Exact Solutions of Teukolsky Master Equations


Exact Solutions of Teukolsky Master Equations





Primorsko, 11 June 2007

P. P. Fiziev

Department of

Theoretical Physics

University of Sofia

Exact Solutions of Teukolsky

Master 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:


In r variable RWE reads:

The ansatz:

reduces the RWE to a specific type of 1889 Heun equation:


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:

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, ).

QNM on [0, ) by Maple 10:

Using the



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,

PPF JournalPhys. 66, 0120016, 2006.


one introduces interior time:

and interior radial variable: .



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 :


• 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 potential weels –> two series:

Two series: n=0,…,6; and

n=7,… exist. The eigenvalues

In them are placed around the


and .

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


Dirichlet boundary

Condition at :

The solution:

Physical meaning:

Total reflection

of the waves at

the surface with

area radius :

The simplest model of a compact object

The Spectral


Numerical results: The trajectories in of

The trajectory of the basic



and the BH QNM (black dots):

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, - its dual


- Density for

the Euler



- Density for

the Chern - Pontryagin



and - Two independent

algebraic invariants

Then the differential invariants:




gtt =1 - 2M / , where M is the BH mass

For gtt = 0.7, 0.0, -0.1, -0.3,

-0.5, -1.5, -3.0, - :

Linear perturbations of Kerr metric

S. Teukolsky, PRL, 29, 1115 (1972):

Separation of the


A trivial dependence on the Killing directions - .

(!) :

From stability reasons one MUST have:

1972 Teukolsky master equations (TME):

The angular equation:

The radial equation:




are two



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 regular solutions of the

angular Teukolsky equation are:

An obvious symmetry:

The regularity of the solutions simultaneously

at the both singular ends of the interval [0,Pi] is:

W [ , ] = 0,

W – THE WRONSKIAN , or explicitly:

It yields the relation:

whith unfortunately explicitly unknown function .

Explicit form of the radial Teukolsky equation

where we are using the standard

• Note the symmetry between and in the radial TME

• and are regular syngular points of the radial TME

• is an irregular singular point of the radial TME

Two independent exact solutions of the radial

Teukolsky equation in outer domain are:

BH boundary conditions

at the event horizon:

The waves can go only into the horizon.


- only the solution

obeys BH BC at the EH.


- only the solution

obeys BH BC at the EH.

=> An additional physical clarification.

Boundary conditions at space

infinity – only going to waves:


, then:


, then:

As a result one has to solve the system of

equations for and : ( )

1) For any :

2) and when :


=> a nontrivial numerical problem.

Making use

of indirect


H. Onozawa,


Singular solutions of the angular

Teukolsky equation

Besides regular solutions the angular TME has singular solutions:


The singularities can be essentially weakened if

one works with Polynomial Heun’s functions

(analogy with Hydrogen atom):

Three terms

recurrence relation:

Polynomial solutions with:


Defines symple functions

Examples of Relativistic Jets 1

Examples of Relativistic Jets 2

The distribution of the eigenvalues in the complex

plane for the singular case s=-2, m=1 with



The singular case s=-2, m=1 with

, 2M=1, a/M=0.99

Re(omega) Im(omega)

0.17288 -0.00944

0.18630 -0.05564

0.22508 -0.07692

0.30106 -0.09009

0.33533 -0.09881

0.38281 -0.09909

0.35075 -0.12008

0.27110 -0.13029

0.47609 -0.15200

0.47601 -0.16000

0.60080 -0.18023

0.56077 -0.25076

0.50049 -0.29945

0.40205 -0.37716

Problems in progress:

Imposing BH boundary conditions one can

obtain and improve the known numerical results

=> a more systematic of the QNM in outer


QNM of the Kerr metric in the BH interior.

Novel models of the central engine of GRB

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.


The pink cluster


At present:

32 processors


Up to

128 GFlops

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


• 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

Thank You

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