We present the first proof-of-principle Cauchy evolutions of asymptotically global anti–de Sitter (AdS) spacetimes with no imposed symmetries, employing a numerical scheme based on the generalized harmonic form of the Einstein equations. In this scheme, the main difficulty in removing all symmetry assumptions can be phrased in terms of finding a set of generalized harmonic source functions that are consistent with AdS boundary conditions. In four spacetime dimensions, we detail an explicit set of source functions that achieves evolution in full generality. A similar prescription should also lead to stable evolution in higher spacetime dimensions, in various couplings with matter fields, and on the Poincaré patch. We apply this scheme to obtain the first long-time stable 3 + 1 simulations of four-dimensional spacetimes with a negative cosmological constant, using initial data sourced by a massless scalar field. We present preliminary results of gravitational collapse with no symmetry assumptions, and the subsequent quasinormal mode ringdown to a static black hole in the bulk, which corresponds to evolution toward a homogeneous state on the boundary.
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