Higher-dimensional numerical relativity: Formulation and code tests

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 long-term accurate evolution of the 5D Schwarzschild spacetime is feasible using the so-called puncture approach. In addition, we derive the Landau-Lifshitz pseudotensor 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.

Figure 1

Left-hand panel: Snapshots of ${\tilde{\gamma }}_{xx}$ along the $x$ axis for $\tau /r_h=0.5$, 0.6, 0.7, 0.8, and 0.9. The unit of $x$ is $r_h/2$. The grid resolutions are $\Delta x=0.1$ (×) and 0.05 (⊙). The solid curves denote the analytic solutions, ${\tilde{\gamma }}_{xx}^{(\mathrm{a})}$. Right-hand panel: The averaged error, ${\delta }_{\gamma }$, as a function of $\Delta x$. Here, the average is taken for the data in the range $0\le x\le 5$. The upper short line segment shows the relation of the fourth-order convergence (i.e., a segment with the slope 4).

Figure 2

Left-hand panel: The violation of the Hamiltonian constraint at the time $\tau /r_h=0.9$ for the grid resolutions $\Delta x=0.1$ (×), 0.05 (⊙), and 0.025 (●). Although the violation grows as the 4D hypersurface approaches the singularity, it becomes smaller for a fixed value of $\tau$ as the resolution is increased. Right-hand panel: The average, ${\delta }_H$, as a function of the grid spacing $\Delta x$. The upper short line segment shows the relation of the fourth-order convergence (i.e., a segment with the slope 4).

Figure 3

The evolution of the trace of the extrinsic curvature $K$ at $x=r_h/2$ ($y=z=w=0$) in the puncture gauge. The unit of $t$ is $r_h/2$. The value of $K$ asymptotes to zero.

Figure 4

The values of the lapse $\alpha$ and the shift vector ${\beta }^x$ along the $x$ axis at the time $t=50r_h$. Here, the unit of $x$ is $r_h/2$.

Figure 5

Left-hand panel: The violation of the Hamiltonian constraint at the time $t=50r_h$ for the grid resolutions $\Delta x=0.1$ (×), 0.05 (⊙), and 0.025 (●). After the long-term evolution, the spatial pattern of $H^0$ depends on the resolution, and the error generated at the puncture is $O(1)$. But the general tendency is that the violation becomes smaller as the resolution is increased. Right-hand panel: The average, ${\delta }_H$, in the range $0.5\le x\le 10$ as a function of the grid spacing $\Delta x$. The lower short line segment shows the relation of the fourth-order convergence (i.e., a segment with the slope 4). The convergence is worse than the fourth-order convergence because of the error generated at the puncture.

Figure 6

Left-hand panel: Snapshots of ${\gamma }_{zz}$ along the $x$ axis for $t=1,2,\dots ,10$ for propagation of a gravitational-wave packet. The dotted points (●) and solid curves denote the numerical results and perturbative solutions, respectively. Right-hand panel: The averaged deviation of ${\gamma }_{xx}$ from the analytic perturbative solution (▪) and that from the numerical data with the grid resolution $\Delta x=0.05$ (□) as functions of $\Delta x$. Here, the data at time $t=3$ are used and the average is taken for the data in the range $0\le x\le 6$ and $0\le z\le 6$. The upper short line segment shows the relation of the fourth-order convergence (i.e., a segment with the slope 4).

Figure 7

The integrated energy flux $8\pi G_5{E}_{\mathrm{rad}}{\omega }_0^2$ calculated by the Landau-Lifshitz pseudotensor as a function of the extraction radii $r_{\mathrm{obs}}{\omega }_0$ for the analytic solution (solid curve) and for the numerical results computed with the grid resolutions $\Delta x=0.1$ (⊙), 0.15 (▴), and 0.2 (×). The inset shows the enlarged figure for the region $31\le r_{\mathrm{obs}}{\omega }_0\le 40$.