Petrov type of linearly perturbed type D spacetimes

We show that a spacetime satisfying the linearized vacuum Einstein equations around a type D background is generically of type I, and that the splittings of the Principal Null Directions (PNDs) and of the degenerate eigenvalue of the Weyl tensor are non analytic functions of the perturbation parameter of the metric. This provides a gauge invariant characterization of the effect of the perturbation on the underlying geometry, without appealing to differential curvature invariants. This is of particular interest for the Schwarzschild solution, for which there are no signatures of the even perturbations on the algebraic curvature invariants. We also show that, unlike the general case, the unstable even modes of the Schwarzschild naked singularity deforms the Weyl tensor into a type II one.Comment: 9 page

Hidden symmetries of generalised gravitational instantons

For conformally K\"ahler Riemannian four-manifolds with a Killing field, we develop a framework to solve the field equations for generalised gravitational instantons corresponding to conformal self-duality and to cosmological Einstein-Maxwell. We obtain generic identities for the curvature of such manifolds without assuming field equations. After applying the framework to recover standard solutions, we find conformally self-dual generalisations of the Page-Pope, Plebanski-Demianski, and Chen-Teo solutions, which are neither hyper-K\"ahler nor quaternionic-K\"ahler, giving new self-dual gravitational instantons in conformal gravity.Comment: 30 page

Complex conformal transformations and zero-rest-mass fields

We give a simple prescription for relating different solutions to the zero-rest-mass field equations in conformally flat space-time via complex conformal transformations and changes in reality conditions. We give several examples including linearized black holes. In particular, we show that the linearized Plebanski-Demianski and Schwarzschild fields are related by a complex translation and a complex special conformal transformation. Similar results hold for the linearized Kerr and C-metric fields, and for a peculiar toroidal singularity.Comment: 16 page

Conformal invariance, complex structures and the Teukolsky connection

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.Fil: Araneda, Bernardo Gabriel. Consejo Nacional de Investigaciones Científicas y Técnicas. Centro Científico Tecnológico Conicet - Córdoba. Instituto de Física Enrique Gaviola. Universidad Nacional de Córdoba. Instituto de Física Enrique Gaviola; Argentina. Universidad Nacional de Córdoba. Facultad de Matemática, Astronomía y Física; Argentin

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

We present weighted covariant derivatives and wave operators for perturbations of certain algebraically special Einstein spacetimes in arbitrary dimensions, under which the Teukolsky and related equations become weighted wave equations. We show that the higher dimensional generalization of the principal null directions are weighted conformal Killing vectors with respect to the modified covariant derivative. We also introduce a modified Laplace-de Rham-like operator acting on tensor-valued differential forms, and show that the wave-like equations are, at the linear level, appropriate projections off shell of this operator acting on the curvature tensor; the projection tensors being made out of weighted conformal Killing-Yano tensors. We give off shell operator identities that map the Einstein and Maxwell equations into weighted scalar equations, and using adjoint operators we construct solutions of the original field equations in a compact form from solutions of the wave-like equations. We study the extreme and zero boost weight cases; extreme boost corresponding to perturbations of Kundt spacetimes (which includes near horizon geometries of extreme black holes), and zero boost to static black holes in arbitrary dimensions. In 4D our results apply to Einstein spacetimes of Petrov type D and make use of weighted Killing spinors.Fil: Araneda, Bernardo Gabriel. Consejo Nacional de Investigaciones Científicas y Técnicas. Centro Científico Tecnológico Conicet - Córdoba. Instituto de Física Enrique Gaviola. Universidad Nacional de Córdoba. Instituto de Física Enrique Gaviola; Argentina. Universidad Nacional de Córdoba. Facultad de Matemática, Astronomía y Física; Argentin

Two-dimensional twistor manifolds and Teukolsky operators

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.Fil: Araneda, Bernardo Gabriel. Consejo Nacional de Investigaciones Científicas y Técnicas. Centro Científico Tecnológico Conicet - Córdoba. Instituto de Física Enrique Gaviola. Universidad Nacional de Córdoba. Instituto de Física Enrique Gaviola; Argentina. Universidad Nacional de Córdoba. Facultad de Matemática, Astronomía y Física; Argentin

Kaehler geometry of black holes and gravitational instantons

We obtain a closed formula for the Kaehler potential of a broad class of four-dimensional Lorentzian or Euclidean conformal "Kaehler" geometries, including the Plebanski-Demianski class and various gravitational instantons such as Fubini-Study and Chen-Teo. We show that the Kaehler potentials of Schwarzschild and Kerr are related by a Newman-Janis shift. Our method also shows that a class of supergravity black holes, including the Kerr-Sen spacetime, is Hermitian (but not conformal Kaehler). We finally show that the integrability conditions of complex structures lead naturally to the (non-linear) Weyl double copy, and we give new vacuum and non-vacuum examples of this relation.Comment: 6 page

Simetrías ocultas, twistors, y estabilidad de campos lineales en agujeros negros

Tesis (Doctor en Física)--Universidad Nacional de Córdoba. Facultad de Matemática, Astronomía, Física y Computación, 2018.En el marco del problema de estabilidad de agujeros negros, esta tesis trata diversos aspectos de las ecuaciones de campos libres sin masa sobre espacio-tiempos curvos, con énfasis en espacios algebraicamente especiales que contienen agujeros negros como soluciones particulares. El enfoque central es el estudio de la posible correspondencia entre campos escalares y campos de spin superior, y de la existencia y origen de simetrías ocultas y ciertos mecanismos asociados a la teoría de twistors. Encontramos fórmulas explícitas de esta correspondencia, y mostramos que el patrón de simetrías subyacente se entiende desde el punto de vista de la covariancia conforme y la existencia de estructuras complejas en el espacio-tiempo. Analizamos también aspectos de estabilidad de los campos en el caso de agujeros negros estáticos asintóticamente anti-de Sitter. Estudiamos espacio-tiempos tanto de cuatro como de altas dimensiones.In the context of the black hole stability problem, this thesis deals with several aspects of the massless free field equations on curved spacetimes, with emphasis on algebraically special spaces that contain black hole solutions as particular cases. The main approach is the study of the possible correspondence between scalar fields and higher spin fields, and of the existence and origin of hidden symmetries and certain mechanisms associated to twistor theory. We find explicit formulas for this correspondence, and we show that the underlying symmetry pattern is understood from the point of view of conformal covariance and the existence of complex structures on the spacetime. We also analyze aspects of the stability of the fields in the case of asymptotically anti-de Sitter static black holes. We study spacetimes of both four and higher dimensions