Gravitational observatories in AdS4

A bstract We consider four-dimensional general relativity with a negative cosmological constant in the presence of a finite size boundary, Γ, for both Euclidean and Lorentzian signature. As our boundary condition, we consider the ‘conformal’ boundary condition that fixes the conformal class of the induced metric at Γ and the trace of the extrinsic curvature, K ( x m ). In Lorentzian signature, we must supplement these with appropriate initial data comprising the standard Cauchy data along a spatial slice and, in addition, initial data for a boundary mode that appears due to the presence of the finite size boundary. We perform a linearised analysis of the gravitational field equations for both an S 2 × ℝ as well as a Minkowskian, ℝ 2,1 , boundary. In the S 2 × ℝ case, in addition to the usual AdS 4 normal modes, we uncover a novel linearised perturbation, ω ( x m ), which can exhibit complex frequencies at sufficiently large angular momentum. Upon moving Γ toward the infinite asymptotic AdS 4 boundary, the complex frequencies appear at increasingly large angular momentum and vanish altogether in the strict limit. In the ℝ 2 , 1 case, although we uncover an analogous novel perturbation, we show it does not exhibit complex frequencies. In Euclidean signature, we show that K ( x m ) plays the role of a source for ω ( x m ). When close to the AdS 4 asymptotic boundary, we speculate on the holographic interpretation of ω ( x m ).