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Uniwersytet Jagiellonski Institut Fizyki Cracow 23 May 2007

Uniwersytet Jagiellonski Institut Fizyki Cracow 23 May 2007. P. P. Fiziev. Department of Theoretical Physics University of Sofia. Talk given at May 23, 2007. Cracow. Exact Solutions of Regge-Wheeler and Teukolsky Equations.

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Uniwersytet Jagiellonski Institut Fizyki Cracow 23 May 2007

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  1. UniwersytetJagiellonski Institut Fizyki Cracow23 May 2007

  2. P. P. Fiziev Department of Theoretical Physics University of Sofia Talk given at May 23, 2007 Cracow

  3. 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. • Exact solutions of Teukolsky master equations (TME)

  4. 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

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

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

  7. 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 .

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

  9. Notation in use: (based on the in-out properties of the solutions) Limits: Important justification:

  10. General local solutions: : X=0, 1, Transition coefficients : The main problem: Unfortunately at present the transition coefficients are not known explicitly !

  11. 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, ).

  12. BH boundary problems on [1, ): • Quasi-normal modes: • Left mixed modes: • Normal modes: • Right mixed modes:

  13. 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:

  14. Perturbations of the BH interior Matzner (1980), PPF gr-qc/0603003 For one introduces interior time: and interior radial variable: . Then: where:

  15. The role of the BH interior -recent articles: • V. Balasubramanian, D. Marolf, M. Rosali, hep-th/0604045 (in quantum gravity) • L. Baiotti, L. Rezzola gr-qc/0608113 (in numerical calculations)

  16. Local solutions in : and For they have the symmetry property:

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

  18. 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 .

  19. 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

  20. 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 .

  21. Dependence of eigenvalues on the mixing angle – NEW PHENOMENONAttraction and Repulsion of the Levels: The mixing angle alpha describes the ratio of the amplitudes of thewaves, going in and going out of the horizon: alpha=0 – no outgoing waves alpha= Pi/2 – no ingoing waves

  22. A typical behavior of the eigenfunctions of the discrete imaginary spectrum: - Abasis for Laplace expansion of perturbations of general form in the BH interior The dependence of the eigenvalues on the angular momentum l

  23. 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.

  24. 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

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

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

  27. 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

  28. 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

  29. Linear perturbations of Kerr metric1972 Teukolsky master equations (TME): The angular equation: Spin: S=-2,-1,0,1,2. The radial equation: and are two independent parameters

  30. 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.

  31. Two independent exact solutions of the angular Teukolsky equation are: Symmetry: The regularity of the solutions at both singular ends of the interval yields the relation:

  32. 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

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

  34. Problems in progress: • Imposing BH boundary conditions one can obtain the known numerical results => a more systematic study of BH QNM in outer domain. • QNM of the Kerr metric interior. • ImposingDirichlet 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 (Pawel Mazur).

  35. Some basic conclusions: • Heun’s functions are a powerful tool for study of all types of solutions of the Regge-Wheer and the Teukolsky masterequations. • 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 reallyholein the space-time ? => resolution of the problem of the real existence of BH

  36. FOURTH ADVANCED RESEARCH WORKSHOP GRAVITY, ASTROPHYSICS, AND STRINGS @ THE BLACK SEA Bulgaria, Kiten, June 10-16, 2007 http://tcpa.uni-sofia.bg/conf/2007/gas/confind.html Organizing Committee P. Fiziev (chairman) R. Borissov P. BozhilovH. DimovE. NissimovR. RashkovM. TodorovR. Tsenov

  37. Thank You

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