Solving Einstein’s equations with dual coordinate frames

A method is introduced for solving Einstein’s equations using two distinct coordinate systems. The coordinate basis vectors associated with one system are used to project out components of the metric and other fields, in analogy with the way fields are projected onto an orthonormal tetrad basis. These field components are then determined as functions of a second independent coordinate system. The transformation to the second coordinate system can be thought of as a mapping from the original inertial coordinate system to the computational domain. This dual-coordinate method is used to perform stable numerical evolutions of a black-hole spacetime using the generalized harmonic form of Einstein’s equations in coordinates that rotate with respect to the inertial frame at infinity; such evolutions are found to be generically unstable using a single rotating-coordinate frame. The dual-coordinate method is also used here to evolve binary black-hole spacetimes for several orbits. The great flexibility of this method allows comoving coordinates to be adjusted with a feedback control system that keeps the excision boundaries of the holes within their respective apparent horizons.

Figure 1

Constraint violations in evolutions of a Schwarzschild black hole using the generalized harmonic system and coordinates that rotate with angular velocity $\Omega$.

Figure 2

Constraint violations in dual-coordinate-frame evolutions of Schwarzschild, with comoving coordinates that rotate uniformly with angular velocity $\Omega =0.2/M$. Angular filtering (see text) is performed on the inertial-frame components.

Figure 3

Time dependence of the inertial-frame components of two cross sections (equatorial and polar) of a vector field expressed as functions of rotating coordinates. At $t=0$ the vector field is radial, but the ${\partial }_{\overline{x}}$ and ${\partial }_{\overline{y}}$ basis vectors rotate with angular velocity $\Omega$. The vectors on the right side of the polar cross section figure at $t=\pi /(2\Omega )$ are pointed into the plane of the figure, while those on the left point out.

Figure 4

Tensor-spherical-harmonic components of the Schwarzschild spatial three-metric expressed in rotating coordinates. Dashed curves show the components expressed in rotating-frame tensor spherical harmonics (only the $\ell =0$ component is nonzero); solid curves show the inertial-frame tensor-spherical-harmonic components.

Figure 5

Constraint violations in dual-coordinate-frame evolutions of Schwarzschild, with comoving coordinates that rotate uniformly with angular velocity $\Omega =0.2/M$. These evolutions use the new rotating-frame tensor-spherical-harmonic filtering algorithm. The highest resolution case (dashed curve) uses more angular basis functions ($\ell \le 13$ instead of $\ell \le 11$) as a more stringent test of convergence.

Figure 6

Evolution of the apparent horizon center, $(x_c,y_c)$, of one of the black holes relative to a uniformly rotating-coordinate frame. The minimum coordinate distance between the apparent horizon and the excision boundary becomes less than 0.003 at $t=45.6M_{\mathrm{ADM}}$, and the evolution is terminated.

Figure 7

Moving coordinate frame representation of an equatorial slice through the apparent horizon of one of the black holes for the evolution of Fig. 6. The evolution is terminated when the apparent horizon approaches the excision boundary (dotted curve) near the right side of this figure.

Figure 8

Evolution of the apparent horizon center of one of the holes for a binary black-hole evolution using a feedback control system that adjusts the moving coordinate frame to control the parameter $[x_c-x_c(0)]/x_c(0)$. Compare to Fig. 6.

Figure 9

Moving coordinate frame representation of an equatorial slice through the apparent horizon of one of the black holes for the evolution of Fig. 8. Evolution is terminated when the apparent horizon approaches the excision boundary near the bottom of this figure.

Figure 10

Evolution of the apparent horizon center of one black hole for a binary black-hole evolution using a feedback system that adjusts the moving coordinate frame to control both the parameters $y_c/x_c$ and $[x_c-x_c(0)]/x_c(0)$. Compare to Figs. 6, 8.

Figure 11

Motion of the inertial-frame coordinates of the center of the apparent horizon $({\overline{x}}_c,{\overline{y}}_c)$ (solid curve) of one of the black holes from the evolution of Fig. 10. The dotted curve shows the equatorial cross section of the apparent horizon at $t=661.4M_{\mathrm{ADM}}$. The binary system has completed about 4.6 orbits.

Figure 12

Constraint violations $\Vert \mathcal{C}\Vert$ for the binary black-hole evolution of Fig. 10. The two curves represent two numerical resolutions, the upper curve has about ${64}^3$ collocation points, while the lower curve has about ${76}^3$ collocation points.

Figure 13

Equatorial cross sections, in comoving coordinates, of the apparent horizon of one hole for the evolution of Fig. 10. The horizon is shown both at the beginning and at the end of the evolution. The smaller dotted circle (just inside the initial horizon) shows the excision boundary used for $t\le 554.1M_{\mathrm{ADM}}$. The outer dotted circle shows the excision boundary used for $t>554.1M_{\mathrm{ADM}}$.