Evolution of Binary Black Hole Spacetimes

TL;DR

Early success is described in the evolution of binary black-hole spacetimes with a numerical code based on a generalization of harmonic coordinates capable of evolving binary systems for enough time to extract information about the orbit, merger, and gravitational waves emitted during the event.

Abstract

We describe early success in the evolution of binary black hole spacetimes with a numerical code based on a generalization of harmonic coordinates. Indications are that with sufficient resolution this scheme is capable of evolving binary systems for enough time to extract information about the orbit, merger and gravitational waves emitted during the event. As an example we show results from the evolution of a binary composed of two equal mass, non-spinning black holes, through a single plunge-orbit, merger and ring down. The resultant black hole is estimated to be a Kerr black hole with angular momentum parameter a~0.70. At present, lack of resolution far from the binary prevents an accurate estimate of the energy emitted, though a rough calculation suggests on the order of 5% of the initial rest mass of the system is radiated as gravitational waves during the final orbit and ringdown.

Figures (3)

• Figure 1: A depiction of the orbit for the simulation described in the text (see also Table \ref{['tab_id']}). The figure shows the coordinate position of the center of one apparent horizon relative to the other, in the orbital plane $z=0$. The units have been scaled to the mass $M_0$ of a single black hole, and curves are shown from simulations with three different resolutions. Overlaid on this figure are reference ellipses of eccentricity $0$, $0.1$ and $0.2$, suggesting that if one were to attribute an initial eccentricity to the orbit it could be in the range $0-0.2$.
• Figure 2: The plot to the left shows the net black hole mass of the spacetime in units of the mass $M_0$ of a single initial black hole, calculated from apparent horizon (AH) properties (using (\ref{['smarr']}) with the dynamical horizon estimate for $J$), and from simulations with three different resolutions. The initial sharp increase in mass is due to scalar field accretion, the small "wiggle" at around $20 M_0$ appears to be a gauge effect, and the "jaggedness" around the time of the merger is due to robustness problems in the AH finder that manifest when the AH shapes are highly distorted. To the right the Kerr parameter $a$ of the final black hole is shown (for clarity we only plot the results from a single simulation), calculated using the ratio $C_r$ of polar to equatorial proper circumference of the AH and applying (\ref{['a_cr']}), and using the dynamical horizon framework (curve labeled DH). The loss of mass (and similarly increase in $a$) with time after the merger is due to accumulating numerical error.
• Figure 3: A sample of the gravitational waves emitted during the merger, as estimated by the Newman-Penrose scalar $\Psi_4$ (from the medium resolution simulation). Here, the real component of $\Psi_4$ multiplied by the coordinate distance $r$ from the center of the grid is shown at a fixed angular location, though several distances $r$. The waveform has also been shifted in time by amounts shown in the plot, so that the oscillations overlap. If the waves are measured far enough from the central black hole then the amplitudes should match, and they should be shifted by the light travel time between the locations (i.e. by $25 M_0$ in this example). That we need to shift the waveforms by more than this suggests the extraction points are still too close to the black hole; the decrease in amplitude is primarily due to numerical error as the wave moves into regions of the grid with relatively low resolution.