Generalized wave operators, weighted Killing fields, and perturbations of higher dimensional spacetimes

Araneda, Bernardo

doi:10.1088/1361-6382/aab06a

Cited by 7 publications

(22 citation statements)

References 40 publications

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“…Finally, we mention that the generalized Teukolsky connection and the closely related 'weighted Killing fields' found in [3] for perturbations of higher dimensional spacetimes, can be shown to follow the same principle as the one exploited here, namely conformal and GHP covariance. However, the question about conformal invariance of field equations in higher dimensions is much more subtle than in the 4-dimensional case (in particular, Maxwell fields in d = 4 are not conformally invariant).…”

Section: Discussionsupporting

confidence: 57%

“…(1.6) Interestingly enough, it was found in [3] that (1.6) solves the equation 7) and therefore it can be considered as a Killing spinor with respect to the Teukolsky connection (in [3] these objects were called 'weighted' Killing spinors). The ordinary Killing spinor K AB = Ψ −1/3 2 o (A ι B) of type D spacetimes mentioned above is just a particular case of (1.6)-(1.7).…”

Section: Introductionmentioning

confidence: 99%

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Conformal invariance, complex structures and the Teukolsky connection

Araneda¹

2018

Class. Quantum Grav.

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We show that the Teukolsky connection, which defines generalized wave operators governing the behavior of massless fields on Einstein spacetimes of Petrov type D, has its origin in a distinguished conformally and GHP covariant connection on the conformal structure of the spacetime. The conformal class has a (metric compatible) integrable almost-complex structure under which the Einstein space becomes a complex (Hermitian) manifold. There is a unique compatible Weyl connection for the conformal structure, and it leads to the construction of a conformally covariant GHP formalism and a generalization of it to weighted spinor/tensor fiber bundles. In particular, 'weighted Killing spinors', previously defined with respect to the Teukolsky connection, are shown to have their origin in the GHP-Weyl connection, and we show that the type D principal spinors are actually parallel with respect to it. Furthermore, we show that the existence of a conformal Killing-Yano tensor can be thought to be a consequence of the presence of a Kähler metric in the conformal class. These results provide an interpretation of the persistent hidden symmetries appearing in black hole perturbations. We also show that the preferred Weyl connection allows a natural injection of spinor fields into local twistor space and that this leads to the notion of weighted local twistors. Finally, we find conformally covariant operator identities for massless fields and the corresponding wave equations. * 1 i.e. a second order linear differential operator whose principal symbol is the spacetime metric.

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Section: Discussionsupporting

confidence: 57%

Section: Introductionmentioning

confidence: 99%

Conformal invariance, complex structures and the Teukolsky connection

Araneda¹

2018

Class. Quantum Grav.

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“…This form of the gauge potentials was pointed out in [15,16], where Araneda derived the form of gauge potential on a type-D vacuum spacetime with cosmological constant. Our result generalizes it to non-vacuum case.…”

Section: Gauge Potentials Of Maxwell Fieldmentioning

confidence: 74%

“…For s = ±2, even in four dimensions, the separation of variables in the linearised Einstein equation has not been realized so far except in the vacuum case. It is remarkable that Araneda [15,16]…”

Section: Summary and Discussionmentioning

confidence: 99%

Hidden symmetry and the separability of the Maxwell equation on the Wahlquist spacetime

Houri

Tanahashi

Yasui

2020

Class. Quantum Grav.

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We examine hidden symmetry and its relation to the separability of the Maxwell equation on the Wahlquist spacetime. After seeing that the Wahlquist spacetime is a type-D spacetime whose repeated principal null directions are shear-free and geodesic, we show that the spacetime admits three gauged conformal Killing-Yano (GCKY) tensors which are in a relation with torsional conformal Killing-Yano tensors. As a by-product, we obtain an ordinary CKY tensor. We also show that thanks to the GCKY tensors, the Maxwell equation reduces to three Debye equations, which are scalar-type equations, and two of them can be solved by separation of variables. I. INTRODUCTIONHidden symmetry of spacetime has played an important role in the study of black hole physics. In particular, conformal Killing-Yano symmetry, known as hidden symmetry of the Kerr spacetime, has received special attention since it has been crucial to understanding the separability of various equations on a curved spacetime.On the Kerr spacetime, the Hamilton-Jacobi equation for geodesics, the Klein-Gordon equation, and the Dirac equation can be solved by separation of variables, and these separabilities have been understood in terms of a Killing-Yano tensor. For electromagnetic and gravitational perturbations, Teukolsky [1, 2] provided the scalar-type master equations, which can be solved by separation of variables. Cohen and Kegeles [3] showed that, if a spacetime is algebraically special and its repeated principal null direction (PND) is shear-free and geodesic, 1 the Maxwell equation reduces to a scalar-type master equation called the Debye equation. They also showed that when their method is applied to the Kerr spacetime, the Debye equation coincides with the Teukolsky equation. Moreover, they extended their results to massless Dirac fields on algebraically special spacetimes and gravitational perturbations on vacuum spacetimes [4]. After a while, Benn, Charlton, and Kress [5] unveiled the underlying structure of the works by Cohen and Kegeles from the viewpoint of GCKY symmetry. So far, while the GCKY symmetry has been related to obtaining the scalar-type master equations from the Dirac, Maxwell, and linearized Einstein equations, the separability of those equations has not been well understood. To fill the gap is one of the aims in this paper.In this paper, we examine hidden symmetry of the Wahlquist spacetime [6][7][8][9][10][11][12], which is a stationary, axially symmetric solution to the Einstein equation with rigidly rotating perfect fluids. According to the Goldberg-Sachs theorem, the repeated PNDs on a type-D vacuum spacetime are shear-free and geodesic. However, since the Wahlquist metric is a non-vacuum type-D spacetime, it is interesting to ask if the repeated PNDs are shear-free and geodesic. According to [3,5], if they are so, we obtain GCKY tensors with a certain condition and hence the Maxwell equation reduces to the Debye equation. Since the Wahlquist spacetime is known to admit a torsional conformal Killing-Yano (TCKY) tensor [12], there may...

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“…The separation of variables in higher-dimensional Maxwell equations remained an open problem for a long time. A partial success was achieved in [24] where it was demonstrated that such equations can be decoupled, using the higher-dimensional generalization of the Newman-Penrose formalism [25], provided that the background spacetime is Kundt, i.e., it admits a null geodesic congruence which is all: shear-free, twist-free, and expansion-free. Unfortunately, the higher-dimensional rotating black hole spacetimes do not belong to this class.…”

Section: Introductionmentioning

confidence: 99%

Separation of variables in Maxwell equations in Plebański-Demiański spacetime

2018

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A new method for separating variables in the Maxwell equations in four-and higher-dimensional Kerr-(A)dS spacetimes proposed recently by Lunin is generalized to any off-shell metric that admits a principal Killing-Yano tensor. The key observation is that Lunin's ansatz for the vector potential can be formulated in a covariant form-in terms of the principal tensor. In particular, focusing on the fourdimensional case we demonstrate separability of Maxwell's equations in the Kerr-NUT-(A)dS and the Plebański-Demiański family of spacetimes. The new method of separation of variables is quite different from the standard approach based on the Newman-Penrose formalism.

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Generalized wave operators, weighted Killing fields, and perturbations of higher dimensional spacetimes

Cited by 7 publications

References 40 publications

Conformal invariance, complex structures and the Teukolsky connection

Conformal invariance, complex structures and the Teukolsky connection

Hidden symmetry and the separability of the Maxwell equation on the Wahlquist spacetime

Separation of variables in Maxwell equations in Plebański-Demiański spacetime

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