Compact classical linear oscillating systems such as finite
strings, membranes, or cavities filled with electromagnetic
radiation have preferred time harmonic states of motion (
is real):
if dissipation is neglected. (We assume
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,
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 (
may be a vector)
where
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,
becomes a self-adjoint operator on the appropriate Hilbert space
and has a pure point spectrum. The eigenfunctions and eigenvalues
determine the periodic solutions (1
).
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
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 (2) 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 (2
). 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
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,
Cauchy data
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 (1) for any real
and by forming appropriate superpositions one can construct
solutions which are ``almost eigenfunctions''. (In the case
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 (5) for potentials with
which vanish for
. Then in this case all solutions determined by data of compact
support are bounded:
. We can use Laplace transformation techniques to represent such
solutions. The Laplace transform
(s
>0 real) of a solution
is
and satisfies the ordinary differential equation
where
is the homogeneous equation. The boundedness of
implies that
is analytic for positive, real
s, and has an analytic continuation onto the complex half plane
Re
(s)>0.
Which solution
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
and
, and define
where
W
(s) is the Wronskian of
and
. If we denote the inhomogeneity of (7
) by
j, a solution of (7
) is
We still have to select a unique pair of solutions
. Here the information that the solution in spacetime is bounded
can be used. The definition of the Laplace transform implies that
is bounded as a function of
x
. Because the potential
V
vanishes for
, the solutions of the homogeneous equation (8
) for
are
The following pair of solutions
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
is exponentially decaying for large
x
and
is exponentially decaying for small
x
. For small
x
however,
will be a linear combination
which will in general grow exponentially. Similar behavior is
found for
.
Quasi-Normal mode frequencies
can be defined as those complex numbers for which
that is the two functions become linearly dependent, the
Wronskian vanishes and the Green function is singular! The
corresponding solutions
are called quasi eigenfunctions.
Are there such numbers
? From the boundedness of the solution in spacetime we know that
the unique Green function must exist for
Re
(s)>0. Hence
are linearly independent for those values of
s
. However, as solutions
of the homogeneous equation (8
) they have a unique continuation to the complex
s
plane. In [35
] 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
and the corresponding quasi-normal functions
? First of all we should note that because of
Re
(s)<0 the function
grows exponentially for small and large
x
! The corresponding spacetime solution
is therefore not a physically relevant solution, unlike the
normal modes.
If one studies the inverse Laplace transformation and
expresses
as a complex line integral (a
>0),
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
Here we assume that
,
. The approximation
means that if we choose
,
,
and
then there exists a constant
such that
holds for
,
,
with
independent of
t
. The constants
depend only on the data [35
]! 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
, i.e. slowest damping. On finite intervals and for late times
the solution is approximated by a finite sum of quasi
eigenfunctions (15
).
It is presently unclear whether one can strengthen (16) to a statement like (2
), 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 [42
].
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 [117]. Hence we can proceed as before. The first new point concerns
the definitions of
. It can be shown that the homogeneous equation (8
) has for each real positive
s
a unique solution
such that
holds and correspondingly for
. 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
. (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
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 (15) in a time range
. This is the behavior seen in numerical calculations. The
situation is similar in the case of
-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
.
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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 Problems/Comments to livrev@aei-potsdam.mpg.de |