Inspiralling, nonprecessing, spinning black hole binary spacetime via asymptotic matching

We construct a new global, fully analytic, approximate spacetime which accurately describes the dynamics of nonprecessing, spinning black hole binaries during the inspiral phase of the relativistic merger process. This approximate solution of the vacuum Einstein’s equations can be obtained by asymptotically matching perturbed Kerr solutions near the two black holes to a post-Newtonian metric valid far from the two black holes. This metric is then matched to a post-Minkowskian metric even farther out in the wave zone. The procedure of asymptotic matching is generalized to be valid on all spatial hypersurfaces, instead of a small group of initial hypersurfaces discussed in previous works. This metric is well suited for long term dynamical simulations of spinning black hole binary spacetimes prior to merger, such as studies of circumbinary gas accretion which requires hundreds of binary orbits.

Figure 1

A schematic diagram detailing the different zones for the approximate analytic spacetime looking down the $z$-axis at a particular instant in time. The black dots on the $x$-axis represent the two BHs, with an orbital separation of $r_{12}$. The cyan shells indicate the BZs—regions where two adjacent metrics have overlapping regions of validity. The outermost shell is the NZ-FZ BZ, with both the IZ1-NZ and IZ2-NZ BZs labeled around the individual BHs. The IZ, NZ, and FZ are denoted as the regions contained by the BZs. Note that the circular nature of the BZs is not physical, only schematic, and in general it is expected that they will have some distortions. (This was also presented in Refs. [33, 56, 57].)

Figure 2

Comparison of the volume element, $\sqrt{-g}$, for the global metric for differing values of the spin parameter $\chi$, where both black holes have spin aligned with the orbital angular momentum of the binary.

Figure 3

The absolute value of the Ricci scalar along the $x$ coordinate, at an initial separation of $20M$. We compare here the violations of the Ricci scalar for the spinning BHB metric against the nonspinning (NS) BHB first and second order metrics [33]. It is interesting to note that the new spinning BHB metric has violations of the same order of magnitude as the previous nonspinning BHB metric. In this plot and the following plots, the horizon is denoted by the grey dashed vertical line, discussed in more detail in Fig. 5. The ISCO is the orange dashed vertical line (see Appendix pp3 for details). The green dash-dotted vertical lines indicate the boundaries of the different zones, and are consistent for all of the subsequent plots. It is good to note here that the zone boundaries do change for differing spins, though not by much, so here we picked the fiducial zone boundary for the ${\chi }_i=0.9$ (highly spinning) and aligned case.

Figure 4

The absolute value of the Ricci scalar along $x$, at an initial separation of $20M$. The spin parameter is varied in this plot. We see that there is little qualitative variation in changing ${\chi }_i$.

Figure 5

The absolute value of the Ricci scalar along $x$, for aligned spins, zoomed in on the inner region near the horizon. The horizon is denoted by the grey dashed vertical line, roughly at a position of $x_h=x_{\mathrm{BH},1}\pm M/2$ on the $x$-axis, and is easier to distinguish here than in the previous plots. The ISCO is the orange dashed vertical line (see Appendix pp3 for details). The inset shows the behavior close to the horizon.

Figure 6

The absolute value of the Ricci scalar along $x$, at an initial separation of $20M$. This plot shows the spin parameter ${\chi }_i$ varied for antialigned spins.

Figure 7

The absolute value of the Ricci scalar along $x$, zoomed in near the BH to show the violation near the horizon. This plot shows the spin parameter ${\chi }_i$ varied for antialigned spins. The inset shows the behavior close to the horizon.

Figure 8

The absolute value of the Ricci scalar along $x$, at an initial separation of $20M$, $12M$, and $8M$, for spins ${\chi }_1={\chi }_2=0.0$ (top) and ${\chi }_1={\chi }_2=0.9$ aligned (bottom). Note that the violation increases smoothly as we decrease in orbital separation. This gives a good indication that the dynamics is not introducing any spurious error into the metric.

Figure 9

The Kretschmann invariant calculated for the BHB spacetime for differing values of spin, all of which are aligned with the orbital angular momentum of the binary, at an initial separation of $r_{12}=20M$. The $1/r^6$ behavior seen far from the two BHs ($\sim 40M$) is consistent with the value of the Kretschmann in the single Schwarzschild BH case: $K=48M^2/r^6$, where $M$ is the total mass of the binary centered on the origin. The inset shows the behavior close to the horizon, where the spin effects become noticeable. The Kretschmann becomes large as it approaches a BH, because the invariant blows up at a true singularity.

Figure 10

The relative Kretschmann for an initial separation of $r_{12}=20M$, with a grid resolution of 0.0125. In this plot, we are plotting aligned spinning BHs, and increasing the dimensionless spin parameter ${\chi }_i$ from nonspinning to highly spinning. Observe the normalized behavior that these plots exhibit. The violation to this invariant is very good close to the horizon due to the way that we are normalizing. See Fig. 9 for the normalization function.

Figure 11

Zoomed-in view of Fig. 10 around the IZ. The inset shows the behavior close to the horizon.

Figure 12

The relative Kretschmann for an initial separation of $r_{12}=20M$, with a grid resolution of 0.0125 for varying values of the dimensionless spin parameter ${\chi }_i$. In this plot, we are looking at what the relative Kretschmann does for antialigned BHs with high ${\chi }_i$ values. Observe the normalized behavior that these plots exhibit. The violation to this invariant is very good close to the horizon, due to the way that we are normalizing. See Fig. 9 for the normalization function.

Figure 13

Zoomed-in view of Fig. 12 around the IZ. The inset shows the behavior close to the horizon.

Figure 14

Convergence factor along $x$. Each of the panels show the Ricci scalar convergence factor $Q$ at $t=0$ along the coordinate $x$-axis for different intervals in $x$, depending on the set of mesh spacing used in the convergence factor computation. On all panels the red dotted line represents the convergence factor for the low resolution mesh spacing set used in that panel, while the blue dashed and black solid lines represent the medium and high resolution mesh spacing set, respectively. The green horizontal dashed line at $Q=16.0$ is the theoretical solution at infinite resolution $h=0$. On the top panel, the low resolution mesh set, $h_L/M=\{0.1,0.05,0.025\}$, is used to compute $Q_L$, while the medium and high resolution ones are $h_M/M=\{0.05,0.025,0.0125\}$ and $h_H/M=\{0.025,0.0125,0.00625\}$, respectively. We masked out $Q_H$ outside the interval $[7M,13M]$ to avoid noise due to round-off precision errors. On the second panel from above, the low resolution set is $h_L/M=\{0.8,0.4,0.2\}$, while the medium and high ones are $h_M/M=\{0.4,0.2,0.1\}$ and $h_H/M=\{0.2,0.1,0.05\}$, respectively. We also mask out $Q_H$ and $Q_M$ for $x>25M$ and $x>35M$, respectively, to avoid round-off noise. On the third panel, we use $h_L/M=\{6.4,3.2,1.6\}$, $h_M/M=\{3.2,1.6,0.8\}$ and $h_L/M=\{1.6,0.8,0.4\}$. We mask out $Q_H$ and $Q_M$ for $x>90M$ and $x>140M$, respectively. Finally on the bottom panel, we use $h_L/M=\{51.2,25.6,12.8\}$, $h_M/M=\{25.6,12.8,6.4\}$ and $h_L/M=\{12.8,6.4,3.2\}$. These different sets of resolutions were used here to emphasize the mesh spacings required to be in the convergence regime for different zones.

Figure 15

The top panel is the orbital hang-up effect shown by plotting the individual trajectories of BH1. Note that the orbits get bunched up as the spin is increased from ${\chi }_1={\chi }_2=0.0$ to 0.9. Both of the BH spins are aligned with the orbital angular momentum of the binary. The bottom panel shows the orbital hang-up effect as a decrease in the orbital separation as a function of time for varying aligned spin parameters, with the fiducial spin values chosen to coincide with the spins used in the Ricci and relative Kretschmann analyzes. For both the top and bottom panels, the evolution was terminated at $14000M$ in time.