Conformal invariance, complex structures, and the Teukolsky connection

TL;DR

This paper reveals that the Teukolsky connection arises from a special conformally covariant connection on Einstein spacetimes, linking conformal geometry, complex structures, and hidden symmetries in black hole perturbations.

Contribution

It introduces a conformally covariant GHP formalism, connecting the Teukolsky connection to the conformal and complex structures of Einstein spaces, and explains hidden symmetries in black hole physics.

Findings

Teukolsky connection originates from a conformally covariant GHP connection.

Type D principal spinors are parallel with respect to this connection.

Existence of conformal Killing-Yano tensor relates to Kahler metrics in the conformal class.

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 Kahler 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.