Numerical relativity forDdimensional axially symmetric space-times: Formalism and code tests

Abstract

The numerical evolution of Einstein's field equations in a generic background has the potential to answer a variety of important questions in physics: from applications to the gauge-gravity duality, to modeling black hole production in TeV gravity scenarios, to analysis of the stability of exact solutions, and to tests of cosmic censorship. In order to investigate these questions, we extend numerical relativity to more general space-times than those investigated hitherto, by developing a framework to study the numerical evolution of $D$ dimensional vacuum space-times with an $SO(D\ensuremath{-}2)$ isometry group for $D\ensuremath{\ge}5$, or $SO(D\ensuremath{-}3)$ for $D\ensuremath{\ge}6$. Performing a dimensional reduction on a ($D\ensuremath{-}4$) sphere, the $D$ dimensional vacuum Einstein equations are rewritten as a $3+1$ dimensional system with source terms, and presented in the Baumgarte, Shapiro, Shibata, and Nakamura formulation. This allows the use of existing $3+1$ dimensional numerical codes with small adaptations. Brill-Lindquist initial data are constructed in $D$ dimensions and a procedure to match them to our $3+1$ dimensional evolution equations is given. We have implemented our framework by adapting the Lean code and perform a variety of simulations of nonspinning black hole space-times. Specifically, we present a modified moving puncture gauge, which facilitates long-term stable simulations in $D=5$. We further demonstrate the internal consistency of the code by studying convergence and comparing numerical versus analytic results in the case of geodesic slicing for $D=5$, 6.