Conformal invariance, complex structures and the Teukolsky connection

Abstract

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.

Two-dimensional twistor manifolds and Teukolsky operators

“…We also mentioned in the introduction that it arises in the study of the Teukolsky equations: there exists a covariant derivative D a (the 'Teukolsky connection') whose square D a D a is the Teukolsky operator, and certain spinor fields involved in the equations satisfy (2.3). (See the introduction in [9].) This fact is actually one of the main motivations for the present work.…”

“…ω a is the usual GHP connection form, and the 1-form B a was originally considered in [2] (for a choice of conformal weights different to (3.13) B a has to be modified, for details see [9]). Now consider a section ∈ (S k,0 l,0 ), and project its indices on the frame (ξ A , η A ) and its dual, so that one gets a bunch of components.…”

“…admits non-trivial solutions. This can be formulated in a way closer to the theory of integrable systems 9 . More precisely, consider a primed spin frame (o A , ι A ) such that CA o B = 0 and CA ι B = 0, and introduce the following operators acting on (S A ( p)[w]):…”

“…For the construction of symmetry operators, we can obtain solutions of the Teukolsky and Fackerell-Ipser equations starting from SD or ASD, admit a null geodesic congruence that is shear-free (referred to as SFR along the text), and we have also assumed that the corresponding spinor field is a twofold principal null direction of the SD curvature. As mentioned, our main motivation for studying this problem was the recent result [9] that the Teukolsky equations (that are central to the black hole stability problem) are intimately related to a combination of conformal, complex and spinor geometry, which is a natural territory of twistor theory.…”

“…Further interest in this question arises when taking into account the result found in [8] that certain spinor fields involved in the equations satisfy the twistor equation with respect to the Teukolsky derivative. The problem of uncovering the underlying geometry was addressed in [9], where, by using spinor methods, it was found that it can be understood from consideration of conformal and complex structures in the spacetime. Now, since the combination of conformal, complex and spinor geometry in four dimensions is a natural arena for twistor theory, the appearance in the same problem of (versions of) the twistor equation together with conformal and complex structures suggests that more profound aspects of twistor theory could be involved in the problem.…”