Numerical relativity approach to the initial value problem in asymptotically anti–de Sitter spacetime for plasma thermalization: An ADM formulation

Abstract

This article studies a numerical relativity approach to the initial value problem in Anti-de Sitter spacetime relevant for dual nonequilibrium evolution of strongly coupled non-Abelian plasma undergoing Bjorken expansion. In order to use initial conditions for the metric obtained in [G. Beuf, M. P. Heller, R. A. Janik, and R. Peschanski, J. High Energy Phys. 10 (2009) 043.] we introduce new, ADM formalism-based scheme for numerical integration of Einstein's equations with negative cosmological constant. The key novel element of this approach is the choice of lapse function vanishing at fixed radial position, enabling, if needed, efficient horizon excision. Various physical aspects of the gauge theory thermalization process in this setup have been outlined in our companion article [M. P. Heller, R. A. Janik, and P. Witaszczyk, Phys. Rev. Lett. 108, 201602 (2012).]. In this work we focus on the gravitational side of the problem and present full technical details of our setup. We discuss, in particular, the ADM formalism, the explicit form of initial states, the boundary conditions for the metric on the inner and outer edges of the simulation domain, the relation between boundary and bulk notions of time, the procedure to extract the gauge theory energy-momentum tensor and nonequilibrium apparent horizon entropy, as well as the choice of point for freezing the lapse. Finally, we comment on various features of the initial profiles we consider.