Superposed metric for spinning black hole binaries approaching merger

We construct an approximate metric that represents the spacetime of spinning binary black holes (BBH) approaching merger. We build the metric as an analytical superposition of two Kerr metrics in harmonic coordinates, where we transform each black hole term with time-dependent boosts describing an inspiral trajectory. The velocities and trajectories of the boost are obtained by solving the post-Newtonian (PN) equations of motion at 3.5 PN order. We analyze the spacetime scalars of the new metric and we show that it is an accurate approximation of Einstein’s field equations in vacuum for a BBH system in the inspiral regime. Furthermore, to prove the effectiveness of our approach, we test the metric in the context of a 3D general relativistic magnetohydrodynamical (GRMHD) simulation of accreting minidisks around the black holes. We compare our results with a previous well-tested spacetime construction based on the asymptotic matching method. We conclude that our new spacetime is well-suited for long-term GRMHD simulations of spinning binary black holes on their way to the merger.

Figure 1

Diagram of the BH rest frame, ${\mathcal{O}}^{\prime }$, which is noninertial, and the global frame, $\mathcal{O}$. The coordinates $\{X^a\}$ and $\{x^a\}$ both describe the event $e$.

Figure 2

Determinant of superposed metric for different values of the spin and separation of $r_{12}=20M$. Note that the curves for different spins are very similar. For comparison, we include the determinant of the matching metric for $\chi =0$.

Figure 3

Absolute value of several metric components for the superposed and matching metric in the fiducial configuration. Note that the superposed metric components are much smoother than the matching metric because there are no transition regions.

Figure 4

Ricci scalar for the matching and superposed metric with $r_{12}=20M$, equal mass-ratio, and different values of spin. The dashed green lines denote the location of the horizon and the solid green lines the location of the ISCO for a nonspinning BH.

Figure 5

Equatorial logarithmic plot of the Ricci scalar for the SHPN metric with $r_{12}=20M$, equal masses, and $\chi =0.9$

Figure 6

Ricci violations of the SHPN metric, for $r_{12}=20M$, $\chi =0.5$, and different mass-ratio values.

Figure 7

Fake mass introduced in the spacetime by the SHPN and matching metric approximations for different orbital distances.

Figure 8

The Ricci scalar ($R$), the Hamiltonian ($\mathcal{H}$) and momentum ($\mathcal{M}$) constraint, and the Kretschmann scalar $\mathcal{K}$ of the SHPN metric, for a BBH with $r_{12}=20M$, equal mass, and $\chi =0.9$. We can notice that the Kretschmann scalar follows the decay $\sim 1/r^6$ typical for a single BH. The solid green lines denotes the location of the ISCO for a nonspinning BH.

Figure 9

Equatorial plot of the rest-mass density $\rho$ in logarithmic scale for (right) the superposed metric and (left) the matching metric, at $T=1810M$, which represents approximately $\sim 3$ orbits.

Figure 10

Mass in each BH’s minidisk region for the superposed metric simulation (thick lines) and the matching metric simulation (dashed lines).

Figure 11

Magnetic energy in each BH’s minidisk region for the superposed metric simulation (thick lines) and the matching metric simulation (dashed lines).

Figure 12

Left: ergosphere region of Kerr BH with spin $\chi =0.9$ for Kerr-Schild coordinates (blue) and harmonic coordinates (orange). Note that the surfaces in harmonic coordinates are more oblique compared with the Kerr-Schild coordinates. The radius of the singularity (green) is the same for both coordinate systems. Right: ergosphere regions for a x-boosted harmonic Kerr BH with spin $\chi =0.9$ for different velocities ($v/c=0$, 0.1, 0.5, 0.9). The horizon (red) and singularity (green) are the same in each case, but the ergosphere region increases with increasing velocity.

Figure 13

Convergence factor of our numerical scheme for different regions. In the top panel we use $h/M=0.0125$, in the middle panel $h/M=0.1$, and in the bottom panel $h/M=0.8$.