Asymptotic frame selection for binary black hole spacetimes: Post-Newtonian limit

One way to select a preferred frame from gravitational radiation is via the principal axes of $⟨{\mathcal{L}}_{(a}{\mathcal{L}}_{b)}⟩$, an average of the action of rotation group generators on the Weyl tensor at asymptotic infinity. In this paper we evaluate this time-domain average for a quasicircular binary using approximate (post-Newtonian) waveforms. For nonprecessing unequal-mass binaries, we show the dominant eigenvector of this tensor lies along the orbital angular momentum. For precessing binaries, this frame is not generally aligned with either the orbital or total angular momentum, working to leading order in the spins. The difference between these two quantities grows with time, as the binary approaches the end of the inspiral and both precession and higher harmonics become more significant.

Figure 1

“Comparing trajectories 1: BH-NS binary”: These panels show the trajectories of ${\widehat{L}}_N$ (black, second from bottom), $\widehat{L}$ (green, bottom curve), $\widehat{J}$ (cyan, top curve), $\widehat{V}$ the principal eigendirection of $⟨{\mathcal{L}}_{(a}{\mathcal{L}}_{b)}⟩$ (blue, second from top) and ${\widehat{V}}_h$ the principal eigendirection of $\{{\mathcal{L}}_{(a}{\mathcal{L}}_{b)}\}$ (red, third from top) for a $10+1.4M_{\odot }$ BH-NS binary where the black hole has maximal spin tilted 30° relative to the Newtonian orbital angular momentum at an initial frequency of 40 Hz. The frame is chosen so that the $z$-axis coincides with the initial value of $\widehat{J}$. The top panel shows the first precessional cycle, which spans a frequency range of roughly 40–47 Hz. The bottom panel shows a very late precessional cycle shortly before merger would occur, covering a frequency band of 200–600 Hz.

Figure 2

“Comparing trajectories 2: BH-BH binary”: This plot is the same as Fig. 1, but for a $60+30M_{\odot }$ BH-BH binary with both spins maximal, perpendicular to the Newtonian orbital angular momentum and perpendicular to each other at an initial frequency of 10 Hz. This plot shows the entire inspiral of the binary, reaching a GW frequency of about 85 Hz at the end.

Figure 3

Comparing eigenvalues: The fractional deviation $(\lambda /\overline{\lambda }-1)$ in the dominant eigenvalues (where $\overline{\lambda }$ is the unperturbed eigenvalue and $\lambda$ is the actual eigenvalue) of $⟨{\mathcal{L}}_{(a}{\mathcal{L}}_{b)}{⟩}_l$ (top panels) and $\{{\mathcal{L}}_{(a}{\mathcal{L}}_{b)}{\}}_l$ (bottom panels), for the BH-NS binary orbit described in Fig. 1. The blue curve indicates $l=2$; the red dashed curve shows $l=3$. For reference, a black solid line is shown at 0. Left panels: Eigenvalues of $⟨{\mathcal{L}}_{(a}{\mathcal{L}}_{b)}⟩$ (top) and $\{{\mathcal{L}}_{(a}{\mathcal{L}}_{b)}\}$ (bottom) for the full evolution shown. Right panels: As above, but limited to a short interval immediately preceding the last simulated post-Newtonian orbit, where the minimum-energy condition breaks down.

Figure 4

“Orientations of subspace eigenvectors”: Top panel: A plot of $1-{\widehat{V}}_2·\widehat{V}$ (solid blue) and $1-{\widehat{V}}_{h2}·{\widehat{V}}_h$ (dashed red) versus time for the BH-BH binary orbit described in Fig. 2. Bottom panel: A plot of $1-{\widehat{V}}_3·{\widehat{L}}_N$ (solid blue) and $1-{\widehat{V}}_{h3}·{\widehat{L}}_N$ (dashed red) versus time for the same BH-BH binary. The eigenvectors from the $l=2$ ($l=3$) subspaces are rather close to the eigenvector of the full rotation tensor (${\widehat{L}}_N$).