Numerical relativity in spherical coordinates with the Einstein Toolkit

Numerical relativity codes that do not make assumptions on spatial symmetries most commonly adopt Cartesian coordinates. While these coordinates have many attractive features, spherical coordinates are much better suited to take advantage of approximate symmetries in a number of astrophysical objects, including single stars, black holes, and accretion disks. While the appearance of coordinate singularities often spoils numerical relativity simulations in spherical coordinates, especially in the absence of any symmetry assumptions, it has recently been demonstrated that these problems can be avoided if the coordinate singularities are handled analytically. This is possible with the help of a reference-metric version of the Baumgarte-Shapiro-Shibata-Nakamura formulation together with a proper rescaling of tensorial quantities. In this paper we report on an implementation of this formalism in the Einstein Toolkit. We adapt the Einstein Toolkit infrastructure, originally designed for Cartesian coordinates, to handle spherical coordinates, by providing appropriate boundary conditions at both inner and outer boundaries. We perform numerical simulations for a disturbed Kerr black hole, extract the gravitational wave signal, and demonstrate that the noise in these signals is orders of magnitude smaller when computed on spherical grids rather than Cartesian grids. With the public release of our new Einstein Toolkit thorns, our methods for numerical relativity in spherical coordinates will become available to the entire numerical relativity community.

Figure 1

Diagram depicting the slab transfers involved in the $r_{\min }$ boundary condition.

Figure 2

Diagram showing the slab transfers involved in the ${\theta }_{\min }$ and ${\theta }_{\max }$ boundary conditions.

Figure 3

Comparison of the evolution of the irreducible mass of the BH for SphericalBSSN and McLachlan at two different resolutions each.

Figure 4

Comparison of the evolution of the AH angular momentum for SphericalBSSN and McLachlan at two different resolutions each.

Figure 5

Comparing the evolution of the BH irreducible mass for two SphericalBSSN runs, with and without the excision algorithm described in Sec. 2c.

Figure 6

$|{\mathrm{\Psi }}_4|$ for the even $l=2$ through 8, $m=0$ modes, computed with both SphericalBSSN (black and blue lines) and McLachlan (red lines).

Figure 7

$|{\mathrm{\Psi }}_4|$ for the even $l=2$ through 8, $m=0$ modes (left panel), and the odd $l=3$ through 7, $m=0$ modes (right panel), computed with SphericalBSSN.

Figure 8

Fits for the QNM ringdown, with even modes $l=2$ through 8 in the left panel, and odd modes $l=3$ through 7 in the right panel. The solid lines represent the numerical results for $|{\mathrm{\Psi }}_4|$ as computed with SphericalBSSN, while the points are a result of a fit (33) to these numerical data.