Numerical relativity for $D$ dimensional axially symmetric space-times: Formalism and code tests

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 $SO(D-2)$ isometry group for $D\ge 5$, or $SO(D-3)$ for $D\ge 6$. Performing a dimensional reduction on a ($D-4$) 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.

Figure 1

$D$ dimensional representation, using coordinates $(t,x^1,x^2,\dots ,x^{D-3},x^{D-2},z)$, of two types of BH collisions: (left) head-on for spinless BHs, for which the isometry group is $SO(D-2)$; (right) non–head-on, with motion on a single 2-plane, for BHs spinning in that same plane only, for which the isometry group is $SO(D-3)$. The figures make manifest the isometry group in both cases.

Figure 2

The BSSN variable $\chi$ (left) and the quasimatter momentum $K_{\zeta }$ (right) are shown along the axis of collision for a head-on collision at times $t=0$, 5, 20, 40, and $256\mu$. Note that $K_{\zeta }=0$ at $t=0$.

Figure 3

Constraints at time $t=28\mu$, for the evolution of a single Tangherlini BH in five dimensions.

Figure 4

Numerical values versus analytical plot (solid lines) for various values of $\tau$, for the single Tangherlini BH in five dimensions. The horizontal axes are labeled in units of $\mu$.

Figure 5

Constraints at time $t=8\mu$, for the evolution of a single Tangherlini BH in six dimensions.

Figure 6

Numerical values versus the semianalytic expression of ${\tilde{\gamma }}_{xx}$ (cf. Appendix ) along the $x$ axis for the single Tangherlini BH in six dimensions.