Generalized complexity and dynamical response in holographic Vaidya spacetimes

Abstract

We investigate "Complexity=Anything" for smooth Vaidya geometries, using the Weyl squared functional as the most common candidate for our study. Our numerical analysis of candidates in 4- and 5-dimensional Reissner-Nordström (RN) and 5-dimensional Gauss-Bonnet (GB) shows evolution in both $r$ and $v$ ("doubled complexity"), unlike static solutions that depend only on $r$. We then study the difference between the complexity of static and dynamical solutions in the asymptotically AdS regime using a Fefferman-Graham expansion. By expanding the bulk metric, extremal embedding, induced metric, normal vector, and extrinsic curvature simultaneously, we show that the first four FG coefficients cancel between the two geometries. In contrast, the first nonvanishing contribution occurs at the fifth coefficient and is controlled by the boundary stress tensor and its derivatives. During the Vaidya quench, the derivative contribution encodes the time-dependent response of the boundary state. In linear response, this response is governed by the retarded stress-tensor correlator and, through generalized Kramers-Kronig relations, can be represented in terms of its spectral density. We thus identify a connection between generalized holographic complexity and the dynamical stress-tensor response, which in the linear-response regime can be represented in terms of the corresponding stress-tensor spectral density.