Heun equation, Teukolsky equation, and type-D metrics

Batic, Davide; Schmid, Harald

doi:10.1063/1.2720277

Cited by 43 publications

(53 citation statements)

References 19 publications

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“…The fact that we had, for generic values for the curvature coupling ξ, mass M , angular momentum a = J/M , NUT charge b and cosmological constant Λ, a Fuchsian equation is a remarkable fact in itself. We found further that, while the generic coupling has 5 regular singular points for the radial equation, 3 the conformally coupled case ξ = 1/6 has one apparent singularity, and the ensuing equation is of Heun type [19,20].…”

Section: Jhep07(2014)132 7 Discussionmentioning

confidence: 94%

“…Our computation below show this for the spin zero case, because of the explicit non-minimal coupling, and can be straightforwardly extended for higher spin cases. The reduction to Heun has also been shown true for s = 1 2 , 1, 2 perturbations of all type-D metrics with cosmological constant [20].…”

Section: Kerr-nut-(a)ds Casementioning

confidence: 90%

“…This is because r = ∞ in (2.10) becomes a removable singularity. In this section, we apply the transformations used in [20] for a scalar…”

Section: Heun Equation From Conformally Coupled Kerr-nut-(a)dsmentioning

confidence: 99%

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Isomonodromy, Painlevé transcendents and scattering off of black holes

Novaes

Cunha

2014

J. High Energ. Phys.

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Abstract:We apply the method of isomonodromy to study the scattering of a generic Kerr-NUT-(A)dS black hole. For generic values of the charges, the problem is related to the connection problem of the Painlevé VI transcendent. We review a few facts about Painlevé VI, Garnier systems and the Hamiltonian structure of flat connections in the Riemann sphere. We then outline a method for computing the scattering amplitudes based on Hamilton-Jacobi structure of Painlevé, and discuss the implications of the generic result to black hole complementarity.

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Section: Jhep07(2014)132 7 Discussionmentioning

confidence: 94%

Section: Kerr-nut-(a)ds Casementioning

confidence: 90%

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Isomonodromy, Painlevé transcendents and scattering off of black holes

Novaes

Cunha

2014

J. High Energ. Phys.

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“…They try to express their solutions in terms of hypergeometric functions, by going to the asymptotics, to the extreme or to the near extreme limit, or putting the solution into a conic box, by changing the energy momentum term if necessary, but keeping the thermodynamic potentials same. A long endeavor was necessary to label Teukolsky Master Equations as belonging to the Heun class [58]. Only recently the equation given by 't Hooft [139] was shown to belong to the Heun class if it were not modified [140].…”

Section: Resultsmentioning

confidence: 99%

Heun Functions and Their Uses in Physics

Hortaçsu

2012

Mathematical Physics

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Most of the theoretical physics known today is described using a small number of differential equations. For linear systems, different forms of the hypergeometric or the confluent hypergeometric equations often suffice to describe the system studied. These equations have power series solutions with simple relations between consecutive coefficients and/ or can be represented in terms of simple integral transforms. If the problem is nonlinear, one often uses one form of the Painlevé equations. There are important examples, however, where one has to use higher order equations. Heun equation is one of these examples, which recently is often encountered in problems in general relativity and astrophysics. Its special and confluent forms take names as Mathieu, Lamé and Coulomb spheroidal equations. For these equations whenever a power series solution is written, instead of a two way recursion relation between the coefficients in the series, we find one between three or four different ones. An integral transform solution using simpler functions also is not obtainable.The use of this equation in physics and mathematical literature exploded in the later years, more than doubling the number of papers with these solutions in the last decade, compared to time period since this equation was introduced in 1889 up to 2010. We use SCI data to conclude this statement, which is not precise, but in the correct ballpark. An integral transform solution using simpler functions also is not obtainable. Here this equation will be introduced and examples for its use, especially in general relativity literature will be given. * This paper is a revised and few times updated version of the conference talk by the same author, published in

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“…An important point in the history of this equation and its confluent forms is the "Centennial Workshop on Heun Equations: Theory and Applications, Sept. [3][4][5][6][7][8]1989, Schloss Ringberg". The presentations at this workshop were printed in a book edited by A. Ronveaux [2].…”

Section: Introductionmentioning

confidence: 99%

Quantum field theory applications of heun type functions

Birkandan

Hortaçsu

2017

Reports on Mathematical Physics

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After a brief introduction to Heun type functions we note that the actual solutions of the eigenvalue equation emerging in the calculation of the one loop contribution to QCD from the Belavin-Polyakov-Schwarz-Tyupkin instanton and the similar calculation for a Dirac particle coupled to a scalar CP 1 model in two dimensions can be given in terms of confluent Heun equation in their original forms. These equations were previously modified to be solved by more elementary functions. We also show that polynomial solutions with discrete eigenvalues are impossible to find in the unmodified equations.

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Heun equation, Teukolsky equation, and type-D metrics

Abstract: Starting with the whole class of type-D vacuum backgrounds with cosmological constant we show that the separated Teukolsky equation for zero rest-mass fields with spin s = ±2 (gravitational waves), s = ±1 (electromagnetic waves) and s = ±1/2 (neutrinos) is an Heun equation in disguise.

Cited by 43 publications

References 19 publications

Isomonodromy, Painlevé transcendents and scattering off of black holes

Isomonodromy, Painlevé transcendents and scattering off of black holes

Heun Functions and Their Uses in Physics

Quantum field theory applications of heun type functions

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