<p>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, <i>K</i>(<i>x</i>^<i>m</i>). 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 <i>S</i>² × ℝ as well as a Minkowskian, ℝ^2,1, boundary. In the <i>S</i>² × ℝ case, in addition to the usual AdS₄ normal modes, we uncover a novel linearised perturbation, <Emphasis Type="BoldItalic">ω</Emphasis>(<i>x</i>^<i>m</i>), which can exhibit complex frequencies at sufficiently large angular momentum. Upon moving Γ toward the infinite asymptotic AdS₄ boundary, the complex frequencies appear at increasingly large angular momentum and vanish altogether in the strict limit. In the ℝ^2<i>,</i>1 case, although we uncover an analogous novel perturbation, we show it does not exhibit complex frequencies. In Euclidean signature, we show that <i>K</i>(<i>x</i>^<i>m</i>) plays the role of a source for <Emphasis Type="BoldItalic">ω</Emphasis>(<i>x</i>^<i>m</i>). When close to the AdS₄ asymptotic boundary, we speculate on the holographic interpretation of <Emphasis Type="BoldItalic">ω</Emphasis>(<i>x</i>^<i>m</i>).</p>