User:Nikodem0/Project-Kerr

= Title: Dynamics [of fields] around rotating black holes [?] =


 * At one of the courses it was recommended to organize an advisory board for the proposal writing phase. One can obtain financial support for meetings back and one has experts on own side which can strengthen the proposal. We should consider this... (Pi:good idea)

= Carter's constant =

Positive "energy" on Kerr spacetime

 * Killing-Yano tensor $$Y_a^b$$ and Killing-Yano tensor $$K_{ab}\equiv Y_a^c Y_b^d g_{bd}$$ --> 2nd order operator $$K\equiv K^{ab}\nabla_a \nabla_b$$ commuting with $$\Box\equiv g^{ab}\nabla_a \nabla_b$$ --> eigenvalue = separation constant for the wave equation (~Carter constant = separation constant for Hamilton-Jacobi geodesic equation)

Evolution of the Carter constant from radiation at infinity

 * The idea is to find flow through a closed surface (at infinity) which corresponds to the "radiation" of the Carter's constant. This would follow from a 4-current associated to the conservation of the Carter's constant if such could be found.


 * See also "5.8 Adiabatic evolution of Carter constant for orbit with small eccentricity and small inclination angle around a Kerr black hole" in "Analytic Black Hole Perturbation Approach to Gravitational Radiation" (Living Reviews) by Misao Sasaki and Hideyuki Tagoshi (2010)

[Pi: How is this related to the Dirac derivation of the Lorentz-Dirac equation in classical electrodynamics (local momentum/angular-momentum radiated by accelerating charge)?]

= Connection problem for the radial ODE (type: confluent Heun) [Pi] =

Outline of the problem

 * Scalar wave eq. on the Kerr spacetime separates to two ODE in θ (spin weighted harmonics?) and r (confluent Heun eq=spheroidal wave eq).


 * We propose to investigate the radial part $$F_{l \omega}$$ of the solution $$ \Psi_{\omega lm}=e^{-i\omega t} Y_{lm} F_{l \omega}(r) $$ corresponding to arbitrary frequencies $$\omega$$.


 * Direct motivation: analytic formulas for scattering amplitudes of scalar waves in Schwarzschild geometry (interesting in itself)

$$ \omega_2(t,\vec x,s,\vec y)\sim \int d\omega \sum_{l m} \overline{ \Psi (t,x)} \Psi(s,y) $$
 * Deeper motivation: in quantum field theory in Schwarzschild spacetime, integrals over the frequencies of the solutions are essential building blocks of two-point functions of (various states of) quantum fields. For instance the two-point function of the ground state is

low-frequency asymptotics of the two-point functions are important for the behavior of negative-energy-density phenomena; there are indications that this behavior is universal for spacetimes with Schwarzschild exterior. This conclusion, however, depends on the universality of the transmission amplitudes for waves in such geometries.

Specifics
$$ x(x+1) F'' +(2x+1) F' +k^2 \frac{(x+1)^3}{x} F=0 $$  where $$ x=\frac{r}{R}-1 $$ and $$ k=\omega R $$. P(φ)2
 * In the simplest (Schwarzschild, $$\ell=0$$) case the radial equation has the form
 * This equation has two regular singular points (-1 and 0, the latter one corresponding to the horizon), and an irregular one at $$\infty $$.

positive potential decaying to zero at both ends.
 * Indicial equation has the form $$ \alpha^2=-k^2 $$ at $$ x=0$$; in the region corresponding to the exterior of the BH, x>0, one has a classical scattering problem with a


 * With current understanding the solutions of this radial ODE are expressed in an infinite series, each term of which is a hypergeometric function; for many applications (e.g. QFT) this is quite unsatisfactory.


 * In the limit $$\omega x $$<<1 one gets a hypergeometric equation, while in the limit $$ x$$ >>1 it reduces to the confluent hypergeometric equation (the solutions of which are Coulomb waves). Both limits are not mutually exclusive, and in some approaches to the problem one estimates the connection coefficients with the method of asymptotic matching. Currently, only lowest-order matching is employed (in the QFT context, ).


 * Relation to general results for low $$ k $$ scattering (Landau Lifshitz III, ch. 132 ) and inverse scattering methods

Expected results

 * We hope that it might turn out possible to express the connection coefficients by known special functions, or at least to obtain integral symmetries among them.


 * Of special interest is the analytic structure of the connection coefficients in the limit k-->0; a thorough understanding of this issue is lacking at present; in the context of asymptotic matching one assumes analyticity in k, while there are indications of non-analyticity (see )


 * It should at least be possible to derive connection coefficients from a higher order asymptotic matching and thus provide sub-leading asymptotics of the quantum-field-theoretical two-point functions

= Decay & asymptotics on Kerr [N]=

[N: tu troche za duzo wkleilem - musze to przeredagowac]

Stability and decay for nonlinear wave equations

 * A very important and lively discussed problem of mathematical physics and geometric analysis, the nonlinear stability of solutions to the Einstein equations (like black holes) is strongly based on techniques developed for more general systems of nonlinear wave equations and systems of PDEs. During the last years I have been working on the nonlinear stability of static solutions against small perturbations and developed techniques giving optimal pointwise decay estimates for linear and semilinear wave equations. These pointwise estimates contain a lot of information about the solutions, among others, the late-time asymptotics. Simultaneously, they are very sensitive to the structure of the nonlinear terms (e.g. null or weak null structure) what offers much insight into what terms cause what effects, e.g. which terms behave singular, which are responsible for scattering tails and which preserve the Huygens principle.


 * Our present results apply to solutions in 3+1 dimensions when the nonlinearity contains no derivatives [Szp08a, SBCR07], and are currently restricted to spherical symmetry if terms nonlinear in derivatives are present [Szp08b]. We are working on generalization of these methods beyond spherical symmetry, combining two techniques: the commuting vector fields method, being powerful and robust, but loosing information on the optimal decay rates, and pointwise methods based on integral representations of the fundamental solution for multipole-decomposed solutions, being exact, but much less robust. In the next step we want to study quasilinear wave equations which arise from geometric wave equations on non-trivial background geometries.

Asymptotics of propagating nonlinear gravitational waves

 * Einstein equations which describe propagation of the metric perturbations, i.e. gravitational waves, are essentially nonlinear, however, it is convenient to approximate small amplitude waves as solutions to the linearized system. A natural question arises whether propagating waves experience nonlinear effects when travelling at large distances and what is the asymptotic behaviour of propagating such nonlinear waves. First, one has to realize that smallness argument is not sufficient to treat nonlinear terms perturbatively. One can easily write equations very similar to the Einstein equations which show a strong self-interaction. Only the very special structure of the nonlinear terms in the Einstein eqs. (some combinations of terms which miraculously cancel out in the asymptotic regime) prevents this bad, generic behavior. Although known estimates make some use of the special structure of Einstein equations, they concentrate mainly on the qualitative behavior needed for proving stability (i.e. that small perturbations of the flat space remain small). But there are no results guaranteeing that the waves cannot get distorted during a long evolution. This requires a better understanding of the structure of the nonlinear terms in the Einstein equations.


 * The experience we have made studying perturbations of nonlinear wave equations, and techniques which we have developed (analytic and numerical) shed new light on what structures of nonlinear terms cause what effects (e.g. some of them are responsible for developing tails travelling behind the wave fronts, some other prevent the Huygens structure).


 * The asymptotic analysis of propagating nonlinear waves is a very much unexplored area. The importance of its mathematical analysis is enhanced by the fact that its solution escapes the standard numerical methods usually being successful in treating similar problems of PDEs originating in theoretical physics. This program combines purely mathematical tools, approximation techniques as well as numerical evidence.

Late-time asymptotics (tails) for wave equations

 * In a few recent or still ongoing projects, we have been studying the issue of late-time asymptotics (tails) for the wave equations due to potential or nonlinear scattering in the weak-field regime (small perturbations around a static solution). We have developed analytic [Szp08a] and numerical [SBCR07] techniques allowing to predict the exact power and the amplitude of the power-law tail even for complicated nonlinear wave equations. The techniques involve convergent iteration and perturbation schemes in weighted Sobolev spaces, Green’s function based methods, operator analytic approach to the evolution of weak solutions and lot of insight from numerical simulations.


 * Recently, we have applied these methods to Einstein equations and perturbations of black holes (stationary solutions) calculating precisely the form of the purely nonlinear 1/t3 late-time tail behind the wave front for self-gravitating scalar waves [SBCR].


 * Also recently, we were able to generalize a technique of Lindblad based on some scaling properties of the nonlinear wave equations to obtain closed asymptotic formulas for the leading order expansion of the solution at late times in terms of (small) Cauchy data. This result has been reported at the Mittag-Leffler Institute Workshop (Sept. 2008). A written publication is currently in preparation.


 * The problem of late-time tails can be also studied for big data in models with a positive definite energy guaranteeing global existence of solutions and their decay. Here, weak estimates following from boundedness of energy can be successively improved by pointwise estimates developed for small data. We are already quite advanced in this project (with Roger Bieli, AEI) and expect complete results in the following months.

Quasinormal Modes / Resonances

 * The general idea is a representation of solutions to the wave equations $$\partial_t^2 \phi = A \phi$$, where A is some perturbation of the Laplacian, by means of resonances of A. One expects, at least for late times, to obtain a sum of decaying modes, called quasinormal modes (QNMs). They play an important role in the analysis of stability of black holes and have been observed numerically. Unfortunately, the mathematical theory is still very unsatisfactory.


 * Our interest and experience in the analysis of QNMs ranges from the fundamental mathematical theory and general properties like existence, convergence, completeness, stability w.r.t. perturbations of the elliptic operator, through rigorous derivation of the QNM-representation for solutions to the Cauchy problem with estimation of the convergence region in the space-time, using methods based on the Laplace transform, Green’s function approximation and spectral methods [Szp04]; to numerical calculation of QNMs using the following methods: shooting-to-a-fitting-point for a corresponding ODE, complex scaling, analytic continuations, complex path integrals along anti-Stokes lines, continued fractions, resummation of divergent series (Borel, Pade), superasymptotics, hyperasymptotics, etc. [Szp99].


 * We have discovered a new type of modes which are necessary for the completeness when one wants to include the description of the intermediate signal propagating along characteristics and in their vicinity. I have also put up the idea that due to the Stokes phenomenon there exist QNMs, necessarily purely damped (i.e. non-oscillating), which do not correspond to purely outgoing waves, as is usually conjectured, but are still present in the dynamics. However, both these issues have not been studied in a wider contex beyond simple model examples yet, however, two publications are currently in preparation.


 * The late time dynamics for such systems is typically driven by two effects: exponentially decaying QMNs related to resonances and power-law tails related to complex branch cuts of the resolvent. Intriguingly, there is always an intermediate regime of times when the solution is very near to its QNM–decomposition while the power-law tail eventually takes over the leading role at very late times. Interestingly, the same exponential to-power-law behavior has been observed in the description of atomic systems, like evolution of an excited atomic state. We have shown (again by the method of Laplace transform) that here, too, resonances and complex branch cuts are responsible for this universal phenomenon [MS05].