3 Quasi-Normal Modes of Black Quasi-Normal Modes of Stars and 1 Introduction

2 Normal Modes - Quasi-Normal Modes - Resonances 

Before discussing quasi-normal modes it is useful to remember what normal modes are!

Compact classical linear oscillating systems such as finite strings, membranes, or cavities filled with electromagnetic radiation have preferred time harmonic states of motion (tex2html_wrap_inline2624 is real):

  equation37

if dissipation is neglected. (We assume tex2html_wrap_inline2626 to be some complex valued field.) There is generally an infinite collection of such periodic solutions, and the ``general solution'' can be expressed as a superposition,

  equation41

of such normal modes. The simplest example is a string of length L which is fixed at its ends. All such systems can be described by systems of partial differential equations of the type (tex2html_wrap_inline2626 may be a vector)

  equation46

where tex2html_wrap_inline2632 is a linear operator acting only on the spatial variables. Because of the finiteness of the system the time evolution is only determined if some boundary conditions are prescribed. The search for solutions periodic in time leads to a boundary value problem in the spatial variables. In simple cases it is of the Sturm-Liouville type. The treatment of such boundary value problems for differential equations played an important role in the development of Hilbert space techniques.

A Hilbert space is chosen such that the differential operator becomes symmetric. Due to the boundary conditions dictated by the physical problem, tex2html_wrap_inline2632 becomes a self-adjoint operator on the appropriate Hilbert space and has a pure point spectrum. The eigenfunctions and eigenvalues determine the periodic solutions (1Popup Equation).

The definition of self-adjointness is rather subtle from a physicist's point of view since fairly complicated ``domain issues'' play an essential role. (See [43] where a mathematical exposition for physicists is given.) The wave equation modeling the finite string has solutions of various degrees of differentiability. To describe all ``realistic situations'', clearly tex2html_wrap_inline2636 functions should be sufficient. Sometimes it may, however, also be convenient to consider more general solutions.

From the mathematical point of view the collection of all smooth functions is not a natural setting to study the wave equation because sequences of solutions exist which converge to non-smooth solutions. To establish such powerful statements like (2Popup Equation) one has to study the equation on certain subsets of the Hilbert space of square integrable functions. For ``nice'' equations it usually happens that the eigenfunctions are in fact analytic. They can then be used to generate, for example, all smooth solutions by a pointwise converging series (2Popup Equation). The key point is that we need some mathematical sophistication to obtain the ``completeness property'' of the eigenfunctions.

This picture of ``normal modes'' changes when we consider ``open systems'' which can lose energy to infinity. The simplest case are waves on an infinite string. The general solution of this problem is

  equation57

with ``arbitrary'' functions A and B . Which solutions should we study? Since we have all solutions, this is not a serious question. In more general cases, however, in which the general solution is not known, we have to select a certain class of solutions which we consider as relevant for the physical problem.

Let us consider for the following discussion, as an example, a wave equation with a potential on the real line,

  equation60

Cauchy data tex2html_wrap_inline2642 which have two derivatives determine a unique twice differentiable solution. No boundary condition is needed at infinity to determine the time evolution of the data! This can be established by fairly simple PDE theory [116].

There exist solutions for which the support of the fields are spatially compact, or - the other extreme - solutions with infinite total energy for which the fields grow at spatial infinity in a quite arbitrary way!

From the point of view of physics smooth solutions with spatially compact support should be the relevant class - who cares what happens near infinity! Again it turns out that mathematically it is more convenient to study all solutions of finite total energy. Then the relevant operator is again self-adjoint, but now its spectrum is purely ``continuous''. There are no eigenfunctions which are square integrable. Only ``improper eigenfunctions'' like plane waves exist. This expresses the fact that we find a solution of the form (1Popup Equation) for any real tex2html_wrap_inline2624 and by forming appropriate superpositions one can construct solutions which are ``almost eigenfunctions''. (In the case tex2html_wrap_inline2646 these are wave packets formed from plane waves.) These solutions are the analogs of normal modes for infinite systems.

Let us now turn to the discussion of ``quasi-normal modes'' which are conceptually different to normal modes. To define quasi-normal modes let us consider the wave equation (5Popup Equation) for potentials with tex2html_wrap_inline2648 which vanish for tex2html_wrap_inline2650 . Then in this case all solutions determined by data of compact support are bounded: tex2html_wrap_inline2652 . We can use Laplace transformation techniques to represent such solutions. The Laplace transform tex2html_wrap_inline2654 (s >0 real) of a solution tex2html_wrap_inline2658 is

  equation69

and satisfies the ordinary differential equation

  equation74

where

  equation80

is the homogeneous equation. The boundedness of tex2html_wrap_inline2626 implies that tex2html_wrap_inline2662 is analytic for positive, real s, and has an analytic continuation onto the complex half plane Re (s)>0.

Which solution tex2html_wrap_inline2662 of this inhomogeneous equation gives the unique solution in spacetime determined by the data? There is no arbitrariness; only one of the Green functions for the inhomogeneous equation is correct!

All Green functions can be constructed by the following well known method. Choose any two linearly independent solutions of the homogeneous equation tex2html_wrap_inline2670 and tex2html_wrap_inline2672, and define

  equation88

where W (s) is the Wronskian of tex2html_wrap_inline2676 and tex2html_wrap_inline2678 . If we denote the inhomogeneity of (7Popup Equation) by j, a solution of (7Popup Equation) is

  equation97

We still have to select a unique pair of solutions tex2html_wrap_inline2682 . Here the information that the solution in spacetime is bounded can be used. The definition of the Laplace transform implies that tex2html_wrap_inline2662 is bounded as a function of x . Because the potential V vanishes for tex2html_wrap_inline2650, the solutions of the homogeneous equation (8Popup Equation) for tex2html_wrap_inline2650 are

  equation104

The following pair of solutions

  equation108

which is linearly independent for Re (s)> 0, gives the unique Green function which defines a bounded solution for j of compact support. Note that for Re (s)>0 the solution tex2html_wrap_inline2678 is exponentially decaying for large x and tex2html_wrap_inline2676 is exponentially decaying for small x . For small x however, tex2html_wrap_inline2678 will be a linear combination tex2html_wrap_inline2712 which will in general grow exponentially. Similar behavior is found for tex2html_wrap_inline2676 .

Quasi-Normal mode frequencies tex2html_wrap_inline2716 can be defined as those complex numbers for which

  equation117

that is the two functions become linearly dependent, the Wronskian vanishes and the Green function is singular! The corresponding solutions tex2html_wrap_inline2718 are called quasi eigenfunctions.

Are there such numbers tex2html_wrap_inline2716 ? From the boundedness of the solution in spacetime we know that the unique Green function must exist for Re (s)>0. Hence tex2html_wrap_inline2724 are linearly independent for those values of s . However, as solutions tex2html_wrap_inline2724 of the homogeneous equation (8Popup Equation) they have a unique continuation to the complex s plane. In [35Jump To The Next Citation Point In The Article] it is shown that for positive potentials with compact support there is always a countable number of zeros of the Wronskian with Re (s)<0.

What is the mathematical and physical significance of the quasi-normal frequencies tex2html_wrap_inline2716 and the corresponding quasi-normal functions tex2html_wrap_inline2678 ? First of all we should note that because of Re (s)<0 the function tex2html_wrap_inline2678 grows exponentially for small and large x ! The corresponding spacetime solution tex2html_wrap_inline2744 is therefore not a physically relevant solution, unlike the normal modes.

If one studies the inverse Laplace transformation and expresses tex2html_wrap_inline2626 as a complex line integral (a >0),

  equation123

one can deform the path of the complex integration and show that the late time behavior of solutions can be approximated in finite parts of the space by a finite sum of the form

  equation131

Here we assume that tex2html_wrap_inline2750, tex2html_wrap_inline2752 . The approximation tex2html_wrap_inline2754 means that if we choose tex2html_wrap_inline2756, tex2html_wrap_inline2758, tex2html_wrap_inline2760 and tex2html_wrap_inline2762 then there exists a constant tex2html_wrap_inline2764 such that

  equation137

holds for tex2html_wrap_inline2766, tex2html_wrap_inline2768, tex2html_wrap_inline2770 with tex2html_wrap_inline2764 independent of t . The constants tex2html_wrap_inline2776 depend only on the data [35Jump To The Next Citation Point In The Article]! This implies in particular that all solutions defined by data of compact support decay exponentially in time on spatially bounded regions. The generic leading order decay is determined by the quasi-normal mode frequency with the largest real part tex2html_wrap_inline2778, i.e. slowest damping. On finite intervals and for late times the solution is approximated by a finite sum of quasi eigenfunctions (15Popup Equation).

It is presently unclear whether one can strengthen (16Popup Equation) to a statement like (2Popup Equation), a pointwise expansion of the late time solution in terms of quasi-normal modes. For one particular potential (Pöschl-Teller) this has been shown by Beyer [42Jump To The Next Citation Point In The Article].

Let us now consider the case where the potential is positive for all x, but decays near infinity as happens for example for the wave equation on the static Schwarzschild spacetime. Data of compact support determine again solutions which are bounded [117Jump To The Next Citation Point In The Article]. Hence we can proceed as before. The first new point concerns the definitions of tex2html_wrap_inline2782 . It can be shown that the homogeneous equation (8Popup Equation) has for each real positive s a unique solution tex2html_wrap_inline2672 such that tex2html_wrap_inline2788 holds and correspondingly for tex2html_wrap_inline2676 . These functions are uniquely determined, define the correct Green function and have analytic continuations onto the complex half plane Re (s)>0.

It is however quite complicated to get a good representation of these functions. If the point at infinity is not a regular singular point, we do not even get converging series expansions for tex2html_wrap_inline2782 . (This is particularly serious for values of s with negative real part because we expect exponential growth in x).

The next new feature is that the analyticity properties of tex2html_wrap_inline2782 in the complex s plane depend on the decay of the potential. To obtain information about analytic continuation, even use of analyticity properties of the potential in x is made! Branch cuts may occur. Nevertheless in a lot of cases an infinite number of quasi-normal mode frequencies exists.

The fact that the potential never vanishes may, however, destroy the exponential decay in time of the solutions and therefore the essential properties of the quasi-normal modes. This probably happens if the potential decays slower than exponentially. There is, however, the following way out: Suppose you want to study a solution determined by data of compact support from t =0 to some large finite time t = T . Up to this time the solution is - because of domain of dependence properties - completely independent of the potential for sufficiently large x . Hence we may see an exponential decay of the form (15Popup Equation) in a time range tex2html_wrap_inline2812 . This is the behavior seen in numerical calculations. The situation is similar in the case of tex2html_wrap_inline2814 -decay in quantum mechanics. A comparison of quasi-normal modes of wave equations and resonances in quantum theory can be found in the appendix, see section  9 .



3 Quasi-Normal Modes of Black Quasi-Normal Modes of Stars and 1 Introduction

image Quasi-Normal Modes of Stars and Black Holes
Kostas D. Kokkotas and Bernd G. Schmidt
http://www.livingreviews.org/lrr-1999-2
© Max-Planck-Gesellschaft. ISSN 1433-8351
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