Higher-dimensional numerical relativity: Formulation and code tests

TL;DR

This paper develops a formalism and numerical codes for simulating five-dimensional spacetimes in numerical relativity, extending existing methods and validating them through stable, accurate simulations of higher-dimensional black holes and gravitational waves.

Contribution

It introduces a higher-dimensional extension of the BSSN formalism and the cartoon method, enabling stable 5D numerical relativity simulations for the first time.

Findings

Stable long-term evolution of 5D Schwarzschild spacetime demonstrated.

Numerical results agree with analytic solutions and converge correctly.

The Landau-Lifshitz pseudo tensor effectively computes gravitational wave energy flux.

Abstract

We derive a formalism of numerical relativity for higher-dimensional spacetimes and develop numerical codes for simulating a wide variety of five-dimensional (5D) spacetimes for the first time. First, the Baumgarte-Shapiro-Shibata-Nakamura formalism is extended for arbitrary spacetime dimensions $D \ge 4$, and then, the so-called cartoon method, which was originally proposed as a robust method for simulating axisymmetric 4D spacetimes, is described for 5D spacetimes of several types of symmetries. Implementing 5D numerical relativity codes with the cartoon methods, we perform test simulations by evolving a 5D Schwarzschild spacetime and a 5D spacetime composed of a gravitational-wave packet of small amplitude. The numerical simulations are stably performed for a sufficiently long time, as done in the 4D case, and the obtained numerical results agree well with the analytic solutions: The numerical solutions are shown to converge at the correct order. We also confirm that a longterm accurate evolution of the 5D Schwarzschild spacetime is feasible using the so-called puncture approach. In addition, we derive the Landau-Lifshitz pseudo tensor in arbitrary dimensions, and show that it gives a robust tool for computing the energy flux of gravitational waves. The formulations and methods developed in this paper provide a powerful tool for studying nonlinear dynamics of higher-dimensional gravity.