Conformal thin-sandwich solver for generic initial data

We present a new scheme for constructing initial data for the Einstein field equations using the conformal thin-sandwich formulation that does not assume conformal flatness or approximate Killing vectors. This includes a method for determining free data based on superposition, as well as a way to handle black hole singularities without excision. We numerically solve the constraint equations using a multigrid algorithm with mesh refinement. We demonstrate the efficacy of the method with initial data solutions for several applications: a quasicircular binary black hole merger, a dynamical capture black hole-neutron star merger, and an ultrarelativistic collision.

Figure 1

The conformal factor $\Psi$ from BH-BH ID. Upper: $\Psi$ on the $x$ axis, which lies on the orbital plane and goes through the centers of the black holes. Lower: Differences in $\Psi$ with resolution on the $x$ axis, scaled assuming second-order convergence.

Figure 2

Differences in the shift component ${\beta }^y$ with resolution on the $x$ axis from BH-BH ID, scaled assuming second-order convergence.

Figure 3

Coordinate separation of the centers of the two black holes fitted to a function $(A-B(t-t_0){)}^{1/4}+C\cos (\omega (t-t_0)+\varphi )$. This function combines the decaying orbit attributable to quadrupole radiation with the effects of eccentricity, given by $e=C/d(t=t_0)\approx 0.05$. Because of early-time gauge effects (a transition from harmonic to damped harmonic gauge) we exclude the first $t=40M_{\mathrm{irr}}$ from the fit.

Figure 4

The real and imaginary components of the $l=2$, $m=2$ spin-weight $-2$ spherical harmonic of $r{\Psi }_4$ extracted at a radius of $105M$. Time is measured from the beginning of the simulation.

Figure 5

The $L^2$ norm of the constraint violation, $C_a\equiv H_a-\square x_a$, in units of $1/M$ for the $d=15M$ BH-NS merger in the $100M\times 100M$ region around the center of mass in the equatorial plane (i.e., $\sqrt{\int \Vert C_a{\Vert }^2{d}^{2}x/\int d^{2}x}$). This is shown for low, medium, and high resolutions where the latter two are scaled assuming second-order convergence.

Figure 6

The log of the magnitude of the $l=2$, $m=2$ spin-weight $-2$ spherical harmonic of $r{\Psi }_4$ for BH-NS simulations with different initial separations $d$. The value of ${\Psi }_4$ was extracted on a sphere of radius $100M$ and is shown starting at the beginning of the simulation and continuing past merger. The waveforms have been aligned so that the peaks occurs at time 0.

Figure 7

Various quantities from ultrarelativistic collision ID with $\gamma =10$ made using the superposition method for constructing free data. From top to bottom the quantities shown are the maximum (over the entire domain) difference of the conformal factor from unity, the maximum coordinate velocity of the fluid, the total rest mass, and the ADM mass. All quantities are shown as a function of $d$, the coordinate separation between the two stars. For all these cases the maximum of $|\Psi -1|$ occurs for values of $\Psi$ that are less than unity. One might expect these quantities to approach their infinite separation limits as $1/d$ for large $d$; the dotted lines show such $1/d$ curves for each quantity matched to the $d/M=50$ point.

Figure 8

Same as Fig. 7 but with conformally flat data. For all these cases the maximum of $|\Psi -1|$ occurs for values of $\Psi$ that are greater than unity.