Abstract
Abstract The Teukolsky master equation and its associated spin-weighted spheroidal harmonic decomposition simplify considerably the study of linear gravitational perturbations of the Kerr(-AdS) black hole. However, the formulation of the problem is not complete before we assign the physically relevant boundary conditions. We find a set of two Robin boundary conditions (BCs) that must be imposed on the Teukolsky master variables to get perturbations that are asymptotically global AdS, i.e. that asymptotes to the Einstein Static Universe. In the context of the AdS/CFT correspondence, these BCs allow a non-zero expectation value for the CFT stress-energy tensor while keeping fixed the boundary metric. When the rotation vanishes, we also find the gauge invariant differential map between the Teukolsky and the Kodama-Ishisbashi (Regge-Wheeler-Zerilli) formalisms. One of our Robin BCs maps to the scalar sector and the other to the vector sector of the Kodama-Ishisbashi decomposition. The Robin BCs on the Teukolsky variables will allow for a quantitative study of instability timescales and quasinormal mode spectrum of the Kerr-AdS black hole. As a warm-up for this programme, we use the Teukolsky formalism to recover the quasinormal mode spectrum of global AdS-Schwarzschild, complementing previous analysis in the literature.
Highlights
To study linear gravitational perturbations of a black hole we need to solve the linearized Einstein equation
We find a set of two Robin boundary conditions (BCs) that must be imposed on the Teukolsky master variables to get perturbations that are asymptotically global AdS, i.e. that asymptotes to the Einstein Static Universe
In this paper we are interested in linear gravitational perturbations of the Kerr-AdS black hole, with a focus on its boundary conditions (BCs)
Summary
The Kerr-AdS geometry was originally written by Carter in the Boyer-Lindquist coordinate system {T, r, θ, φ} [50]. Where Ξ is to be defined in (2.3) In this coordinate system the Kerr-AdS black hole line element reads [51]. The Chambers-Moss coordinate system {t, r, χ, φ} has the nice property that the line element treats the radial r and polar χ coordinates at an almost similar footing One anticipates that this property will naturally extend to the radial and angular equations that describe gravitational perturbations in the Kerr-AdS background. The Kerr-AdS black hole obeys Rμν = −3L−2gμν, and asymptotically approaches global AdS space with radius of curvature L This asymptotic structure is not manifest in (2.2), one of the reasons being that the coordinate frame {t, r, χ, φ} rotates at infinity with angular velocity Ω∞ = −a/(L2Ξ). Further properties of the Kerr-AdS spacetime are discussed in appendix A of [52]
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