Dubna, August 2009

International Bogoliubov Conference

PROBLEMS IN THEORETICAL AND MATHEMATICAL PHYSICS

Generalized Teukolsky-Starobinsky Identities

Plamen FizievDepartment of Theoretical Physics

University of Sofia

Talk at

The International Bogoliubov ConferencePROBLEMS IN THEORETICAL AND

MATHEMATICAL PHYSICS

25 August 2009

The Nonlinear Mechanics and Wave Mechanics

Born in Weisbaden April 3, 1859Died in Karsruhe January 10, 1929

Heun’s DifferentialEquation:

Zur Theorie der Riemann'schen Functionen zweiter Ordnung mit Vier Verzweigungs-punkten

Math. Ann. 31 (1889) 161-179

A KEYfor

HugeamountofPhysicalProblemsfoundby

Confluent Heun Equation:

Frobenius solution aound z = 0 :

- a recurrence relation

Novel relations for confluent Heun’s functions and their Derivatives, PF: arXiv:0904.0245 [math-ph]

Self-adjoint form of confluent Heun’s operator:

The comutator:

Chain of confluent Heun’s operators:

The basic general relation:

The - condition

=> =>

Note that=>

= N-polynomial

A Novel Identity:

Teukolsky Master Equation:

x

Separatipon of the variables:

xTAE:

TRE: x

Small perturbations of spin-weightss =-2,-3/2,-1,-1/2 0,1/2, 1, 3/2, 2 of Kerr and Schwarzschild,backgroundin terms of Weylinvariants

Schwarzschild: (a=0)Kerr:

Universal Form of the Exact Solutions of TAE, TRE and Regge-Wheeler Eq.

PF: arXiv:0902.1277, arXiv:0906.5108 [gr-qc]

For TAE and: For TRE and:

x

Regge-Wheeler Equation:

,

Since the geodesic equations are solved in elliptic functions

Universal form of the Teukolsky-Starobinsky Identities

For the above special values of the parameters all solutions turn to be -solutions. As a result the universal identities take place: PF: arXiv:0906.5108 [gr-qc]

GeneralizedTeukolsky-StarobinskyIdentities:

As a result of amazing new symmetry for N+1=2|s| :

if is a solutions with spin-weight +s, then is a solution of TE with –s !

+

The Explicit Form of TSI for all -solutions to TRE:

Starobinsky

Constant

The Explicit Form of TSI for all -solutions to TAE:

Starobinsky

Constant

Disentangled form of TSI for TAE:

The Explicit Form of TSI for all -solutions to RWE:

Starobinsky

Constant

Note that here

New effective method for calculation of Starobinsky constant for all spin-weights s

In the case of -solutions:

Starobinsky constants for different s coincide up to known factor with the for Taylor series for confluent Heun’s function .

Hence, we can calculate Starobinsky constants using recurrence relation :

Thank you for your attention !