Approximate black hole binary spacetime via asymptotic matching

We construct a fully analytic, general relativistic, nonspinning black hole binary spacetime that approximately solves the vacuum Einstein equations everywhere in space and time for black holes sufficiently well separated. The metric is constructed by asymptotically matching perturbed Schwarzschild metrics near each black hole to a two-body post-Newtonian metric far from them and a two-body post-Minkowskian metric farther still. Asymptotic matching is done without linearizing about a particular time slice, and thus it is valid dynamically and for all times, provided the binary is sufficiently well separated. This approximate global metric can be used for long dynamical evolutions of relativistic magnetohydrodynamical, circumbinary disks around inspiraling supermassive black holes to study a variety of phenomena.

Figure 1

L2 norm of the Ricci scalar as a function of time computed with metrics of two different perturbation orders (see Sec. 2). The norms are computed using a physical domain extent of $[0,-80,-0.15]$ to [159.95, 79.95, 0.1] in units of total mass $M$. The black hole horizons are excised and a mesh spacing of $0.05M$ is used. The black, dotted, vertical lines indicate instants of time when the binary separations reach $14M$, $12M$, $10M$ and $8M$, from left to right, respectively. The evolution is initialized at $20M$.

Figure 2

Schematic diagram of the spacetime on a spatial hypersurface. BH1 and BH2 are denoted by filled, solid black circles, where the orbital separation is $r_{12}$. The BZs are denoted with cyan shells, the outer one representing the FZ-NZ BZ and the two inner ones representing the NZ-IZ BZs (see also Table 1). The circular nature of these BZs is only schematic; in practice, the BZs should be distorted. The IZ, NZ and FZ are also shown in this figure.

Figure 3

Left: Comparison of the volume element, $\sqrt{-g}$, for the several different metric pieces composing the global, second-order matched metric, viz., the IZ, the NZ and the FZ metrics. Right: Absolute value of Ricci scalar computed with these metric pieces. The vertical dotted lines mark the boundaries of each zone.

Figure 4

Left: Convergence factor $Q^h$ computed using three different grid resolutions: $h=0.05M$, $h=0.025M$ and $h=0.0125M$, respectively, corresponding to 10, 20 and 40 points across the horizon radius. Observe that this factor is almost everywhere close to 16, which corresponds to fourth-order convergence. The magenta regions in this panel show areas that converge faster that fourth-order, while red regions show areas that converge slower, possibly due to zero-crossings of the solution error function. Right: L2 norm of the Ricci scalar difference for different levels of refinement as a function of time. The L2 norm is calculated over a sector of the simulation domain that ranges from $[-12,-12,-0.75]$ to [12, 12, 0.75] with uniform resolutions following a $2:1$ Cartesian grid spacing ratio, where the coarsest grid has a mesh spacing of $0.05M$.

Figure 5

Absolute value of the Ricci scalar along the $x$ axis (left) using the first and second-order metrics at a separation of $20M$ and (right) using the second-order metric at different separations. The mesh spacing is $dx=0.0125M$, roughly 40 grid points along the horizon radius (unless indicated otherwise, the same resolution is used in all other figures). The thick purple vertical lines show the location of the event horizon, while the black ones show the boundary of the different zones. Observe how smoothly the violations to the Einstein equations increase as the orbital separation shrinks.

Figure 6

Comparison of the absolute value of the Ricci scalar calculated with the first order (left) and the second order (right) metrics at $t=10756.60M$, corresponding to an orbital separation of $14.13M$, when the initial separation was $20M$. The color scale is logarithmic and fixed to emphasize the regions that saturate the interval $[{10}^{-7},{10}^{-1}]$. The concentric dotted lines centered around the black hole and the origin indicate where the IZ-NZ BZ and the NZ-FZ BZ start and end respectively.

Figure 7

Absolute value of the Ricci scalar calculated with the first-order (left) and the second-order (right) global metric at an initial separation of $20M$ (top) and after $10756.60M$ of evolution, when the separation is $14.13M$ (bottom). The color scale is logarithmic and fixed to emphasize the regions that saturates the interval $[{10}^{-7},{10}^{-1}]$. The concentric dotted and dashed lines around each black hole indicate where the IZ-NZ BZ starts and where the ISCO would be located for an individual black hole, respectively.

Figure 8

Left: L2 norm of the Ricci scalar along the $x$ axis computed with the second-order metric for different choices of $s$ in the NZ-FZ BZ transition function. Observe that an appropriate choice of transition function parameter leads to a smaller “bump” in the Ricci scalar in the NZ-FZ BZ. Right: Behavior of $f_{\text{far}}=f(r,\lambda /5,\lambda ,1,s)$, $d{f}_{\text{far}}/dr$ and $|d^2{f}_{\text{far}}/d{r}^2|$ for various values of $s$. Observe that the choice $s=1.4$ leads to a better behaved transition.