Exact Solutions of Teukolsky Master Equations

Gravity

Astrophysics

and

Strings

Primorsko, 11 June 2007

P. P. Fiziev

Department **of**

Theoretical Physics

University **of** S**of**ia

**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:

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:

Different types **of** boundary problems:

I. BH boundary problems: two-singular-points boundary.

Ґ

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

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,

PPF JournalPhys. 66, 0120016, 2006.

For

one introduces interior time:

and interior radial variable: .

Then:

where:

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 potential weels –> two series:

Two series: n=0,…,6; and

n=7,… exist. The eigenvalues

In them are placed around the

lines

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

PPF,

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

condition:

Numerical results: The trajectories in **of**

The trajectory **of** the basic

eigenvalue

in

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

Let

- Density for

the Euler

characteristic

class

- Density for

the Chern - Pontryagin

characteristic

class

and - Two independent

algebraic invariants

Then the differential invariants:

CAN LOCALLY SEE

-The TWO HORIZONS

-The ERGOSPHERE

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

variables:

A trivial dependence on the Killing directions - .

(!) :

From stability reasons one MUST have:

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 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.

Consequence:

- only the solution

obeys BH BC at the EH.

If

- only the solution

obeys BH BC at the EH.

=> An additional physical clarification.

Boundary conditions at space

infinity – only going to waves:

If

, then:

If

, then:

As a result one has to solve the system **of**

equations for and : ( )

1) For any :

2) and when :

or

=> a nontrivial numerical problem.

Making use

**of** indirect

methods:

H. Onozawa,

1996

Singular solutions **of** the angular

**Teukolsky** equation

Besides regular solutions the angular TME has singular solutions:

and

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:

and

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

F(z)=z

F(z)=1/z

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

domain.

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.

Physon:

The pink cluster

http://physon.phys.uni-s**of**ia.bg/IndexPage

At present:

32 processors

Performance:

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

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

Thank You