Tensor Harmonics in Canonical Form for Gravitational Radiation and Other Applications

Zerilli, Frank J.

doi:10.1063/1.1665380

Cited by 131 publications

(142 citation statements)

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“…This remarkable property followed from a special relation (the equivalent for asymptotically flat spacetimes of our equation (19)) between the potentials and the behavior of W at the boundaries. However, as one can see in tables 3, 4 and 5 there is a isospectrality breaking between odd and even perturbations in Schwarzschild anti-de Sitter spacetime.…”

Section: On the Isospectrality Breaking Between Odd And Even Perturbamentioning

confidence: 99%

“…We use the same perturbations as originally given by Regge and Wheeler [18], retaining their notation. After a decomposition in tensorial spherical harmonics (see Zerilli [19] and Mathews [20]), these fall into two distinct classes -odd and even -with parities (−1) l+1 and (−1) l respectively, where l is the angular momentum of the particular mode. While working in general relativity one has some gauge freedom in choosing the elements h ab (x ν ) and one should take advantage of that freedom in order to simplify the rather lengthy calculations involved in computing (8).…”

Section: Gravitational Perturbationsmentioning

confidence: 99%

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Quasinormal modes of Schwarzschild–anti-de Sitter black holes: Electromagnetic and gravitational perturbations

Cardoso

Lemos²

2001

Phys. Rev. D

332

363

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We study the quasi-normal modes (QNM) of electromagnetic and gravitational perturbations of a Schwarzschild black hole in an asymptotically Anti-de Sitter (AdS) spacetime. Some of the electromagnetic modes do not oscillate, they only decay, since they have pure imaginary frequencies. The gravitational modes show peculiar features: the odd and even gravitational perturbations no longer have the same characteristic quasinormal frequencies. There is a special mode for odd perturbations whose behavior differs completely from the usual one in scalar and electromagnetic perturbation in an AdS spacetime, but has a similar behavior to the Schwarzschild black hole in an asymptotically flat spacetime: the imaginary part of the frequency goes as 1 r + , where r + is the horizon radius. We also investigate the small black hole limit showing that the imaginary part of the frequency goes as r 2 + . These results are important to the AdS/CFT conjecture since according to it the QNMs describe the approach to equilibrium in the conformal field theory.

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Section: On the Isospectrality Breaking Between Odd And Even Perturbamentioning

confidence: 99%

Section: Gravitational Perturbationsmentioning

confidence: 99%

Quasinormal modes of Schwarzschild–anti-de Sitter black holes: Electromagnetic and gravitational perturbations

Cardoso

Lemos²

2001

Phys. Rev. D

332

363

View full text Add to dashboard Cite

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“…These harmonics are normalized so that A basis for 2-tensor fields on the sphere can also be constructed in a similar way [33] and it is formed by three types of objects: pure-trace tensors can be decomposed using ab Y m l ; antisymmetric tensors can be expanded using ab Y m l ; and finally symmetric traceless tensors can be expanded using…”

Section: A Gs Notation For Harmonicsmentioning

confidence: 99%

“…III. Section IV introduces the Regge-Wheeler-Zerilli harmonics [28,33] and generalizes them to arbitrary rank tensors. Formulas for their products are given, based on the representation matrices of the rotation group.…”

Section: Introductionmentioning

confidence: 99%

Second- and higher-order perturbations of a spherical spacetime

2006

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The Gerlach and Sengupta (GS) formalism of coordinate-invariant, first-order, spherical and nonspherical perturbations around an arbitrary spherical spacetime is generalized to higher orders, focusing on second-order perturbation theory. The GS harmonics are generalized to an arbitrary number of indices on the unit sphere and a formula is given for their products. The formalism is optimized for its implementation in a computer-algebra system, something that becomes essential in practice given the size and complexity of the equations. All evolution equations for the second-order perturbations, as well as the conservation equations for the energy-momentum tensor at this perturbation order, are given in covariant form, in Regge-Wheeler gauge.

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“…The linearized stability of the Schwarzschild black hole follows by combining the ReggeWheeler-Zerilli-Moncrief decomposition of gravitational perturbations of the Schwarzschild metric 16,19,13 with a result by Kay and Wald 12 that proves the boundedness of all solutions of the wave equation corresponding to C ϱ -data of compact support. The proof of the last rests on the positivity of the conserved energy.…”

Section: General Introductionmentioning

confidence: 99%

A note on the Klein–Gordon equation in the background of a rotating black hole

Beyer

2009

Journal of Mathematical Physics

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This short paper should serve as a basis for further analysis of a previously found new symmetry of the solutions of the wave equation in the gravitational field of a Kerr black hole. Its main new result is the proof of essential self-adjointness of the spatial part of a reduced normalized wave operator of the Kerr metric in a weighted L 2 -space. As a consequence, it leads to a purely operator theoretic proof of the well posedness of the initial value problem of the reduced Klein-Gordon equation in that field in that L 2 -space and in this way generalizes a corresponding result of Kay ͓"The double-wedge algebra for quantum fields on Schwarzschild and Minkowski spacetimes," Commun. Math. Phys. 100, 57 ͑1985͔͒ in the case of the Schwarzschild black hole. It is believed that the employed methods are applicable to other separable wave equations. © 2009 American Institute of Physics. ͓DOI: 10.1063/1.3037327͔ I. GENERAL INTRODUCTIONThe linearized stability of the Schwarzschild black hole follows by combining the ReggeWheeler-Zerilli-Moncrief decomposition of gravitational perturbations of the Schwarzschild metric 16,19,13 with a result by Kay and Wald 12 that proves the boundedness of all solutions of the wave equation corresponding to C ϱ -data of compact support. The proof of the last rests on the positivity of the conserved energy.The question of the linearized stability of the Kerr black hole is still an open problem whose outcome is of major importance to general relativity. In comparison to the case of the Schwarzschild black hole, the solution to this problem is considerably more complicated. Mainly, this is due to two facts. First, a decomposition comparable to that of Regge-Wheeler-Zerilli-Moncrief does not yet exist in this case, although the recent finding in Ref. 6 gives hope that such a decomposition might exist. In contrast, a partial decomposition based on the Newman-Penrose formalism depends on the choice of a tetrad field, i.e., is gauge dependent even under "small" coordinate transformations. 2 Second, a conserved energy for the solutions of the wave equation exists, but the energy density is negative inside the ergosphere. This fact excludes at least a direct application of the so-called "energy methods" to a proof of stability of the solutions. The total energy could be finite while the field still might grow exponentially in parts of the space-time. But recently a local stability result has been proved that the restrictions of the solutions of the wave equation to compact subsets K in space are elements of L C ϱ ͑K͒ with a norm converging to zero for t → ϱ. 9 Because of the absence of a decomposition of the Regge-Wheeler-Zerilli-Moncrief type, the question of applicability of the last and other similar results to the question of linearized stability of the Kerr metric is still open.As mentioned above, Ref. 6 contains the surprising find of a new symmetry operator that a͒ Electronic

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Tensor Harmonics in Canonical Form for Gravitational Radiation and Other Applications

Cited by 131 publications

References 3 publications

Quasinormal modes of Schwarzschild–anti-de Sitter black holes: Electromagnetic and gravitational perturbations

Quasinormal modes of Schwarzschild–anti-de Sitter black holes: Electromagnetic and gravitational perturbations

Second- and higher-order perturbations of a spherical spacetime

A note on the Klein–Gordon equation in the background of a rotating black hole

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