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P . P . Fiziev

P . P . Fiziev. Nis 28.12.2007. Department of Theoretical Physics University of Sofia. Exact Solutions of Regge-Wheeler and Teukolsky Equations.

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P . P . Fiziev

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

  2. Exact Solutions of Regge-Wheeler and Teukolsky Equations • The Regge-Wheeler (RW) equation describes the axial perturbations of Schwarzschild metric in linear approximation. • TheTeukolsky 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 SchwarzschildBH interior. • The exact solutions of Teukolsky master equations (TME). • New singular exact solutions of TME and their application to the theory • of therelativistic jets.

  3. Linear perturbations of Schwarzschild metric1957 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

  4. The standard ansatz separates variables. 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!

  5. Exact mathematical treatment: PPF, In r variable RWE reads: The ansatz: reduces the RWE to aspecific type of 1889 Heun equation: with

  6. Thus one obtains a confluent Heun equation with: 2regular singular points: r=0 and r=1, and 1irregular singular point: in the complex plane Note that after all the horizon r=1turns 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 SchwarzschildRiemannian 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 .

  7. The local solutions (one regular + one singular) around the singular points:X=0, 1, Frobenius type of solutions: Tome (asymptotic) type of solutions:

  8. Different types of boundary problems: I. BH boundary problems: two-singular-points boundary. Up to recently onlythe 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, ).

  9. 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:

  10. Perturbations of theBHinteriorMatzner (1980), PPF gr-qc/0603003, PPF JournalPhys. 66, 0120016, 2006. For one introduces interior time: and interior radial variable: . Then: where:

  11. The continuous spectrum Normal modes in Schwarzschild BH interior: A basis for Fourier expansion of perturbations of general form in the BH interior

  12. 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 .

  13. 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

  14. Numerical resultsFor 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 .

  15. Perturbations of Kruskal-Szekeres manifold In this case the solution can be obtained from functions imposing the additionalcondition 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.

  16. 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

  17. The Spectral condition: Numerical results: The trajectories in of The trajectory of thebasic eigenvalue in and the BH QNM(blackdots):

  18. TheKerr (1963) Metric In Boyer - Lindquist (1967) - {+,-,-,-} coordinates:

  19. The Kerr solution yieldsmuch more complicated structures then the Schwarzschild one: The event horizon, the ergosphere, the Cauchy horizon and the ring singularity The event horizon, the Cauchyhorizon and the ring singularity

  20. Simple algebraic and differential invariants for the Kerr solution:Let is theWeyltensor, - 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

  21. gtt =1 - 2M /, where M is theBHmassForgtt = 0.7, 0.0, -0.1, -0.3, -0.5, -1.5, -3.0, - :

  22. Linear perturbations of Kerr metric S. Teukolsky, PRL, 29, 1115 (1972): Separation of thevariables: A trivial dependence on the Killing directions - . (!) : From stability reasons one MUST have:

  23. 1972 Teukolsky master equations (TME): The angular equation: Spin: S=-2,-1,0,1,2. The radial equation: and are two independent parameters

  24. Up to now only numerical results and approximate methods were studied • First results: • S. Teukolsky, PRL, 29, 1115 (1972). • W Press, S. Teukolsky,AJ185, 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.

  25. Two independent exactregular solutions of the angular Teukolsky equation are: An obvious symmetry:

  26. 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 explicitlyunknown function .

  27. Explicit form of the radial Teukolsky equation where we are using the standard • Note the symmetry between and in the radial TME • and are regularsyngular points of the radial TME • is an irregular singular point of the radial TME

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

  29. BHboundary 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.

  30. Boundary conditions at space infinity – only going to waves: If , then: If , then:

  31. As a result one has to solve the system of equations forand : () 1) For any : 2) and when : or => a nontrivial numerical problem.

  32. Making use of indirect methods: H. Onozawa, 1996

  33. TheRelativistic 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 ???

  34. Jets from GRB

  35. A hyper nova 08.09.05 (distance 11.7 bills lys) FormationofWHAT???: BH???,OR ???VU6APFLG.mov Series ofexplosions observed!

  36. The Jet from M87 2006 News Jets from GRB060418 and GRB060607A: ~ 200 Earth masses with velocity0.999997 c

  37. 3C321 Jet :Black Hole Fires at Neighboring Galaxy

  38. Other observed jets:

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

  40. Another Model – accretion of material from companion star

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

  42. The singularities can be essentially weakened if one works with PolynomialHeun’s functions(analogy with Hydrogen atom): Three terms recurrence relation: Polynomial solutions with: and Defines symple functions

  43. Examples of Relativistic Jets 1

  44. Examples ofRelativistic Jets2

  45. Some animations of our jet model

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

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

  48. 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

  49. The singular cases=-2, m=1with,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

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