Holographic complexity in Vaidya spacetimes. Part II

In this second part of the study initiated in https://doi.org/10.1007/JHEP06(2018)046, we investigate holographic complexity for eternal black hole backgrounds perturbed by shock waves, with both the complexity$=$action (CA) and complexity$=$volume (CV) proposals. In particular, we consider Vaidya geometries describing a thin shell of null fluid with arbitrary energy falling in from one of the boundaries of a two-sided AdS-Schwarzschild spacetime. We demonstrate how known properties of complexity, such as the switchback effect for light shocks, as well as analogous properties for heavy ones, are imprinted in the complexity of formation and in the full time evolution of complexity. Following our discussion in https://doi.org/10.1007/JHEP06(2018)046, we find that in order to obtain the expected properties of the complexity, the inclusion of a particular counterterm on the null boundaries of the Wheeler-DeWitt patch is required for the CA proposal.