1 Introduction

Superradiance on the Reissner-Nordstrøm metric


Laurent DI MENZA111Laboratoire de Mathématiques de Reims, FR CNRS 3399, UFR SEN, Moulin de la Housse, BP 1039, 51687 Reims Cedex 2, France.
E-mail: Laurent.Di-Menza@univ-reims.fr
and Jean-Philippe NICOLAS222LMBA, UMR CNRS n 6205, Université de Brest, 6 avenue Le Gorgeu, 29238 Brest cedex 3, France.
E-mail: Jean-Philippe.Nicolas@univ-brest.fr

Abstract

In this article, we study the superradiance of charged scalar fields on the sub-extremal Reissner-Nordstrøm metric, a mechanism by which such fields can extract energy from a static spherically symmetric charged black hole. A geometrical way of measuring the amount of energy extracted is proposed. Then we investigate the question numerically. The toy-model and the numerical methods used in our simulations are presented and the problem of long time measurement of the outgoing energy flux is discussed. We provide a numerical example of a field exhibiting a behaviour analogous to the Penrose process : an incoming wave packet which splits, as it approaches the black hole, into an incoming part with negative energy and an outgoing part with more energy than the initial incoming one. We also show another type of superradiant solution for which the energy extraction is more important. Hyperradiant behaviour is not observed, which is an indication that the Reissner-Nordstrøm metric is linearly stable in the sub-extremal case.

1 Introduction

Although the existence of black holes was already conjectured in the XVIII century by Mitchell and Laplace, it is in the XX century, within the framework of general relativity, that these objects finally acquired an unambiguous mathematical status, first as explicit solutions of the Einstein equations (the Schwarzschild solution in 1917, the Kerr solution in 1963) then as inevitable consequences of the evolution of the universe (assuming matter satisfies the dominant energy condition and a sufficient amount of it is concentrated in a small enough domain, see S. Hawking and R. Penrose, 1970 [14]). It is in the neighbourhood of black holes that general relativity reveals its most striking aspects. One of the remarkable phenomena occurring in such regions is superradiance : the possibility for bosonic fields to extract energy from the black hole. This energy extraction can result from the interaction of the charges of the black hole and the field, or from the tidal effects of rotation. Superradiance has very different features in each of these two cases. For rotation induced superradiance, the energy extraction is localized in a fixed neighbourhood of the horizon called the ergosphere. In the case of charge interaction, the region where energy extraction takes place varies with the physical parameters of the field (mass, charge, angular momentum) and may even cover the whole exterior of the black hole. We shall refer to it as the “effective ergosphere”, following G. Denardo and R. Ruffini [8] who first introduced this notion333The effective, or generalized, ergosphere should not be confused with the dyadosphere, introduced by G. Preparata, R. Ruffini and Xue S.-S. [20], which is the region outside the horizon of an electromagnetic black hole where the electromagnetic field exceeds the critical value for electron-positron pair production.. A question of crucial importance concerning the stability of black hole spacetimes is the amount of energy that can be extracted. If the process turned out not to be limited and fields could extract an infinite amount of energy, this would indicate that when taking the back reaction of the field on the metric into account, the evolution could be unstable.

In the case of rotationally induced superradiance, although the problems are very delicate, the clear-cut geometrical nature of the ergosphere allows the use of geometric-analytic methods such as vector field techniques in order to study the evolution of fields. Moreover, in the case of Kerr black holes, the separability of the equations provides a convenient access to the frequency analysis of their solutions. Hence, superradiance outside a Kerr black hole has been directly or indirectly studied in various papers. The pioneering work was done by S.A. Teukolsky and W.H. Press in [22], the third in a series of papers analyzing the perturbations of a Kerr black hole by spin , and fields ([26, 21, 22] in chronological order), using the Newman-Penrose formalism and the separability of the equations ; it contained a precise description and numerical analysis of superradiant scattering processes. In 1983, S. Chandrasekhar’s remarkable book [5] provided a complete and up-to-date account of the theory of black hole perturbations, with in particular calculations of scattering matrices (reflection and transmission coefficients) at fixed frequencies for Reissner-Nordstrøm and Kerr black holes. The last decade or so has seen a rather intensive analytical study of superradiance around rotating black holes (see for example [1, 7, 10, 11]). It is now clear that in the slowly rotating case, the amount of energy extracted by scalar fields is finite and controlled by a fixed multiple of the initial energy of the field. Recent results [2] show that the phenomenon is similar for Maxwell fields. In a different spirit, conditions on the stress-energy tensor under which fields or matter outside a Kerr black hole can extract rotational energy are derived in a recent paper by J.-P. Lasota, E. Gourgoulhon, M. Abramowicz, A. Tchekhovskoy, R. Narayan [15] ; these extend the conditions for energy extraction in the Penrose process.

In the case of charge-induced superradiance, we lose the covariance of the equations and the niceties of a strong geometrical structure. To our knowledge, the only example of a theoretical approach to the question is a work of A. Bachelot [3] where a general spherically symmetric situation is considered, a particular case of which is charged scalar fields outside a De Sitter-Reissner-Nordstrøm black hole. A rigorous definition of two types of energy extracting modes is provided : superradiant modes, where the energy extracted is finite, and hyperradiant modes corresponding to the extraction of an infinite amount of energy. We give an excerpt from [3] which is interesting for the clear definition of the two types of modes but also for the last sentence as we shall see below : “when is a superradiant mode, is strictly larger than , but finite, this is the phenomenon of superradiance of the Klein-Gordon fields. […] and diverge at the hyperradiant modes444 and are the transmission and the reflection coefficients for a full scattering experiment, i.e. both incoming and outgoing conditions are considered at the horizon.. The situation differs for the Dirac or Schrödinger equations, for which the reflection is total in the Klein zone.” A. Bachelot’s work also provides a criterion characterizing the presence of hyperradiant modes. Unfortunately, it involves the existence of zeros of a particular analytic function and in the De Sitter-Reissner-Nordstrøm case seems difficult to apply due to the complexity of the function to study.

When it comes to numerical investigation of superradiance by charged black holes, very few results are available in the literature. M. Richartz and A. Saa [23] work in the frequency domain, i.e. via a Fourier analysis in time. The scattering process is then described at a given frequency by the scattering matrix, whose coefficients are the reflection and transmission coefficients. Their paper provides an expression of the transmission coefficient for spin and spin fields outside a Reissner-Nordstrøm black hole, in the small frequency regime. Very recently, a thorough review on superradiance has appeared, by R. Brito, V. Cardoso and P. Pani [4], with in particular a frequency domain analysis of the superradiance of charged scalar fields by a Reissner-Nordstrøm black hole. Another interesting approach is to study superradiance in the time domain, i.e. to follow the evolution of the field without performing a Fourier transform in time. With this approach, the natural thing to look for is a field analogue of the Penrose process. Let us recall here that the Penrose process is a thought experiment proposed by R. Penrose [18] then discussed in more details in R. Penrose and R.M. Floyd [19], by which a particle is sent towards a rotating black hole, enters the ergosphere, disintegrates there into a particle with negative energy, which falls into the black hole, and another particle, which leaves the ergosphere again with more energy than the incoming one. A field analogue of this mechanism would involve wave packets instead of particles, the rest being essentially unchanged. In particular the incoming wave packet needs, when entering the ergosphere, to split spontaneously into a wave packet with negative energy (which is then bound to fall into the black hole or at least stay in the region where negative energy is allowed) and another one which propagates to infinity with more energy than the initial incoming one. To this day and to our knowledge, charge-induced superradiance has not been studied in the time-domain, but there are some important recent results that use this approach in the Kerr case, due to P. Czismadia, A. László and I. Rácz [6] and to A. László and I. Rácz [16]. What the authors observe, in the case of the wave equation outside a rather fast rotating Kerr black hole, is that the phenomenon described above does not seem to occur ; instead their simulations show that wave packets that are a priori in the superradiant regime, undergo an almost complete reflection as they enter the ergosphere. This is in sharp contrast with the excerpt from [3] in the previous paragraph. But of course one must be careful when comparing things that are not directly comparable ; A. Bachelot in [3] focuses on a general spherically symmetric setting, while in [6, 16] P. Czismadia, A. László and I. Rácz work in the case of a rotating black hole which is not even slowly rotating and cannot therefore be seen as a small perturbation of a spherically symmetric situation.

The present paper focuses on a case of charge-induced superradiance : the evolution of charged scalar fields outside a sub-extremal Reissner-Nordstrøm black hole, which we study in the time domain. It is organized as follows. Section 2 contains a description of superradiance purely as a problem of analysis of partial differential equations and a reduction to a toy model. The natural conserved energy is not positive definite and the region where the energy density is allowed to become negative depends on the choice of physical parameters. In section 3, after a short account of Reissner-Nordstrøm metrics and the definition of the charged D’Alembertian on such backgrounds, we construct a conserved energy current. The field equation that we study is not covariant because the electromagnetic field is not influenced in return by the scalar field, so there is no way of constructing a conserved stress-energy tensor. However, we introduce a simple modification of the stress-energy tensor for a Klein-Gordon field, which, though not conserved, leads to a conserved current when we contract it with the global timelike Killing vector field. This gives us a geometrical means of measuring the amount of energy extracted by a given field. Section 4 contains the numerical investigation. The algorithm is described, with a particular attention paid to the treatment of the boundary. This is followed by a detailed study of the toy model, with the hyperradiant and superradiant cases. Then the full problem is tackled. Our observations indicate the absence of hyperradiant modes and the existence of superradiant modes. We exhibit two types of superradiant solutions. The first kind is the field analogue of the Penrose process : incoming wave packets splitting in the correct manner and coming out of the effective ergosphere with more energy than they brought in. Our results show an energy extraction of nearly (i.e. a gain, or reflection coefficient, close to ). We do not observe the total reflection seen in [6, 16] and our simulations are consistant with the general spherically symmetric analysis in [3] and the numerical simulations in [4]. The second kind is given by solutions whose data are located within the effective ergosphere but on which the energy is nevertheless positive definite ; we call them flare-type initial data because the field vanishes at but not its time derivative. This second kind of solution provides much larger energy gains. For both these classes of solutions, the gain always stabilizes to a finite value, ruling out hyperradiant behaviour. As a test of the robustness of our numerical scheme, we push it into the high energy regime ; we observe exactly as expected the concentration of the propagation of the field along the null geodesics as well as the fact that superradiance is a low energy phenomenon. We discuss our results in the conclusion.

2 Analytic description and toy model

The evolution of a charged scalar field with charge , mass and angular momentum (meaning that ) outside a Reissner-Nordstrøm black hole is described by the equation on

(1)

where and , are the charge and mass of the black hole, is determined in terms of by where is an analytic function555It is the reciprocal function of defined in (14) and (15) in the sub-extremal and extreme cases. on , strictly increasing, such that

and the function is given by

The region corresponds to the neighbourhood of the event horizon localized at  ; the function is positive on and vanishes at .

This equation has a natural conserved quantity given by

(2)

Since vanishes at , unless the field is uncharged the potential

(3)

becomes negative near the horizon and the conserved quantity is therefore not positive definite.

Definition 2.1.

The effective ergosphere is the region where the potential (3) becomes negative.

The localization of the effective ergosphere depends on the respective values of , , , and in a complicated way. In the worst cases, it covers a neighbourhood of infinity, maybe even the whole real axis. As soon as the mass of the field is positive, the potential is positive in a neighbourhood of infinity and the effective ergosphere is localized in a compact region around the black hole. This is also the case when the mass of the field vanishes and its angular momentum is large enough, namely . In the case where and (the case is not interesting for us since there is no superradiance), the effective ergosphere covers a neighbourhood of infinity.

Note that the mass of the field, when it is large enough, prevents the occurrence of super-radiance in the sense that there exists a positive definite conserved energy. This is well-known and easy to prove but does not seem to be present in the literature, apart from a remark in [3] p. 1207. We state and prove the result here.

Lemma 2.1.

For , there is no superradiance in the sense that there exists another conserved energy that is positive definite outside the black hole.

Proof. Putting

we find that satisfies the equation

whose natural conserved quantity

is positive definite if and only if . Indeed, we have (using and the fact that )

This quantity is positive in the neighbourhood of if and only if . In this case it is positive everywhere outside the black hole, since . All the other terms are clearly positive except . However, outside the black hole, , whence

Equation (1) is of the general form

(4)

with

A usual toy model for superradiance is obtained by taking equation (4) with potentials and that are constant outside of a fixed interval, say , . A toy model for the massless case is given by

(5)

whereas for the massive case we choose for and

(6)
(7)

The extreme situation often considered as a toy model is the case when and as step functions, i.e. . We shall see in the numerical study that this changes the situation drastically since hyperradiant modes are then observed.

On the toy model, there is a clear-cut way of measuring the gain of energy, because there is a fixed zone () where the energy is positive definite. Consequently, taking initial data supported in and letting them evolve, the gain at time is given by

(8)

where

(9)

We define the asymptotic gain as the following limit when it exists :

(10)

In the physical case of charged Klein-Gordon fields outside a Reissner-Nordstrøm black hole, for fixed and , the region where the potential is negative depends on the physical parameters of the field and can spread out to the whole exterior of the black hole. If we use a similar approach to the one described above for the toy model, we need to change the region of integration for the energy and the localization of the data when we change these parameters. This makes the comparison of gains difficult for different values of , and because the same initial data may not be suitable for all cases. Moreover, the extreme situations where the potential is negative everywhere cannot be treated this way. It is much better to find a geometrical way of measuring an energy flux that does not depend on a reference positive definite energy. We shall see when we come to numerical experiments that this geometrical approach is also more convenient for the toy model.

3 Construction of a geometric conserved quantity

Our approach is to define a Poynting vector, i.e. an energy current, whose flux across a hypersurface for a given angular momentum is exactly the conserved energy (2). This is typically done by considering a conserved stress-energy tensor and contracting it with the global timelike Killing vector field. In our case however, the lack of covariance of the equation prevents the existence of such a tensor. Nevertheless, we find a modification of the construction for the Klein-Gordon equation, involving the Maxwell potential, that gives rise to an adequate conserved Poynting vector. We first recall the essential features of Reissner-Nordstrøm metrics and the definition of the charged wave equation on such backgrounds.

3.1 Charged scalar fields around Reissner-Nordstrøm black holes

An asymptotically flat universe containing nothing but a charged static spherical black hole is described, in Schwarzschild coordinates , by the manifold

equipped with the Reissner-Nordstrøm metric

(11)

where is the mass of the black hole and its charge. Two types of singularities appear in the expression (11) of  : is a true curvature singularity while the spheres where vanishes, called horizons, are mere coordinate singularities ; they can be understood as smooth null hypersurfaces by means of Kruskal-type coordinate transformations (see for example [13]). The fact that these hypersurfaces are null reveals that a horizon can be crossed one way, but requires a speed greater than the speed of light to be crossed the other way, hence the name “event horizon”. The black hole is the part of space-time situated beyond the outermost horizon. There are three types of Reissner-Nordstrøm metrics, depending on the respective importance of and .

  1. Sub-extremal case : for , the function has two real roots

    (12)

    The sphere is the outer horizon, or horizon of the black hole and is the inner or Cauchy horizon.

  2. Extreme case : for , is the only root of and there is only one horizon.

  3. Super-extremal case : for , the function has no real root. There are no horizons, the space-time contains no black hole and the singularity is naked (i.e. not hidden beyond a horizon).

We consider only the sub-extremal case but will mention some striking aspects of the extreme case in this section. On such backgrounds, we study the charged Klein-Gordon equation from the point of view of a distant observer, whose experience of time is described by and whose perception is limited to the domain of outer communication (i.e. the exterior of the black hole)

(13)

The horizon of the black hole is perceived by distant observers as an asymptotic region : a light ray, or a massive object, falling towards the black hole, will only reach the horizon as . It is convenient to introduce a new radial coordinate , referred to as the Regge-Wheeler tortoise coordinate, such that

The radial null geodesics travel at constant speed in this new variable. This will simplify the form of the wave equation as a perturbation of a -dimensional wave equation with constant coefficients.

In the case where , is given by ( being an arbitrary real number)

The surface gravity (resp. ) at the horizon of the black hole (resp. the inner horizon), is defined by

Thus, the expression of can be further simplified as

(14)

In the extreme case, we have

(15)

In both cases, the map is an analytic diffeomorphism from onto , corresponds to and corresponds to . Moreover,

and as ,

We see that there is a radical change of behaviour of the function between the sub-extremal case and the extreme case. For the Klein-Gordon equation on these metrics, this means that the mass term will be exponentially decreasing in at the horizon in the sub-extremal case, whereas in the extreme case it will be a long range term falling-off like .

In coordinates , the Reissner-Nordstrøm metric has the form

(16)

and the domain of outer communication is described as

(17)

We shall denote by the level hypersurfaces of the variable outside the black hole

(18)

The charged Klein-Gordon equation on a Reissner-Nordstrøm metric reads

(19)

where

being the electromagnetic vector potential (expressed here as a one-form)

A short calculation gives the following explicit form of (19) :

Changing the unknown function to simplifies the radial part of the equation and we obtain

(20)

and for a given angular momentum this gives equation (1) with .

3.2 Geometric conserved current

It is well-known that a charged scalar field on a fixed charged background does not admit a conserved stress-energy tensor. This is due to the fact that in this model, the electromagnetic field influences the scalar field but is not influenced in return ; only for the coupled Maxwell-Klein-Gordon system do we have a good conservation of stress-energy. However, if we modify in a simple way the stress-energy tensor of a Klein-Gordon equation to include the electromagnetic vector-potential, we obtain a new non-conserved tensor which has the advantage of inducing a conserved current. The flux of this current through a spacelike slice for a solution with angular momentum exactly corresponds to the conserved energy (2).

The stress-energy tensor for the Klein-Gordon equation

(21)

is given by

and satisfies which is zero whenever is a solution of (21). Now if we consider a Killing vector , i.e. a vector field whose flow is an isometry, which is characterized by or equivalently by the Killing equation

(22)

the contraction gives a conserved current :

In particular, taking

we see that the energy current measured by an observer static at infinity is divergence-free :

For our charged, and necessarily complex, Klein-Gordon field, we modify the stress-energy tensor as follows. Let us consider the symmetric tensor defined in terms of and by

(23)

We use it to construct a Poynting vector by contracting with

(24)

The following lemma summarizes the properties of  ; its proof is given in appendix A.

Lemma 3.1.

The tensor does not satisfy , but the Poynting vector field is divergence-free :

(25)
Remark 3.1.

We shall establish in the next section that the flux of across a hypersurface is exactly the conserved energy (2) for a given angular momentum .

3.3 Energy fluxes across two types of hypersurfaces

Given a piecewise oriented hypersurface, a vector field transverse to compatible with the orientation of , a normal vector field to such that , the flux of a vector field across reads

and is independent of the choice of and satisfying the above properties. For the flux of the energy current across a hypersurface , we may choose and the flux is therefore given by

where

Across a hypersurface

orientated by , we can take and we get

since and . We give the expressions of these fluxes in terms of and . The energy flux across becomes

Developing the second term for , we get

the last term vanishes for smooth compactly supported functions and therefore also for functions in the finite energy Hilbert space by a standard density argument. This entails the following expression of the energy flux through  :

which, once restricted to an angular momentum , is exactly the expression (2) of the conserved energy .

The outgoing energy flux across in terms of has the form

(26)

since gives the right orientation.

3.4 Practical measure of energy gain

Given initial data belonging to a subspace on which the energy is positive definite, large enough so that the support of the initial data is entirely contained in , the quantity we wish to measure is the ratio of the total energy radiated away through at time , to the energy of the data. We shall observe superradiance if we can find such initial data for which, as becomes large, this ratio stabilizes to a value that is strictly larger than . With our previous construction of conserved energy current, is easy to evaluate for any solution whose initial data are compactly supported ; it is given as follows

(27)

The asymptotic gain, which is the theoretical quantity of interest here, is defined as the following limit, if it exists :

(28)

Note that it is not at all obvious that this limit should exist. The fact that is well-defined and independent of chosen as above is a scattering property that we observe numerically.

4 Numerical study of superradiance

We now use numerical tools in order to observe superradiance for the toy-model as well as for the Reissner-Nordstrøm model.

4.1 Discretization of the problem

Equation (4) can be expressed as the following first-order system

(29)

with and . This is a linear evolution system that can be numerically solved using a finite difference scheme both in time and space. We first consider a semi-implicit time discretization of the problem : denoting the prescribed timestep, we set , , and write that (29) is satisfied at the midpoint time . We respectively approximate time derivatives and with and . Expressing and with the second order approximations and , (29) reduces to the simplified system

” denoting the Identity matrix. The space operator is then discretized using the standard point approximation as follows : when considering a uniform spatial mesh with space step , i.e. for all , we have that for any function depending on

(30)

up to a second order error term. Dealing with a bounded numerical domain, say , we are faced with unknowns and . This leads to the following finite dimensional linear system to solve at each iteration :

with , , and , where stands for the tridiagonal matrix arising when considering the discretization of given by (30).

This scheme is known to be convergent under the assumption that the Courant-Friedrichs-Lewy condition is fulfilled : this grants the numerical stability of the scheme and the Lax-Richtmyer convergence Theorem then gives the convergence (see for example [25] for technical details of the numerical analysis of finite differences schemes). This means that taking well-adapted time and space meshgrid parameters leads to an accurate approximation of the exact solution of the wave equation. For all the tests presented in this paper, we have checked that the numerical solutions that are obtained do not depend on the choice of the grid parameters.

4.2 The boundary conditions

Since we restrict ourselves to a bounded spatial domain, often referred to as the computational domain, we have to be very careful with the numerical treatment of the boundary. Indeed, for sufficiently large times, the solution reaches the frontier and ill-adapted boundary conditions may lead to unphysical solutions that cannot represent the restriction of the correct solution to the domain under consideration. As a consequence, waves may be reflected by the boundary and propagate in the wrong direction. This problem occurs even for the homogeneous wave equation

(31)

that admits the general solution involving two classes of solutions that propagate at speeds . In this case, it is possible to derive transparent conditions that enable to compute a numerical solution that will not be affected by the boundary. In the last three decades, there has been a sustained effort to extend these boundary conditions to more general equations. Transparent conditions are delicate to use in the general case, since they involve nonlocal operators as well as possibly infinite expansions. For a comprehensive study of these conditions as well as the search for approximate conditions, one can refer to the pioneering works [9], [12] and [17] (note that such conditions have been studied for other problems such as diffusion or Schrödinger equations, for which propagation at finite velocity no longer holds). It can be shown that for the model equation

(32)

the transparent condition that has to be considered at the right extremity of the segment simply reads

This is a local condition that can easily be implemented in the finite difference discretization. For , one recognizes the condition which selects propagation at speed across the right boundary. The symmetric condition

is set at the left boundary. The expression of these boundary conditions can be obtained by means of the algebraic identity which holds when and are differential operators.

We now illustrate the influence of the boundary condition on the calculated solution, for several values of constant potentials and . In all our simulations, the timestep and the spatial mesh are taken in order to fulfill the Courant-Friedrichs-Lewy condition that grants the numerical stability and the convergence of the scheme. We first perform computations for the homogeneous case  : we deal with the wave equation (31) for initial data

where . The associated solution solves the transport equation and propagates at constant speed  ; it is an incoming wave packet with zero frequency. We performed our computations on the space domain discretized with , until final computational time .

As shown in Figure 1, the transparent condition enables the solution to go outside the numerical domain and the boundary has no effect on long-time dynamics. The computed solution mimics the profile of the one calculated on a larger domain, whereas taking Dirichlet homogeneous conditions on the boundary gives birth to a reflected wave that propagates to the right.

Figure 1: Solution computed first with transparent boundary conditions, second on a larger spatial domain and shown only on the smaller domain, third with Dirichlet boundary conditions.

We now perform the same computations for constant potentials and , using the same computational parameters as before and starting from the same Cauchy data. The same conclusions can be drawn, as can be seen in Figure 2 where the amplitude is now plotted, since the solution is now complex-valued : once again, the computed solution is not affected by the boundaries. Let us mention here that as opposed to the homogeneous case, the support of the solution spreads out through time and the transparent conditions at both extremities are required, even for incoming wave packet Cauchy data.

Figure 2: Amplitude of the solution computed first with transparent boundary conditions, second on a larger domain and shown only on the smaller domain, third with Dirichlet boundary conditions.

In the case , the problem becomes more difficult to handle : the transparent boundary condition is much more costly to implement, since it involves the evaluation of a pseudodifferential operator that is non local in time. Indeed the factorization

shows that the boundary condition now involves an operator with a square root symbol. Nonlocal boundary conditions are delicate to implement because first, they require the full storage of the solution at the boundary from the initial computational time until current time and also, the computation of the integral boundary operator may lead to an additional error. For this reason, we prefer to deal with a splitting strategy in order to get rid of this difficulty and to use local exact transparent conditions for the resolution of the homogeneous equation (that is for ). Between two consecutive time increments, the initial problem (4) is decomposed into the two elementary problems

This first equation is numerically solved as above with the use of the transparent conditions, whereas the second one that reduces to an ODE is integrated using a classical implicit scheme. Simulations have been performed with the same discretization parameters as before, for the case and and up to final time . The classical second-order Strang splitting (see [24]) is used in order to get a global second-order method in time. The results displayed in Figure 3 show that once again transparent conditions produce a correct approximation of the solution on a larger domain (on which the boundary has no influence), even if small differences with the reference solution can be observed as a consequence of the use of the splitting algorithm.

Figure 3: Amplitude of the solution computed first with transparent boundary conditions, second on a larger domain and shown only on the smaller domain, third with Dirichlet boundary conditions.

The problem of the boundary treatment is crucial for the computation of the energy gain at large times when seeking superradiance evidence, as will be seen later.

4.3 Numerical results for the toy model

We now study the toy model (4) for potentials and defined as follows :

Such a potential is a smooth approximation of the limit case , where stands for the Heaviside function. Once is prescribed, we set . Using such a relation, we have that , for and , for .

Once again, we consider incoming wave packet initial data

(33)
(34)

where is fixed and the frequency can be varied to observe the high and low energy behaviours. We analyze the behaviour of the solution of equation (4) associated with such data. The gain is measured using formula (8).

We first investigate the influence of the smoothing parameter on the gain of energy, recalling that in the limit case the two potentials are discontinuous if not identically zero. We only consider the case