Two-dimensional twistor manifolds and Teukolsky operators

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.

Linear Stability of Black Holes and Naked Singularities

“…The analysis of the stability of black hole outer regions, having passed decades ago the test of modal linear stability, has had, after a long period of little activity, a remarkable progress in the last few years. An incomplete list of recent advances related to the stability of the Schwarschild and Kerr black holes follow: (i) for Λ ≥ 0 Schwarzschild black holes the nonmodal linear stability was established in [9,36]; (ii) in the Λ = 0 case, the decay in time of generic linear perturbations of the Schwarzschild black hole, leaving a "slowly" rotating Kerr black hole was proved in [43]; (iii) the conditional stability of the Λ < 0 Schwarzschild black hole, and the breaking of the even/odd symmetry mediated by the Chandrasekhar operators (93) and (94) was studied in [29]; (iv) the non-linear stability of the Schwarzschild de Sitter black hole was proved in [7]; (v) a preprint is now available with a proof of the non-linear stability of the Λ = 0 Schwarzschild black hole [8]; (vi) pointwise decay estimates for solutions of the linearized Einstein's equations on the outer region of a Kerr black hole were obtained in [13]; (vii) the role of hidden symmetries (see the review [44]) in type D spacetimes, and the reconstruction of (gravitational, Maxwell and spinor) perturbation fields from "Debye potentials" (first introduced in [45,46]), was studied in depth and made clear in the series of papers [47][48][49][50].…”

Linear Stability of Black Holes and Naked Singularities

“…where ǫ is the perturbation parameter in (5). The above equation is telling us that, no matter the ω value (whether real or complex), the perturbation is inconsistent unless we select the boundary condition α = 0.…”

“…The non-analytical character of the splitting (as a function of the perturbation parameter ǫ), discussed in some detail in [15], can be avoided by a re-parametrization of the familiy of solutions g ab (ǫ) in (5).…”

“…The analysis of the stability of black hole outer regions, having passed decades ago the test of modal linear stability, has had, after a long period of little activity, a remarkable progress in the last few years. An incomplete list of recent advances related to the stability of the Schwarschild and Kerr black holes follow: (i) for Λ ≥ 0 Schwarzschild black holes the nonmodal linear stability was established in [24,25]; (ii) in the Λ = 0 case, the decay in time of generic linear perturbations of the Schwarzschild black hole, leaving a "slowly" rotating Kerr black hole was proved in [17]; (iii) the conditional stability of the Λ < 0 Schwarzschild black hole, and the breaking of the even/odd symmetry mediated by the Chandrasekhar operators (93) and (94) was studied in [6]; (iv) the non-linear stability of the Schwarzschild de Sitter black hole was proved in [32]; (v) a preprint is now available with a proof of the non-linear stability of the Λ = 0 Schwarzschild black hole [18]; (vi) pointwise decay estimates for solutions of the linearized Einstein's equations on the outer region of a Kerr black hole were obtained in [1]; (vii) the role of hidden symmetries (see the review [28]) in type D spacetimes, and the reconstruction of (gravitational, Maxwell and spinor) perturbation fields from "Debye potentials" (first introduced in [37,47]), was studied in depth and made clear in the series of papers [2][3][4][5].…”

On the geometry of Petrov type II spacetimes

“…This means that a primed spinor field ψ A ′ can be thought of as a "1-form" in T * Σ. Therefore, restricting to β-surfaces, the operator CA ′ is CA ′ : Γ(S {p,0} [w]) → Γ(T * Σ ⊗ S {p+1,0} [w]), from which we see that it can be regarded as a connection, in the usual sense, on conformally and GHP-weighted vector bundles over β-surfaces on CM, see [6].…”