Asymptotics of linear waves and resonances
with applications to black holes
We apply the results of [Dy13] to describe asymptotic behavior of linear waves on stationary Lorentzian metrics with -normally hyperbolic trapped sets, in particular Kerr and Kerr–de Sitter metrics with and . We prove that if the initial data is localized at frequencies , then the energy norm of the solution is bounded by , for , where is a natural dynamical quantity. The key tool is a microlocal projector splitting the solution into a component with controlled rate of exponential decay and an remainder; this splitting can be viewed as an analog of resonance expansion. Moreover, for the Kerr–de Sitter case we study quasi-normal modes; under a dynamical pinching condition, a Weyl law in a band holds.
The subject of this paper are decay properties of solutions to the wave equation for the rotating Kerr (cosmological constant ) and Kerr–de Sitter () black holes, as well as for their stationary perturbations. In the recent decade, there has been a lot of progress in understanding the upper bounds on these solutions, producing a polynomial decay rate for Kerr and an exponential decay rate for Kerr–de Sitter (the latter is modulo constant functions). The weaker decay for is explained by the presence of an asymptotically Euclidean infinite end; however, this polynomial decay comes from low frequency contributions.
We instead concentrate on the decay of solutions with initial data localized at high frequencies ; it is related to the geometry of the trapped set , consisting of lightlike geodesics that never escape to the spatial infinity or through the event horizons. The trapped set for both Kerr and Kerr–de Sitter metrics is -normally hyperbolic, and this dynamical property is stable under stationary perturbations of the metric – see §3.6. The key quantities associated to such trapping are the minimal and maximal transversal expansion rates , see (2.9), (2.10). Using our recent work [Dy13], we show the exponential decay rate , valid for (Theorem 1). This bound is new for the Kerr case, complementing Price’s law.
Our methods give a more precise microlocal description of long time propagation of high frequency solutions. In Theorem 2, we split a solution into two approximate solutions to the wave equation, and , with the rate of decay of between and and bounded from above by , all modulo errors. This splitting is achieved using the Fourier integral operator constructed in [Dy13] using the global dynamics of the flow.
For the case, we furthermore study resonances, or quasi-normal modes, the complex frequencies of solutions to the wave equation of the form . Under a pinching condition which is numerically verified to be true for a large range of parameters (see Figure 2(a)), we show existence of a band of quasi-normal modes satisfying a Weyl law – Theorem 3. In particular, this provides a large family of exact high frequency solutions to the wave equation that decay no faster than . We finally compare our theoretical prediction on the imaginary parts of resonances in the band with the exact quasi-normal modes for Kerr computed by the authors of [BeCaSt], obtaining remarkable agreement – see Figure 2(b).
Theorems 1–3 are related to the resonance expansion and quantization condition proved for the slowly rotating Kerr–de Sitter in [Dy12]. In this paper, however, we use dynamical assumptions stable under perturbations, rather than complete integrability of geodesic flow on Kerr(–de Sitter), and do not recover the precise results of [Dy12].
Statement of results. The Kerr(–de Sitter) metric, described in detail in §3.1, depends on three parameters, (mass), (speed of rotation), and (cosmological constant). We assume that the dimensionless quantities and lie in a small neighborhood (see Figure 1(a)) of either the Schwarzschild(–de Sitter) case,
or the subextremal Kerr case
Our results apply as long as certain dynamical conditions are satisfied, and likely hold for a larger range of parameters, see the remark following Proposition 3.2. To facilitate the discussion of perturbations, we adopt the abstract framework of §2.2, with the spacetime and a Lorentzian metric on which is stationary in the sense that is Killing. The space slice is noncompact because of the spatial infinity and/or event horizon(s); to measure the distance to those, we use a function , such that is compact for each . For the exact Kerr(–de Sitter metric), the function is defined in (3.6).
We study solutions to the wave equation in ,
with and the time variable shifted so that the metric continues smoothly past the event horizons – see (3.45). To simplify the statements, and because our work focuses on the phenomena driven by the trapped set, we only study the behavior of solutions in for some small . Define the energy norm
Fix , , and let be the Kerr(–de Sitter) metric with near one of the cases (1.1) or (1.2), or its small stationary perturbation as discussed in §3.6 (the maximal size of the perturbation depending on ).
Here we say that is localized at frequencies , if for each coordinate neighborhood in and each , the Fourier transforms in the corresponding coordinate system satisfy for each ,
where is a constant independent of . For the proof, it is more convenient to use semiclassical rescaling of frequencies , where is the semiclassical parameter, and the notion of -wavefront set . The requirement that is microlocalized at frequencies is then equivalent to stating that is contained in a fixed compact subset of , with denoting the zero section; see §2.1 for details.
The main component of the proof of Theorem 1 is the following
The decomposition is achieved in §2.4 using the Fourier integral operator constructed for -normally hyperbolic trapped sets in [Dy13]. The component enjoys additional microlocal properties, such as localization on the outgoing tail and approximately solving a pseudodifferential equation – see the proof of Theorem 4 in §2.4 and [Dy13, §8.5]. We note that (1.9) gives a lower bound on the rate of decay of the approximate solution , if is not too small compared to , and the existence of a large family of solutions with the latter property follows from the construction of .
We remark that Theorems 1 and 2 are completely independent from the behavior of linear waves at low frequency. In fact, we do not even use the boundedness in time of solutions for the wave equation, assuming merely that they grow at most exponentially (which is trivially true in our case); this suffices since remainders overcome such growth for . If a boundedness statement is available, then our results can be extended to all times, though the corresponding rate of decay stays fixed for because of the term.
To formulate the next result, we restrict to the case , or its small stationary perturbation. In this case, the metric has two event horizons and we consider the discrete set of resonances, as defined for example in [Va10]. As a direct application of [Dy13, Theorems 1 and 2], we obtain two gaps and a band of resonances in between with a Weyl law (see Figure 1(b)):
1. For small enough, there are no resonances in the region
and the corresponding semiclassical scattering resolvent, namely the inverse of the operator (3.56), is bounded by for in this region.
The pinching condition (1.12) is true for the non-rotating case , since there (see Proposition 3.8). However, it is violated for the nearly extremal case , at least for small enough; in fact, as , stays bounded away from zero, while converges to zero – see Proposition 3.9 and Figure 2(a).
Note that is the size of the resonance free strip and thus gives the minimal rate of exponential decay of linear waves on Kerr–de Sitter, modulo terms coming from finitely many resonances, by means of a resonance expansion – see for example [Va10, Lemma 3.1].
To demonstrate the sharpness of the size of the band of resonances , we use the exact quasi-normal modes for the Kerr metric computed (formally, since one cannot meromorphically continue the resolvent in the case; however, one could consider the case of a very small positive ) by Berti–Cardoso–Starinets [BeCaSt]. Similarly to the quantization condition of [Dy12], these resonances are indexed by three integer parameters (depth), (angular energy), and (angular momentum). The parameter roughly corresponds to the real part of the resonance and the parameter , to its imaginary part. We define
We compare with and plot the supremum of the relative error over for different values of ; the error decays like – see Figure 2(b).
Previous work. We give an overview of results on decay and non-decay on black hole backgrounds; for a more detailed discussion of previous results on normally hyperbolic trapped sets and resonance asymptotics, see the introduction to [Dy13].
The study of boundedness of solutions to the wave equation for the Schwarzschild () black hole was initiated in [Wa, KaWa] and decay results for this case have been proved in [BlSt, DaRo09, MMTT, Lu10a]. The slowly rotating Kerr case () was considered in [AnBl, DaRo11, DaRo08, Ta, TaTo, To, MeTaTo, Lu10b], and the full subextremal Kerr case () in [FKSY, FKSYErr, DaRo10, DaRo12, Sh1, Sh2] – see [DaRo12] for a more detailed overview. In either case the decay is polynomial in time, with the optimal decay rate . A decay rate of , known as Price’s Law, was proved in [DoScSo11, DoScSo12] for linear waves on the Schwarzschild black hole for solutions living on the -th spherical harmonic; the constant in the depends on . Our Theorem 1 improves on these decay rates in the high frequency regime , for times .
The extremal Kerr case () was recently studied for axisymmetric solutions in [Ar12], with a weaker upper bound due to the degeneracy of the event horizon. The earlier work [Ar11a, Ar11b] suggests that one cannot expect the decay to hold in the extremal case. In the high frequency regime studied here, we do not expect to get exponential decay due to the presence of slowly damped geodesics near the event horizon, see Figure 2(a) above.
The Schwarzschild–de Sitter case () was considered in [SBZw, BoHä, DaRo07, MeSBVa], proving an exponential decay rate at all frequencies, a quantization condition for resonances, and a resonance expansion, all relying on separation of variables techniques. In [Dy11a, Dy11b, Dy12], a same flavor of results was proved for the slowly rotating Kerr–de Sitter (). The problem was then studied from a more geometric perspective, aiming for results that do not depend on symmetries and apply to perturbations of the metric – the resonance free strip of [WuZw] for normally hyperbolic trapping, the gluing method of [DaVa], and the analysis of the event horizons and low frequencies of [Va10] together give an exponential decay rate which is stable under perturbations, for , provided that there are no resonances in the upper half-plane except for the resonance at zero. Our Theorem 3 provides detailed information on the behavior of resonances below the resonance free strip of [WuZw], without relying on the symmetries of the problem.
Finally, we mention the the Kerr–AdS case (). The metric in this case exhibits strong (elliptic) trapping, which suggests that the decay of linear waves is very slow because of the high frequency contributions. A logarithmic upper bound was proved in [HoSm11], and existence of resonances exponentially close to the real axis and a logarithmic lower bound were established in [Ga, HoSm13].
Quasi-normal modes (QNMs) of black holes have a rich history of study in the physics literature, see [KoSc]. The exact QNMs of Kerr black holes were computed in [BeCaSt], which we use for Figure 2(b). The high-frequency approximation for QNMs, using separation of variables and WKB techniques, has been obtained in [YNZZZC, YZZNBC, Ho]. In particular, for the nearly extremal Kerr case their size of the resonance free strip agrees with Proposition 3.9; moreover, they find a large number of QNMs with small imaginary parts, which correspond to a positive proportion of the Liouville tori on the trapped set lying close to the event horizon. See [YNZZZC] for an overview of the recent physics literature on the topic. We finally remark that the speed of rotation of an astrophysical black hole (NGC 1365) has recently been accurately measured in [RHMWBCCGHNSZ], yielding a high speed of rotation: at 90% confidence.
Structure of the paper. In §2, we study semiclassical properties of solutions to the wave equation on stationary Lorentzian metrics with noncompact space slices. We operate under the geometric and dynamical assumptions of §2.2; while these assumptions are motivated by Kerr(–de Sitter) metrics and their stationary perturbations, no explicit mention of these metrics is made. The analysis of §2 works in a fixed compact subset of the space slice, and the results apply under microlocal assumptions in this compact subset (namely, outgoing property of solutions to the wave equation for Theorems 1–2 and meromorphic continuation of the scattering resolvent with an outgoing parametrix for Theorem 3) which are verified for our specific applications in §3.4 and §3.5. In §2.3, we reduce the problem to the space slice via the stationary d’Alembert–Beltrami operator and show that some of the assumptions of [Dy13, §§4.1, 5.1] are satisfied. In §2.4, we use the methods of [Dy13] to prove asymptotics of outgoing solutions to the wave equation.
Next, §3 contains the applications of [Dy13] and §2 to the Kerr(–de Sitter) metrics and their perturbations. In §3.1, we define the metrics and establish their basic properties, verifying in particular the geometric assumptions of §2.2. In §3.2, we show that the trapping is -normally hyperbolic, verifying the dynamical assumptions of §2.2. In §3.3, we study in greater detail trapping in the Schwarzschild(–de Sitter) case and in the nearly extremal Kerr case , in particular showing that the pinching condition (1.12) is violated for the latter case; we also study numerically some properties of the trapping for the general Kerr case, and give a formula for the constant in the Weyl law. In §3.4, we study solutions to the wave equation on Kerr(–de Sitter), using the results of §2.4 to prove Theorems 1 and 2. In §3.5, we use the results of [Dy13] and [Va10] to prove Theorem 3 for Kerr–de Sitter. Finally, in §3.6, we explain why our results apply to small smooth stationary perturbations of Kerr(–de Sitter) metrics.
2. General framework for linear waves
2.1. Semiclassical preliminaries
Let be an -dimensional manifold without boundary. Following [Dy13, §3.1], we consider the class of all semiclassical pseudodifferential operators with classical symbols of order . If is noncompact, we impose no restrictions on how fast the corresponding symbols can grow at spatial infinity. The microsupport of a pseudodifferential operator , also known as its -wavefront set , is a closed subset of the fiber-radially compactified cotangent bundle . We denote by the class of all pseudodifferential operators whose wavefront set is a compact subset of (and in particular lies away from the fiber infinity). Finally, we say that microlocally in some open set , if ; similar notions apply to tempered distributions and operators below.
Using pseudodifferential operators, we can study microlocalization of -tempered distributions, namely families of distributions having a polynomial in bound in some Sobolev norms on compact sets, by means of the wavefront set . Using Schwartz kernels, we can furthermore study -tempered operators and their wavefront sets . Besides pseudodifferential operators (whose wavefront set is this framework is the image under the diagonal embedding of the wavefront set used in the previous paragraph) we will use the class of compactly supported and compactly microlocalized Fourier integral operators associated to some canonical relation , see [Dy13, §3.2]; for , is compact.
The -wavefront set of an -tempered family of distributions can be characterized using the semiclassical Fourier transform
We have if and only if there exists a coordinate neighborhood of in , a function with , and a neighborhood of in such that if we consider as a function on using the corresponding coordinate system, then for each ,
The proof is done analogously to [HöIII, Theorem 18.1.27].
One additional concept that we need is microlocalization of distributions depending on the time variable that varies in a set whose size can grow with . Assume that is a family of distributions on , where is fixed and depends on . For , define the shifted function
so that is a distribution on a time interval independent of . We then say that is -tempered uniformly in , if is -tempered uniformly in , that is, for each , there exist constants and such that for all . Next, we define the projected wavefront set , where is the momentum corresponding to and is the fiber-radial compactification of the vector bundle , with part of the fiber, as follows: does not lie in if and only if there exists a neighborhood of in such that
for each compactly supported such that . If is independent of , then is simply the closure of the projection of onto the variables. The notion of makes it possible to talk about being microlocalized inside, or being , on subsets of independent of .
We now discuss restrictions to space slices. Assume that is -tempered uniformly in and moreover, does not intersect the spatial fiber infinity . Then (as well as all its derivatives in ) is a smooth function of with values in , is -tempered uniformly in , and
uniformly in . One can see this using (2.1) and the formula for the Fourier transform of the restriction of to the hypersurface :
2.2. General assumptions
In this section, we study Lorentzian metrics whose space slice is noncompact, and define -normal hyperbolicity and the dynamical quantities in this case.
Geometric assumptions. We assume that:
is an dimensional Lorentzian manifold of signature , and , where , the space slice, is a manifold without boundary;
the metric is stationary in the sense that its coefficients do not depend on , or equivalently, is a Killing field;
the space slices are spacelike, or equivalently, the covector is timelike with respect to the dual metric on ;
The (nonsemiclassical) principal symbol of the d’Alembert–Beltrami operator (without the negative sign), denoted by , is
here denotes a point in and a covector in . The Hamiltonian flow of is the (rescaled) geodesic flow on ; we are in particular interested in nontrivial lightlike geodesics, i.e. the flow lines of on the set , where denotes the zero section.
Note that we do not assume that the vector field is timelike, since this is false inside the ergoregion for rotating black holes. Because of this, the intersections of the sets , invariant under the geodesic flow, with the energy surface need not be compact in the direction, and it is possible that will blow up in finite time along a flow line of , while stays in a compact subset of .111The simplest example of such behavior is , considering the geodesic starting at . We consider instead the rescaled flow
Here never vanishes on by assumption (3). Since , the variable grows linearly with unit rate along the flow . The flow lines of (2.3) exist for all as long as stays in a compact subset of . The flow is homogeneous, which makes it possible to define it on the cosphere bundle , which is the quotient of by the action of dilations. Finally, the flow preserves the restriction of the symplectic form to the tangent bundle of .
We next assume the existence of a ‘defining function of infinity’ on the space slice with a concavity property:
there exists a function such that on , for the set
is compactly contained in , and there exists such that for each flow line of (2.3), and with naturally defined on ,
We now define the trapped set:
Let be a maximally extended flow line of (2.3). We say that is trapped as , if there exists such that for all (and as a consequence, exists for all ). Denote by the union of all trapped as ; similarly, we define the union of all trapped as . Define the trapped set .
If and for some , then it follows from assumption (4) that is not trapped as . Also, if is not trapped as , then and for large enough. It follows that are closed conic subsets of , and .
We next split the light cone into the sets and of positively and negatively time oriented covectors:
Since never vanishes on by assumption (3), we have .
We fix the sign of on the trapped set, in particular requiring that :
Dynamical assumptions. We now formulate the assumptions on the dynamical structure of the flow (2.3). They are analogous to the assumptions of [Dy13, §5.1] and related to them in §2.3 below. We start by requiring that are regular:
for a large constant , are codimension 1 orientable submanifolds of ;
intersect transversely inside , and the intersections are symplectic submanifolds of .
We next define a natural invariant decomposition of the tangent space to at . Let be the symplectic complement of the tangent space to . Since has codimension 2 and is contained in , is a two-dimensional vector subbundle of containing . Since on , we can define the one-dimensional vector subbundles of
Since is a codimension 1 submanifold of and is tangent to , we see that is coisotropic and then are one-dimensional subbundles of ; moreover, since , we find . Since is symplectic, we have
We now formulate the dynamical assumptions on the linearization of the flow with respect to the splitting (2.8). Define the minimal expansion rate in the transverse direction as the supremum of all for which there exists a constant such that
with denoting any smooth -independent norm on the fibers of , homogeneous of degree zero with respect to dilations on . Similarly, define as the infimum of all for which these exists a constant such that
We now formulate the dynamical assumption of -normal hyperbolicity:
, where is the maximal expansion rate of the flow along , defined as the infimum of all for which there exists a constant such that
The large constant determines how many terms we need to obtain in semiclassical expansions, and how many derivatives of these terms need to exist – see [Dy13]. Theorem 3 simply needs to be large (in principle, depending on the dimension), while Theorems 1 and 2 require to be large enough depending on . For exact Kerr(–de Sitter) metrics, our assumptions are satisfied for all , but a small perturbation will satisfy them for some fixed large depending on the size of the perturbation.
2.3. Reduction to the space slice
We now put a Lorentzian manifold satisfying assumptions of §2.2 into the framework of [Dy13]. Consider the stationary d’Alembert–Beltrami operator , , the second order semiclassical differential operator on the space slice obtained by replacing by in the semiclassical d’Alembert–Beltrami operator . The principal symbol of is given by
First of all, we need to understand the solutions in to the equation . Let
be the unique real solution to the equation such that , with the positive time oriented light cone defined in (2.6). The existence and uniqueness of such solution follows from assumption (3) in §2.2, and we also find from the definition of that
We can write as the graph of :
The level sets of are not compact if is not timelike. To avoid dealing with the fiber infinity, we use assumption (5) in §2.2 to identify a bounded region in invariant under the flow and containing the trapped set:
There exists an open conic subset , independent of , such that , the closure of in is contained in , and is invariant under the flow (2.3).
Consider a conic neighborhood of in independent of and such that the closure of is contained in ; this is possible by assumption (5) and since is contained in . Let be the union of all maximally extended flow lines of (2.3) passing through . Then is an open conic subset of containing and invariant under the flow (2.3). It remains to show that each point has a neighborhood that does not intersect . To see this, note that the corresponding trajectory of (2.3) does not lie in (as otherwise, the projection of onto the cosphere bundle would converge to as or , by [Dy13, Lemma 4.1]; it remains to use assumption (5) and the fact that is constant on ). We then see that escapes for both and and does not intersect the closure of and same is true for nearby trajectories; therefore, a neighborhood of does not intersect . ∎
Arguing similarly (using an open conic subset of such that and ), we construct an open conic subset of independent of and such that
and are invariant under the flow (2.3). Now, take small and define
Then are open subsets of convex under the flow (2.3), (note that by assumption (5)), and the closure of is contained in . Moreover, the projections of onto the variables are bounded because are conic and .
Let be the projections of onto the variables, so that
and similarly for . Note that and . Since is bounded, and by (2.12), for small enough and , is the only solution to the equation in .
We now study the Hamiltonian flow of . Since
and for each ,
we see that the flow of is the projection of the rescaled geodesic flow (2.3) on : for ,
We now verify some of the assumptions of [Dy13, §4.1]. We let be an -dimensional manifold containing (for the Kerr–de Sitter metric it is constructed in §3.5) and consider the volume form on related to the volume form on generated by by the formula . The operator is a semiclassical pseudodifferential operator depending holomorphically on and is its semiclassical principal symbol. We do not specify the spaces