Two-dimensional twistor manifolds and Teukolsky operators
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Two-dimensional twistor manifolds and Teukolsky operators Bernardo Araneda1,2 Received: 12 December 2019 / Revised: 29 May 2020 / Accepted: 18 June 2020 © Springer Nature B.V. 2020
Abstract The Teukolsky equations are currently the leading approach for analysing stability of linear massless fields propagating in rotating black holes. It has recently been shown that the geometry of these equations can be understood in terms of a connection constructed from the conformal and complex structure of Petrov type D spaces. Since the study of linear massless fields by a combination of conformal, complex and spinor methods is a distinctive feature of twistor theory, and since versions of the twistor equation have recently been shown to appear in the Teukolsky equations, this raises the question of whether there are deeper twistor structures underlying this geometry. In this work we show that all these geometric structures can be understood naturally by considering a 2-dimensional twistor manifold, whereas in twistor theory the standard (projective) twistor space is 3-dimensional.
1 Introduction Twistor theory [34,35] was originally conceived by Roger Penrose as a possible approach to quantum gravity, in which spacetime is no longer a fundamental entity but it is secondary to a more primitive structure. This structure is twistor space, which is (in its projective version) a three-dimensional complex manifold whose points correspond to ‘totally null 2-surfaces’ in the spacetime. The requirement that the twistor space so defined be three-dimensional forces the conformal curvature to be self-dual (SD) or anti-self-dual (ASD), which unfortunately is of little interest for the classical
The current version of this paper is based upon work supported by the Swedish Research Council under Grant No. 2016-06596 while the author was in residence at Institut Mittag-Leffler in Djursholm, Sweden, during the fall 2019.
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Bernardo Araneda [email protected]
1
Facultad de Matemática, Astronomía, Física y Computación, Universidad Nacional de Córdoba, 5000 Córdoba, Argentina
2
Instituto de Física Enrique Gaviola, CONICET, Ciudad Universitaria, 5000 Córdoba, Argentina
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B. Araneda
Lorentzian curved spacetimes of General Relativity. In this work we study geometric constructions that a two- (rather than three-) dimensional moduli space of totally null 2-surfaces induces on a 4-dimensional conformal structure, and their applications to the description of linear massless fields propagating on an algebraically special space. Our main motivation comes from the apparently unrelated problem of black hole stability. The Teukolsky equations were found in [41,42] and constitute currently the leading approach for analysing linear stability of massless fields propagating in a black hole spacetime. They are scalar, second order, partial differential equations involving only one component (in an appropriate frame) of the linear field under consideration. The original derivation [42] is in terms of the Newman-Penrose (NP) formalism. One ha
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