PP.. P P.. FizievFiziev

Department of Department of Theoretical Physics Theoretical Physics University of SofiaUniversity of Sofia

Nis Nis 28.12.200728.12.2007

Exact Solutions of Regge-Wheeler and Teukolsky Equations

The The Regge-WheelerRegge-Wheeler ( (RWRW) equation describes the axial ) equation describes the axial perturbations of Schwarzschild metric in linear approximation. perturbations of Schwarzschild metric in linear approximation.

TheThe TeukolskyTeukolsky Equations describe perturbations of Equations describe perturbations of Kerr Kerr metric. metric. We present here:We present here:

• Their exact solutions in terms of confluent Their exact solutions in terms of confluent Heun’sHeun’s functions. functions. • The basic properties of the The basic properties of the RWRW general solution. general solution. • Novel analytical approach and numerical techniques for study ofNovel analytical approach and numerical techniques for study ofdifferent boundary problems which correspond to quasi-normaldifferent boundary problems which correspond to quasi-normalmodes of black holes and other simple models of compact objects.modes of black holes and other simple models of compact objects.• The The exact solutionsexact solutions of of RWRW equation in the equation in the SchwarzschildSchwarzschild BHBH

interior.interior. • The The exact solutionsexact solutions of of TeukolskyTeukolsky master equations ( master equations (TMETME).).• New New singular exact solutionssingular exact solutions of TME and their application to the of TME and their application to the

theorytheory

of of thethe relativistic jetsrelativistic jets..

Linear perturbations of Schwarzschild metric 1957 Regge-Wheeler equation (RWE):

The type of perturbations: S=2 - GW, s=1-vector, s=0 – scalar;

The potential:

The tortoise coordinate:

The area radius:

The Schwarzschild radius:

1758 Lambert W(z) function: W exp(W) = z

The standard ansatz

One needs proper boundary conditions (BC).

The “stationary” RWE:

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).V. Ferrari, L. Gualtieri (2007). and some basic results in: S. Chandrasekhar & S. L. Detweiler (1975),E. W. Leaver (1985),N. Andersson (1992), and many others!

separates variables.

In r variable RWE reads:

Exact mathematical treatment:PPF,

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 fiberof 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 asof 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 A basis for FourierFourier expansion expansion of perturbations of general form of perturbations of general form in the in the BHBH interior 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 :

Additional parameter – mixing angle :

Spectral condition – for arbitrary :

“falling at the centre” problem operator with defect

Numerical resultsFor the first 18 eigenvalues one obtains:

Two series: n=0,…,6; andn=7,… exist. The eigenvaluesIn them are placed around thelines and .

For For alpha =0alpha =0 – no outgoing waves: – no outgoing waves: Two potential weels –> two series:Two potential weels –> two series:

Perturbations of Kruskal-Szekeres manifold

In this case the solution can be obtained from In this case the solution can be obtained from functionsfunctions

imposing the additionalimposing the additional condition which may create a spectrum:condition which may create a spectrum:

It annulates the coming from the space-infinity waves. It annulates the coming from the space-infinity waves.

The numerical study for the caseThe numerical study for the case l=s=2 l=s=2 shows that it is shows that it is impossible impossibleto fulfill the last condition and to fulfill the last condition and to haveto have some some nontrivial spectrumnontrivial spectrum of of perturbations perturbations in in Kruskal-Szekeres manifold. .

II. Regular Singular-two-point Boundary Problems at

Dirichlet boundary Condition at :

Physical meaning:Total reflectionof the waves at the surface witharea radius :

The solution:

The simplest model of a compact object

PPF,

The Spectral condition:

Numerical results: The trajectories in of

The trajectory of the basic eigenvalue inand the BH QNM (black dots):

The Kerr (1963) Metric

In In Boyer - LindquistBoyer - Lindquist (1967) - (1967) - {+,-,-,-} {+,-,-,-} coordinates:coordinates:

The Kerr solution yields much more complicated structures then the Schwarzschild one:

The event horizon, the The event horizon, the CauchyCauchy horizonhorizon and the ring singularity and the ring singularity

The event horizon, the ergosphere, The event horizon, the ergosphere, the the CauchyCauchy horizon and horizon and the ring singularity the ring singularity

Simple algebraic and differential invariants for the Kerr solution:Let is the Weyl tensor, - its dual

- Density for- Density forthe the Euler Euler characteristiccharacteristicclassclass

- Density for- Density forthe the Chern - Chern -

PontryaginPontryagin characteristiccharacteristicclassclass

LetLet

andand - Two independent Two independent algebraic invariantsalgebraic invariants

Then the Then the differential invariantsdifferential invariants::CAN CAN LOCALLYLOCALLY SEE SEE-The TWO HORIZONS-The TWO HORIZONS

-The ERGOSPHERE-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 metricS. TeukolskyS. Teukolsky,, PRL, PRL, 2929, 1115 , 1115

(1972):(1972):

Separation of Separation of

thethe variables:variables:

(!) :(!) :

A trivial dependence on the Killing directions - .A trivial dependence on the Killing directions - .

From stability reasons one From stability reasons one MUSTMUST have: have:

1972 Teukolsky master equations (TME):

The angular equation:The angular equation:

The radial equation:The radial equation:

Spin:Spin:S=-2,-1,0,1,2.S=-2,-1,0,1,2.

and and are are two two independentindependentparametersparameters

Up to now only numerical results and approximate methods were studied

First results:First results:• S. TeukolskyS. Teukolsky,, PRL, PRL, 2929, 1115 (1972)., 1115 (1972).• W Press, S. Teukolsky,W Press, S. Teukolsky, AJAJ 185185, 649 (1973)., 649 (1973).• E. Fackerell, R. GrossmanE. Fackerell, R. Grossman,, JMP, 18, 1850 (1977). JMP, 18, 1850 (1977). • E. W. Leaver, Proc. R. Soc. Lond. A 402, 285, (1985).• E. SeidelE. Seidel,, CQG, CQG, 66, 1057 (1989)., 1057 (1989).

For more recent results see, for example: For more recent results see, for example: • H. OnozawaH. Onozawa,, gr-qc/9610048. gr-qc/9610048.• E. Berti, V. CardosoE. Berti, V. Cardoso,, gr-qc/0401052. gr-qc/0401052.

and the references therein.and the references therein.

Two independent exact regular solutions of the angular Teukolsky equation are:

An obvious An obvious symmetry: symmetry:

The The regularity regularity of the solutions of the solutions simultaneouslysimultaneously

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

It yields the relation:It yields the relation:

whith unfortunately whith unfortunately explicitlyexplicitly unknown function .unknown function .

W [ , ] = 0, W – THE WRONSKIAN , or explicitly:

Explicit form of the radial Teukolsky equation

where we are using the standardwhere we are using the standard

• Note the Note the symmetrysymmetry between and in the radial between and in the radial TME TME

• and are and are regularregular syngular pointssyngular points of the radial of the radial

TMETME

• is anis an irregular irregular singular point singular point of the radial of the radial TMETME

Two independent exact solutions of the radial Teukolsky equation in outer domain are:

BH boundary conditions at the event horizon:

The waves can goThe waves can go only into only into the horizon. the horizon. Consequence:Consequence:

IfIf

- only- only the solution the solutionobeys obeys BH BH BC at the BC at the

EH.EH.

- only- only the solution the solutionobeys obeys BH BH BC at the BC at the

EH.EH.

=> An additional physical clarification.=> An additional physical clarification.

Boundary conditions at space infinity – only going to waves:

IfIf

IfIf

, then:, then:

, then:, then:

As a result one has to solve the system of equations for and : ( )

2) and when :2) and when :

oror

=>=> a nontrivial numerical problem. a nontrivial numerical problem.

1) For any : 1) For any :

Making use Making use ofof indirect indirectmethods:methods:H. Onozawa,H. Onozawa,19961996

The Relativistic Jets: The Most Powerful and Misterious Phenomenon

in the Universe, which are observed at different scales:1. Around single neutron star (~10-1000 AU)2. In binary BH–Star, and Star-Star systems3. In Gamma Ray Burst (GRB) (~1 kPs) 4. Around galactic nuclei (~1 MPs)5. Around galactic collisions (~10 MPs) 6. Around galactic clusters (~200 Mps)

=> UNIVERSAL NATURE ???

Jets from GRB

A hyper nova 08.09.05 (distance 11.7 bills lys) Formation of WHAT ???: BH???, OR ??? VU6APFLG.mov

Series of explosions observed!

The Jet from M87

2006 News2006 News Jets fromJets from

GRB060418 GRB060418 andand GRB060607A: GRB060607A:

~ 200 Earth ~ 200 Earth masses with masses with

velocityvelocity

0.999997 0.999997 cc

3C321 Jet :Black Hole Fires at Neighboring Galaxy

Other observed jets:

Today’s theoretical models

Massive Black Hole

AccretionDisk

Relativistic Jet

Molecular Torus

Common feature:Common feature:

Rotating Rotating (Strong)(Strong) Gravitational FieldGravitational Field

Another Model – accretion of material from companion star

Singular solutions of the angular Teukolsky equation

Besides regular solutions the angular TME has Besides regular solutions the angular TME has singular solutions:singular solutions:

andand

The singularities can be essentially weakened if one works with Polynomial Heun’s functions (analogy with Hydrogen atom):

Three terms Three terms recurrence relation:recurrence relation:

Polynomial solutionsPolynomial solutions with:with:

andand

Defines symple Defines symple functionsfunctions

Examples of Relativistic Jets 1

Examples of Relativistic Jets 2

Some animations of our jet model

Double wafes (with different velocities): amplitude wave and phase wave

Regular solution of angular TME with three nodes:Regular solution of angular TME with three nodes:

The phase wave:The phase wave:

The amplitude wave:The amplitude wave:

Jet solutions of the angular TME

Double wafes (with different velocities): amplitude wave and phase wave

The phase wave:The phase wave:

The amplitude wave:The amplitude wave:

The distribution of the eigenvalues in the complex plane for the singular case s=-2, m=1 with

F(z)=zF(z)=z

F(z)=1/F(z)=1/zz

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 clusterhttp://physon.phys.uni-sofia.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 • The exact singular solutions of TME can describe relativistic

jets.

Thank You