the 5 complex scalars

Maxwell’s equations manifest in the NP formalism as a system of equations involving , the derivative operators given by the tetrad basis, , and a set of 12 spin coefficients which are related to the structure constants of the tetrad basis under the bracket operation. The Weyl tensor shares the symmetries of the Riemann tensor and has the further restriction of tracelessness.This gives a similar set of equations this time involving in place of .

The form of these tetrads, corresponding to the Kerr geometry, was discovered by Kinnersly and is given by:

(1.4) |

In the effort to obtain linearized perturbation equations a natural first approach would be to start with the Einstein equation and let for metric perturbation . Expanding the field equations to first order in yields a set of linear equations. In the setting of the Kerr geometry however this approach is complicated. The fewer symmetries, relative to say the Schwarzchild solution, mean that the resulting PDEs in and are not seperable.

Fortunately the NP formalism provides a simpler alternative approach. It can be shown that when studying electromagnetic [11] and gravitational [12] perturbations of the Kerr geometry it suffices to consider the NP scalars and respectively. Further, it was shown by Teukolsky [10] that the linear perturbations of the Kerr BH could be described by a single master equation:

(1.5) |

with and are related as follows:

where .

Further, by Fourier decomposing with the form

Teukolsky was able separate eq 1.5 into the following ODEs for and :

(1.6) |

(1.7) |

The separation constant is constrained when BCs are imposed leading to a complex eigenvalue problem.

### 1.5 Superradiance

We will now outline the theory of superradiant scattering of test fields on a BH background. For concreteness and simplicity we will consider an asymptotically flat spacetime (so not AdS). It should be noted that fluctuations of order in the scalar fields induce a change of order , so one is justified in fixing the BH geometry.

Let us assume that our spacetime is stationary and axisymmetric, as in the case of the Kerr BH. As we have seen above for such a spacetime various types of perturbations can be expressed in terms of a master variable . It can be shown that obeys a Schrodinger-like equation

(1.8) |

with dependent on th curvature of the background and the test field properties. We let be some coordinate which maps . Consider the scattering of a monochromatic wave of frequency with . Supposing is constant at the boundaries, the asymptotics of eq 1.8 give isometries) by dependence given (because of the

(1.9) |

where and . The event horizon imposes the boundary condition of a one-way membrane. We have a wave incident from spatial infinity of amplitude which upon reaching the boundary at gives rise to a transmitted wave of amplitude and reflected wave of amplitude . Superradiance corresponds to the condition that

As a further simplification lets assume that is real. The symmetries of the field equations imply that there is another solution satisfying the complex conjugate of the above BCs. Note and are linearly independent which implies that their Wronskian, , does not depend on . Hence which gives

We see that superradiance occurs when .

## 2 Superradiance and the Kerr BH

In this section we discuss superradiant scattering in the Minkowski background. As we have seen superradiant scattering, like the Penrose Process, is a means of extracting energy from a Kerr BH. For an incident wave of suitable conditions, reflection off of the event horizon occurs and the outgoing wave is more energetic (i.e has greater amplitude) than the ingoing wave. It should be emphasized that there is no change in frequency involved in superradaince; it is not a Doppler phenomenon. Waves of arbitrary spin may be considered by introducing the appropriate field term in the Einstein-Hilbert field action. For our purposes, as we are mainly reviewing the work of [2], we will discuss waves of spin

### 2.1 Perturbations of the Kerr Metric

Consider for a moment and electromagnetic perturbation. The total energy flux per steradian at infinity is given by

Now the Maxwell tensor can be expressed in terms of the NP scalars as follows

Using the Kinnersly tetrad 1.4 we have

with the first term corresponding to an ingoing wave and the second to an outgoing wave. Thus we interpret the terms as

(2.1) |

In the case of gravitational perturbations we can get at the desired energy fluxes by a similar method only with the use of the Landau-Lifshitz pseudotensor. The results obtained are

(2.2) |

We need only consider the scalars and as it turns out that the follow from these. Recalling the discussion of the Teukolsky master equation we use the ansatz

with frequency and angular momentum z-component .

The ODE 1.7 combined with the physically desirable BCs, and yields an eigenvalue problem for

where is some whole number such that . For (either Schwarzchild or wave of zero frequency) one has and the eigenfunctions are weighted spinor spherical harmonics. For is not analytically expressible as a function of

It remains to treat the ODE 1.6. If we introduce the coordinate defined by .

the asymptotic solutions of eq 1.6 are given by

(2.3) |

and

(2.4) |

where . Note that as

(2.5) |

### 2.2 Conditions for Superradiant Scattering

We now present the frequency conditions necessary for superradiance arrived at in [2]. We also discuss the dependence of the strength of the reflection on and the spin number .

Comparing equations 2.3 and 2.4 to the results of our discussion of superradiance in the introduction we see that . The frequency dependent condition for superradiant scattering of an -mode wave is thus

(2.6) |

It follows that the condition is the same for all integral values of s and hence the same for scalar, vector and gravitational waves. The convention that and the observation that Schwarzchild solution corresponds to implies superradiant scattering doesn’t occur for Schwarzchild BH.

As a check of physical plausibility it is a good idea to make sure the condition just stated adheres to the laws of BH thermodynamics. In particular, when superradiance occurs energy is extracted from the BH causing and to decrease. At first glance this might seem ominous as the surface area of the horizon grows monotonically with respect to but the law of BH mechanics requires that increase with time. We will show that and decrease in such a way that actually increases under the process of superradiant scattering.

Let be the energy flux of an incident wave of frequency and multipole order m. Then energy flux of the reflected wave is and

(2.7) |

These expressions makes sense; is just the energy flux gained by the wave which, is the negative of the energy flux lost by the BH (i.e ).

Recall the discussion of the irreducible mass in section 1; specifically the relation

It also follows from the definition of that

By considering the total derivative of , making use of and eq 2.7, it follows that

(2.8) |

We see that the process is reversible, that is , only when (ensures and not extremal) and . We will show below that this implies arbitrarily close to . . So reversibility corresponds to a perfectly reflected wave. It is apparent that we may get as close to reversibility as we wish by choosing

We will now study the behavior of in the neighborhoods of and . The amplification factor can be determined numerically by integrating equations 1.6 and 1.7. If we restrict attention to the low frequency realm the problem has also been solved analytically [2]. In what follows we simply give the analytically obtained expressions for without any prior derivation. See Appendix B of [5] if curious.

For a wave of quantum numbers we have reflection coefficient . It can be shown that is of the from

(2.9) |

with the reflection for the scalar wave given by

where is the temperature of the BH. The above expressions are valid in the region and for any spin . Furthermore the expression is physically valid even when the superradiant condition is not satisfied. In that setting eq 2.9 describes the absorption cross section of a rotating BH.

Note that in when , for any we see that for , . Restricting or focus to those physically relevant case we see . For a given

and

so at most the electromagnetic and gravitational waves are amplified a factor of and times more than the corresponding scalar wave, respectively. Letting , we only need keep the lowest order terms in . Hence we see that for .

Now the case. Consider the quantity . One can show that if then in the region . Further constraining

For the extremal Kerr BH, , as we have two cases. Let

If then

(2.10) |

which is continuous and varies monotonically in the vicinity of .

For on the other hand, in the region we have:

(2.11) |

where is a function involving the argument of the terms. See [2] for the exact form.

Note that is satisfied by the majority of modes. For example if it holds for all and if it holds for all . In the vicinity of the onset of superradiant scattering, the reflection coeffienct has an infinite number of oscillations in the region . Aside from the case when we have these oscillations have small amplitude and can be ignored. In case

and the amplification factor is discontinuous near the onset of superradiance. For suggesting that the barrier can be totally transparent, as one would expect for the region unable to superradiantly scatter.

Switching our attentions to the non-extremal Kerr BH, if but and then is described by equations 2.10 or 2.11, depending on the values of sign of , in the region . In region . Hence is continuous at when .

Using our expressions for , calculations of the magnitude of yield: , in particular for we get . For gravitational waves with we have so reflected gravitational waves can more than double in amplitude!

In general for fixed we get the effect decreases as an mth-power exponential; when

## 3 Kerr-AdS and the Superradiant Instability

Now we shift our focus by studying gravitational perturbations in a Kerr-AdS background. Because of the box like nature of AdS we see that superradiance will tend to lead to instabilities; a superradiantly reflected wave is free to bounce off the boundary and return to the ergoregion in finite time. We will see [1] that general instabilities in Kerr-AdS are always superradiant in nature. Finally we will explore the possible evolution of the superradiantly perturbed Kerr AdS BH.

### 3.1 Kerr-AdS

In this section we give a brief overview of the properties of Kerr-AdS BHs. For purposes of studying instability it is useful to use a variation of Boyer-Lundquist coordinates,, introduced by Chambers and Moss [8] given by

where is the rotation parameter of the solution, is the radius of curvature of the AdS background and . In this coordinate system the Kerr-AdS metric is given by

(3.1) |

where

In this frame the horizon angular velocity and temperature are

The Kerr-AdS BH asymptotically approaches global AdS with radius of curvature . This is not obvious when one looks at eq 3.1 because the coordinate frame rotates at infinity with . If one introduces the coordinate change

and then considers the limit as one gets

which we recognize from the section introducing AdS. Hence the conformal boundary of the bulk spacetime is the Einstein static universe .

The ADM mass and angular momentum of the BH are related to the parameters and by . We can express the angular velocity and temperature in the manifestly globally AdS coordinates in terms of those obtained in Chambers-Moss (CM) coordinates:

As in the Kerr BH the event horizon is located at the largest real root of and is a Killing horizon generated by the KVF . We can express the mass parameter in terms of and as follows .

Any regular BH solution must obey and : which gives us restrictions on

(3.2) |

In discussing superradiance it is useful to parametrize the BH by gauge invariant variables associated with its onset: , where . The extremal curve, where , is given by

Note is just the square root of the area of the spatial section of the EH divided by .

### 3.2 Teukolsky Master Eq

In general, the study of the linearized gravitational perturbations of the Kerr BH involves solving a coupled nonlinear PDE obtained from the linearized Einstein equation for the metric perturbation. This is hard to do. Fortunately, as we have already discussed, in the approach of Teukolsky simplifies the problem immensely. By studying gauge invariant scalar variables we can reduce the problem to solving a single PDE. Furthermore, by making use of harmonic decomposition when can make use of seperation of variables to further reduce the problem to two ODEs.

It should be noted that in the setting of AdS background the curvature slightly alters the terms in the ODEs 1.6 and 1.7. Further, in asymptotically AdS we use the Chambers-Moss null tetrad

(3.3) |

rather than the Kinnersly. Still, the information about gravitational perturbations with spin is encoded in the perturbations of the Weyl scalar . The equation of motion for is given by the Teukolsky master equation [10]. We expect something of the form: