Black holes, hidden symmetries, and complete integrability

Gravitational Perturbations of Rotating Black Holes in Lorenz Gauge

Perturbations of Kerr spacetime are typically studied with the Teukolsky formalism, in which a pair of gauge invariant components of the perturbed Weyl tensor are expressed in terms of separable modes that satisfy ordinary differential equations. However, for certain applications it is desirable to construct the full metric perturbation in the Lorenz gauge, in which the linearized Einstein field equations take a manifestly hyperbolic form. Here we obtain a set of Lorenz-gauge solutions to the linearized vacuum field equations on Kerr-Newman-Unti-Tamburino spacetimes in terms of homogeneous solutions to the spin-2, spin-1, and spin-0 Teukolsky equations. We also derive Lorenz-gauge completion pieces representing mass and angular momentum perturbations of Kerr spacetime.

Hidden Conformal Symmetry in Higher Derivative Dynamics for the Kerr Black Hole

The Kerr/CFT correspondence provides a holographic description of spinning black holes that exist in our universe and the notion of hidden conformal symmetry allows for a formulation of this correspondence that is away from extremality. In this study, we examined how hidden conformal symmetry is manifest when we consider dynamics beyond the Klein–Gordon equation through studying the analytic structure of the higher derivative equations of the motion of a massless probe scalar field on a Kerr background, using the monodromy method. Since such higher derivative dynamics appear in known examples of holographic AdS/logCFT correspondences, we investigated whether or not a Kerr/logCFT correspondence could be possible.

Geodesics for the Painlevé–Gullstrand form of Lense–Thirring Spacetime

Recently, the current authors have formulated and extensively explored a rather novel Painlevé–Gullstrand variant of the slow-rotation Lense–Thirring spacetime, a variant which has particularly elegant features—including unit lapse, intrinsically flat spatial 3-slices, and a separable Klein–Gordon equation (wave operator). This spacetime also possesses a non-trivial Killing tensor, implying separability of the Hamilton–Jacobi equation, the existence of a Carter constant, and complete formal integrability of the geodesic equations. Herein, we investigate the geodesics in some detail, in the general situation demonstrating the occurrence of “ultra-elliptic” integrals. Only in certain special cases can the complete geodesic integrability be explicitly cast in terms of elementary functions. The model is potentially of astrophysical interest both in the asymptotic large-distance limit and as an example of a “black hole mimic”, a controlled deformation of the Kerr spacetime that can be contrasted with ongoing astronomical observations.

Linear Stability of Black Holes and Naked Singularities

A review of the current status of the linear stability of black holes and naked singularities is given. The standard modal approach, that takes advantage of the background symmetries and analyze separately the harmonic components of linear perturbations, is briefly introduced and used to prove that the naked singularities in the Kerr–Newman family, as well as the inner black hole regions beyond Cauchy horizons, are unstable and therefore unphysical. The proofs require a treatment of the boundary condition at the timelike boundary, which is given in detail. The nonmodal linear stability concept is then introduced, and used to prove that the domain of outer communications of a Schwarzschild black hole with a non-negative cosmological constant satisfies this stronger stability condition, which rules out transient growths of perturbations, and also to show that the perturbed black hole settles into a slowly rotating Kerr black hole. The encoding of the perturbation fields in gauge invariant curvature scalars and the effects of the perturbation on the geometry of the spacetime is discussed. These notes follow from a course delivered at the V José Plínio Baptista School of Cosmology, held at Guarapari (Espírito Santo) Brazil, from 30 September to 5 October 2021.

Killing Tensor and Carter Constant for Painlevé–Gullstrand Form of Lense–Thirring Spacetime

Recently, the authors have formulated and explored a novel Painlevé–Gullstrand variant of the Lense–Thirring spacetime, which has some particularly elegant features, including unit-lapse, intrinsically flat spatial 3-slices, and some particularly simple geodesics—the “rain” geodesics. At the linear level in the rotation parameter, this spacetime is indistinguishable from the usual slow-rotation expansion of Kerr. Herein, we shall show that this spacetime possesses a nontrivial Killing tensor, implying separability of the Hamilton–Jacobi equation. Furthermore, we shall show that the Klein–Gordon equation is also separable on this spacetime. However, while the Killing tensor has a 2-form square root, we shall see that this 2-form square root of the Killing tensor is not a Killing–Yano tensor. Finally, the Killing-tensor-induced Carter constant is easily extracted, and now, with a fourth constant of motion, the geodesics become (in principle) explicitly integrable.

Positive Energy Functional for Massless Scalars in Rotating Black Hole Backgrounds of Maximal Ungauged Supergravity

We outline a proof of the stability of a massless neutral scalar field ψ in the background of a wide class of four dimensional asymptotically flat rotating and "electrically charged" solutions of supergravity, and the low energy limit of string theory, known as STU metrics. Despite their complexity, we find it possible to circumvent the difficulties presented by the existence of ergo regions and the related phenomenon of superradiance in the original metrics by following a strategy due to Whiting, and passing to an auxiliary metric admitting an everywhere lightlike Killing field and constructing a scalar field ψ (related to a possible unstable mode ψ by a nonlocal transformation) which satisfies the massless wave equation with respect to the auxiliary metric. By contrast with the case for ψ, the associated energy density of ψ is not only conserved but is also non-negative.

Massive Vector Fields in Rotating Black-Hole Spacetimes: Separability and Quasinormal Modes

We demonstrate the separability of the massive vector (Proca) field equation in general Kerr-NUT-AdS black-hole spacetimes in any number of dimensions, filling a long-standing gap in the literature. The obtained separated equations are studied in more detail for the four-dimensional Kerr geometry and the corresponding quasinormal modes are calculated. Two of the three independent polarizations of the Proca field are shown to emerge from the separation ansatz and the results are found in an excellent agreement with those of the recent numerical study where the full coupled partial differential equations were tackled without using the separability property.

Ricci-flat metrics and Killing-Yano tensors

We consider the problem of constructing Ricci-flat metrics on the total space of the canonical bundle over the del Pezzo surface of rank one. We analyze the so-called ‘orthotoric metric’ and its first-order deformation, whose existence is compatible with the Calabi-Yau theorem.