Department of Theoretical Physics University of Sofia Nis 28.12.2007 P. P. Fiziev Department of Theoretical Physics University of Sofia

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. The exact solutions of Teukolsky master equations (TME). New singular exact solutions of TME and their application to the theory of the relativistic jets.

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

Known Numerical studies and approximate analytical methods for BH BC. 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). 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!

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. 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: -i Using the condition: 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 : Additional parameter – mixing angle : Spectral condition – for arbitrary : “falling at the centre” problem operator with defect

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 Physical meaning: Total reflection of the waves at the surface with area radius : PPF, Dirichlet boundary Condition at : The solution: 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 ergosphere, the Cauchy horizon and the ring singularity The event horizon, the Cauchy horizon and the ring singularity

Simple algebraic and differential invariants Simple algebraic and differential invariants for the Kerr solution: Let is the Weyl tensor, - its dual - Density for the Chern - Pontryagin characteristic class - Density for the Euler characteristic class Let Two independent algebraic invariants and 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

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: Spin: S=-2,-1,0,1,2. The radial equation: 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:

W [ , ] = 0, 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. - only the solution obeys BH BC at the EH. If => 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

The Relativistic Jets:. The Most Powerful and. Misterious Phenomenon 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 systems 3. 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

Series of explosions observed! 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 News Jets from GRB060418 and GRB060607A: ~ 200 Earth masses with velocity 0.999997 c

3C321 Jet : Black Hole Fires at Neighboring Galaxy

Other observed jets:

Today’s theoretical models Relativistic Jet Massive Black Hole Common feature: Rotating (Strong) Gravitational Field Molecular Torus Accretion Disk

Another Model – accretion of material from companion star

Singular solutions of the angular Teukolsky equation Besides regular solutions the angular TME has singular solutions: and

Polynomial solutions with: 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

Some animations of our jet model

Double wafes (with different velocities): amplitude wave and phase wave Regular solution of angular TME with three nodes: The phase wave: The amplitude wave:

Jet solutions of the angular TME Double wafes (with different velocities): amplitude wave and phase wave Jet solutions of the angular TME The phase wave: The amplitude wave:

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-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.

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