Transcript Slide 1

Uniwersytet
Jagiellonski
Institut Fizyki
Cracow
23 May 2007
P. P. Fiziev
Department of
Theoretical Physics
University of Sofia
Talk given at May 23, 2007
Cracow
Exact Solutions of Regge-Wheeler
and Teukolsky Equations
The Regge-Wheeler (RW) equation describes the axial perturbations of
Schwarzschild metric in linear approximation.
The Teukolsky Equations describe perturbations of Kerr metric.
We present here:
• Their exact solutions in terms of confluent Heun’s functions;
• The basic properties of the RW general solution;
• Novel analytical approach and numerical techniques for study of
different boundary problems which correspond to quasi-normal
modes of black holes and other simple models of compact objects.
• The exact solutions of RW equation in the Schwarzschild BH interior.
• Exact solutions of Teukolsky master equations (TME)
Linear perturbations of
Schwarzschild metric
1957 Regge-Wheeler equation (RWE):
The potential:
The type of perturbations: S=2 - GW, s=1-vector, s=0 – scalar;
The tortoise coordinate:
The Schwarzschild radius:
The area radius:
1758 Lambert W(z) function:
W exp(W) = z
The standard ansatz
separates variables.
The “stationary” RWE:
One needs proper boundary conditions (BC).
Known Numerical studies and
approximate analytical methods for BH BC.
See the wonderful reviews:
V. Ferrary (1998),
K. D. Kokkotas & B. G. Schmidt (1999),
H-P. Nollert (1999).
and some basic results in:
S. Chandrasekhar & S. L. Detweiler (1975),
E. W. Leaver (1985),
N. Andersson (1992),
and many others!
Exact mathematical treatment:
PPF,
In r variable RWE reads:
The ansatz:
reduces the RWE to a specific type of 1889 Heun equation:
with
Thus one obtains a confluent Heun equation with:
2 regular singular points: r=0 and r=1, and
1 irregular singular point:
in the complex plane
Note that after all the horizon r=1 turns to be a singular point
in contrary to the widespread opinion.
From geometrical point of view the horizon is indeed
a regular point (or a 2D surface) in the Schwarzschild
Riemannian space-time manifold:
It is a singularity, which is placed in the (co) tangent fiber
of the (co) tangent foliation:
and is “invisible” from point of view of the base
.
The local solutions (one regular + one singular)
around the singular points:
X=0, 1, ¥
Frobenius type of solutions:
Tome (asymptotic) type of solutions:
Notation in use:
(based on the in-out
properties of the solutions)
Limits:
Important justification:
General local solutions:
:
X=0, 1,
Transition coefficients
The main problem:
Unfortunately at present the transition coefficients
are not known explicitly !
¥
:
Different types of boundary problems:
I. BH boundary problems: two-singular-points boundary.
¥
Up to recently only the QNM problem on [1,
), i.e. on the
BH exterior, was studied numerically and using different
analytical approximations.
We present here exact treatment of this problem, as well as
of the problems on [0,1] (i.e. in BH interior), and on [0, ¥ ).
BH boundary problems on [1,
 Quasi-normal modes:
 Left mixed modes:
 Normal modes:
 Right mixed modes:
):
QNM on [0,
) by Maple 10:
Using the
condition:
-i
One obtains by Maple 10 for the first 5 eigenvalues:
and 12 figures - for n=0:
Perturbations of the BH interior
Matzner (1980), PPF gr-qc/0603003
For
one introduces interior time:
and interior radial variable:
Then:
where:
.
The role of the BH interior -
recent articles:
 V. Balasubramanian, D. Marolf, M. Rosali,
hep-th/0604045 (in quantum gravity)

L. Baiotti, L. Rezzola
gr-qc/0608113 (in numerical calculations)
Local solutions in
:
and
For
they
have the symmetry property:
The continuous spectrum
Normal modes in Schwarzschild BH interior:
A basis for Fourier expansion of perturbations of general form
in the BH interior
The special solutions with
:
These:
• form an orthogonal basis with respect to the weight:
• do not depend on the variable
.
• are the only solutions, which are finite at both singular ends
of the interval
.
The discrete spectrum
pure imaginary eigenvalues:
 Ferrari-Mashhoon transformation:
 For
:
“falling at the centre” problem
operator with defect
 Additional parameter – mixing angle
 Spectral condition – for arbitrary
:
:
Numerical results
For the first 18 eigenvalues
one obtains:
For alpha =0 – no outgoing waves:
Two series: n=0,…,6; and
n=7,… exist. The eigenvalues
In them are placed around the
lines
and
Two potential weels –> two series:
.
Dependence of eigenvalues on the mixing angle –
NEW PHENOMENON
Attraction and Repulsion of the Levels:
The mixing angle alpha
describes the ratio
of the amplitudes
of the waves,
going in and going out
of the horizon:
alpha=0
– no outgoing waves
alpha= Pi/2
– no ingoing waves
A typical behavior of the eigenfunctions of
the discrete imaginary spectrum:
- A basis for
Laplace expansion
of perturbations
of general form
in the BH interior
The dependence of the
eigenvalues
on the angular momentum l
Perturbations of Kruskal-Szekeres manifold
In this case the solution can be obtained from functions
imposing the additional condition which may create a spectrum:
It annulates the coming from the space-infinity waves.
The numerical study for the case l=s=2 shows that it is impossible
to fulfill the last condition and to have some nontrivial spectrum of
perturbations in Kruskal-Szekeres manifold.
II. Regular Singular-two-point
Boundary Problems at
PPF,
Dirichlet boundary
Condition at
:
Physical meaning:
Total reflection
of the waves at
the surface with
area radius
:
The solution:
The simplest model of a compact object
The Spectral
condition:
Numerical results:
The trajectory of the basic
eigenvalue
in
and the BH QNM (black dots):
The trajectories in
of
The Kerr (1963) Metric
In Boyer - Lindquist (1967) - {+,-,-,-} coordinates:
The Kerr solution yields much more complicated
structures then the Schwarzschild one:
The event horizon, the Cauchy horizon
and the ring singularity
The event horizon, the ergosphere,
the Cauchy horizon and
the ring singularity
Simple algebraic and differential invariants
for the Kerr solution:
Let
is the Weyl tensor,
- Density for
the Euler
characteristic
class
- its dual
- Density for
the Chern - Pontryagin
characteristic
class
Let
and
Then the differential invariants:
- Two independent
algebraic invariants
CAN LOCALLY SEE
- The two horizons
- The Ergosphere
Linear perturbations of Kerr metric
1972 Teukolsky master equations (TME):
The angular equation:
The radial equation:
Spin:
S=-2,-1,0,1,2.
and
are two
independent
parameters
Up to now only numerical results and
approximate methods were studied
First results:
• S. Teukolsky, PRL, 29, 1115 (1972).
• W Press, S. Teukolsky, AJ 185, 649 (1973).
• E. Fackerell, R. Grossman, JMP, 18, 1850 (1977).
• E. W. Leaver, Proc. R. Soc. Lond. A 402, 285, (1985).
• E. Seidel, CQG, 6, 1057 (1989).
For more recent results see, for example:
• H. Onozawa, gr-qc/9610048.
• E. Berti, V. Cardoso, gr-qc/0401052.
and the references therein.
Two independent exact solutions of the angular
Teukolsky equation are:
Symmetry:
The regularity of the solutions at both singular ends of the interval yields the relation:
Explicit form of the radial Teukolsky equation
where we are using the standard
• Note the symmetry between
and
in the radial TME
•
and
•
is an irregular singular point of the radial TME
are regular syngular points of the radial TME
Two independent exact solutions of the radial
Teukolsky equation in outer domain are:
Problems in progress:
 Imposing BH boundary conditions one can
obtain the known numerical results => a more
systematic study of BH QNM in outer domain.
 QNM of the Kerr metric interior.
 Imposing Dirichlet boundary conditions one can
obtain new models of rotating compact objects.
 More systematic study of QNM of neutron stars.
 Study of the still unknown QNM of gravastars
(Pawel Mazur).
Some basic conclusions:
•
Heun’s functions are a powerful tool for study of all types of
solutions of the Regge-Wheer and the Teukolsky master
equations.
• Using Heun’s functions one can easily study different
boundary problems for perturbations of metric.
• The solution of the Dirichlet boundary problem gives an
unique hint for the experimental study of the old problem:
Whether in the observed in the Nature
invisible very compact objects with strong gravitational fields
there exist really hole in the space-time ?
=> resolution of the problem of the real existence of BH
FOURTH ADVANCED RESEARCH WORKSHOP
GRAVITY, ASTROPHYSICS, AND STRINGS
@ THE BLACK SEA
Bulgaria, Kiten, June 10-16, 2007
http://tcpa.uni-sofia.bg/conf/2007/gas/confind.html
Organizing Committee
P. Fiziev (chairman)
R. Borissov
P. Bozhilov
H. Dimov
E. Nissimov
R. Rashkov
M. Todorov
R. Tsenov
Thank You