dubna, august 2009
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Dubna, August 2009. International Bogoliubov Conference PROBLEMS IN THEORETICAL AND MATHEMATICAL PHYSICS. Generalized Teukolsky-Starobinsky Identities. Plamen Fiziev Department of Theoretical Physics University of Sofia. Talk at The International Bogoliubov Conference - PowerPoint PPT PresentationTRANSCRIPT
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Dubna, August 2009
International Bogoliubov Conference
PROBLEMS IN THEORETICAL AND MATHEMATICAL PHYSICS
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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
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The Nonlinear Mechanics and Wave Mechanics
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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
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Confluent Heun Equation:
Frobenius solution aound z = 0 :
- a recurrence relation
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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:
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The - condition
=> =>
Note that=>
= N-polynomial
A Novel Identity:
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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:
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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
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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 !
+
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The Explicit Form of TSI for all -solutions to TRE:
Starobinsky
Constant
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The Explicit Form of TSI for all -solutions to TAE:
Starobinsky
Constant
Disentangled form of TSI for TAE:
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The Explicit Form of TSI for all -solutions to RWE:
Starobinsky
Constant
Note that here
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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 :
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Thank you for your attention !