A variational principle for block operator matrices and its application to the angular part of the Dirac operator in curved spacetime

Winklmeier, Monika

doi:10.1016/j.jde.2008.07.013

Cited by 4 publications

(9 citation statements)

References 24 publications

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“…We hope this work will also be of benefit to other studies, for example, investigations into (i) the effect of field mass on fermionic quasi-normal ringing of black holes; (ii) the effect of particle mass on the Hawking radiation emission spectrum for temperatures close to the mass of the particle species; and (iii) the interaction of brane-localised fermions with rotating higher-dimensional black holes. In this section we give explicit forms for the Clebsch-Gordan coefficients that appear in integrals (37) and (39), and present simplified expressions for C (1) kk and D (1) kk . The relevant coefficients are…”

Section: Discussionmentioning

confidence: 99%

The massive Dirac field on a rotating black hole spacetime: angular solutions

Dolan

Gair

2009

Class. Quantum Grav.

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The massive Dirac equation on a Kerr-Newman background may be solved by the method of separation of variables. The radial and angular equations are coupled via an angular eigenvalue, which is determined from the Chandrasekhar-Page (CP) equation. Obtaining accurate angular eigenvalues is a key step in studying scattering, absorption and emission of the fermionic field.Here we introduce a new method for finding solutions of the CP equation. First, we introduce a novel representation for the spin-half spherical harmonics. Next, we decompose the angular solutions of the CP equation (the mass-dependent spin-half spheroidal harmonics) in the spherical basis. The method yields a three-term recurrence relation which may be solved numerically via continuedfraction methods, or perturbatively to obtain a series expansion for the eigenvalues. In the case µ = ±ω (where ω and µ are the frequency and mass of the fermion) we obtain eigenvalues and eigenfunctions in closed form. We study the eigenvalue spectrum, and the zeros of the maximally co-rotating mode.We compare our results with previous studies, and uncover and correct some errors in the literature. We provide series expansions, tables of eigenvalues and numerical fits across a wide parameter range, and present plots of a selection of eigenfunctions. It is hoped this study will be a useful resource for all researchers interested in the Dirac equation on a rotating black hole background. PACS numbers: 04.70.-s, 04.62.+v I. INTRODUCTIONThe interaction of a fermionic (Dirac) field with a rotating charged (Kerr-Newman) black hole is of deep theoretical interest. The interaction depends on three couplings: coupling between quantum-mechanical spin and black hole (BH) rotation; coupling between field mass and BH mass; and coupling between field charge and BH charge. Desire for a deeper understanding of these couplings has motivated a range of studies over the last three decades.Over thirty years ago, Chandrasekhar [1] and Page [2] showed that separation of variables is possible for the massive Dirac field on the Kerr-Newman spacetime. In other words, the partial differential equations (PDEs) governing the evolution of the Dirac field may be reduced to a set of coupled ordinary differential equations (ODEs), and the solution can be expressed as a sum over modes and integral over frequency. The radial and angular ODEs are coupled through an angular eigenvalue λ. The eigenvalue is found by solving the so-called Chandrasekhar-Page (CP) equation (see Eq. (4) below). The eigenfunctions of the CP equation -the so-called mass dependent spin-half spheroidal harmonics (MDSHs) -are required for a full reconstruction of the field. In this paper we present a new method for computing both the eigenvalues and eigenfunctions. Our aim is to show that finding MDSHs is no more difficult than finding the spin-weighted spheroidal harmonics for massless fields [3]. Along the way, we review alternative methods and highlight some inaccuracies in the literature.Motivation for this work arose from two s...

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Section: Discussionmentioning

confidence: 99%

The massive Dirac field on a rotating black hole spacetime: angular solutions

Dolan

Gair

2009

Class. Quantum Grav.

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“…In order to implement (3.2), we consider the analytic enclosures derived in references [7,11]. Table 1.…”

Section: Numerical Benchmarksmentioning

confidence: 99%

“…We now determine various numerical approximations of intervals of enclosure for the eigenvalues of the angular Kerr-Newman Dirac operator (1) by means of suitable combinations of ( 12) and (13). In order to implement (13), we consider the analytic enclosures derived in [Win05] and [Win08]. Our purpose here is twofold.…”

Section: Numerical Benchmarksmentioning

confidence: 99%

“…This requires knowing beforehand some rough information about the position of the eigenvalues and the neighbouring spectrum. For this purpose, we have employed a combination of the analytical inclusions found in [Win05] and [Win08], and numerically calculated inclusions determined from (12). This technique allows reducing by roughly two orders of magnitude the length of the segments of eigenvalue inclusion.…”

Section: Numerical Benchmarksmentioning

confidence: 99%

“…A simple explicit expression for the eigenvalues appears to be available only for the case am = ±aω. By invoking an abstract variational principle on the corresponding operator pencil, coarse analytical enclosures for these eigenvalues in the case am = ±aω were found in references [7,11]. Our aim is to sharpen these enclosures by several orders of magnitude via a projection method.…”

Section: Introductionmentioning

confidence: 99%

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Sharp eigenvalue enclosures for the perturbed angular Kerr–Newman Dirac operator

Boulton

Winklmeier

2015

Proc. R. Soc. A.

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We examine a certified strategy for determining sharp intervals of enclosure for the eigenvalues of matrix differential operators with singular coefficients. The strategy relies on computing the secondorder spectrum relative to subspaces of continuous piecewise linear functions. For smooth perturbations of the angular Kerr-Newman Dirac operator, explicit rates of convergence linked to regularity of the eigenfunctions are established. Numerical tests which validate and sharpen by several orders of magnitude the existing benchmarks are also included.

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