Comparing an analytical spacetime metric for a merging binary to a fully nonlinear numerical evolution using curvature scalars

We introduce a new geometrically invariant prescription for comparing two different spacetimes based on geodesic deviation. We use this method to compare a family of recently introduced analytical spacetime representing inspiraling black-hole binaries to fully nonlinear numerical solutions to the Einstein equations. Our method can be used to improve analytical spacetime models by providing a local measure of the effects that violations of the Einstein equations will have on timelike geodesics, and indirectly, gas dynamics. We also discuss the advantages and limitations of this method.

Figure 1

The zones for the analytic metric. The large (green) circle is the outer boundary of the near zone. Immediately inside this circle the metric is exclusively the post-Newtonian near-zone metric, while outside, it is a superposition of the near and far zone metrics. All points in the figure outside this circle are in the near-far buffer zone (the other boundary of this zone is not show). The smaller (cyan) circles denote the inner boundary of the near zone. Inside the envelope of these circles is the near-inner buffer zones. The box (orange) denotes the region inside the near-inner buffer zone where the metric is a superposition of both BH1 and BH2 inner zones, as well as the near zone. Outside the box, the metric is a superposition of the near zone metric and either one of the inner zone metrics. Finally inside the very small (magenta) circles are the two inner zones, where the metric is purely the inner zone perturbed Schwarzschild metric.

Figure 2

Circular geodesics in standard Schwarzschild and transformed Schwarzschild coordinates. While the trajectory is gauge dependent (top), the associated curvature eigenvalues (only one shown) are not (bottom). The differences between the eigenvalues ($\mathbf{Sc}$) calculated in to the two gauges are consistent with roundoff errors.

Figure 3

The differences between one of curvature eigenvalues ($\mathbf{Sc}$) versus time from our new geodesic thorn and the exact values (as determined by a stand-alone code). Here, we denote the resolution of a given simulation by the number of gridpoint, per dimension, from the origin to the outer boundary, and rescale the differences by the ratio of the grid resolution to the fourth power.

Figure 4

The difference between one of the gauge independent curvature eigenvalue ($\mathbf{Sc}$) as calculated using a fully nonlinear numerical evolution of time independent trumpet Schwarzschild data using the EinsteinToolkit, and as calculated using the exact trumpet Schwarzschild metric with the trumpet parameter $R_0=M$. Here, we rescale the differences by the ratio of the grid resolution to the fourth power.

Figure 5

The convergence of the one of the gauge independent curvature eigenvalues ($\mathbf{Sc}$) for a slowly time-dependent Schwarzschild trumpet. The convergence order is still fourth-order.

Figure 6

Second-order convergence of the gauge independent curvature eigenvalues ($\mathbf{Sc}$) as calculated using a fully nonlinear numerical evolution of Kerr data using the EinsteinToolkit, and as calculated using the exact Kerr metric in quasi-isotropic coordinates.

Figure 7

The L2 norm of the constraints for the $D=25M$ configuration. Here the constraints are calculated within the volume outside the two horizons and inside the coordinate sphere $r=30M$.

Figure 8

Separation $D=50M$ results. Here we plot the value of the largest (in magnitude) curvature eigenvalue ($\mathbf{Sc}$) versus time (as evolved using the numerical and analytic metric), as well as plot the coordinate position of the geodesics in a corotating frame (i.e., one where the BH positions are nearly fixed) on the right side of each panel. The vertical lines and circles in these trajectory plots show the location of the various zones described in Sec. 2a. The number $r_0$ (normalized by $M$) is the initial coordinate distance of the geodesic from the nearest BH. For the geodesics close to the BHs, the noise in the numerically evolved spacetime is low compared to the magnitude of the curvature eigenvalues, the opposite is true for the farthest ones.

Figure 9

Separation $D=25M$ and $D=20M$ results. Here we plot the value of the largest (in magnitude) curvature eigenvalue ($\mathbf{Sc}$) versus time (as evolved using the numerical and analytic metric), as well as plot the coordinate position of the geodesics in a corotating frame (i.e., one where the BH positions are nearly fixed) on the right side of each panel. The vertical lines and circles in these trajectory plots show the location of the various zones described in Sec. 2a. The number $r_0$ (normalized by $M$) is the initial coordinate distance of the geodesic from the nearest BH. For the geodesics close to the BHs, the noise in the numerically evolved spacetime is low compared to the magnitude of the curvature eigenvalues, the opposite is true for the farthest ones.

Figure 10

Separation $D=15M$ and $D=10M$ results. Here we plot the value of the largest (in magnitude) curvature eigenvalue ($\mathbf{Sc}$) versus time (as evolved using the numerical and analytic metric), as well as plot the coordinate position of the geodesics in a corotating frame (i.e., one where the BH positions are nearly fixed) on the right side of each panel. The vertical lines and circles in these trajectory plots show the location of the various zones described in Sec. 2a. The number $r_0$ (normalized by $M$) is the initial coordinate distance of the geodesic from the nearest BH. For the geodesics close to the BHs, the noise in the numerically evolved spacetime is low compared to the magnitude of the curvature eigenvalues, the opposite is true for the farthest ones.

Figure 11

(Top two rows) A summary of the results. The plots show the trajectories of the geodesics in a non-corotating frame. The color scale gives the relative differences between the curvature eigenvalues ($\mathbf{Sc}$) as calculated using the numerical (and smoothed by a running average) and analytic metrics. Note that the color changes from blues to reds at 10% relative difference. (Bottom two rows) Plots showing curvature eigenvalues as calculated using the analytical and numerical metrics, as well as a running average of the latter. There are hints here of systematic differences between the analytical and numerical results. However, as can be seen, the noise is much larger than these differences.

Figure 12

Curvature eigenvalues ($\mathbf{Sc}$) for numerical and analytical spacetimes. For the analytical spacetime, we perturb the initial velocity of geodesics by the factors shown in the graphs. The dotted blue curve is the numerical result with velocity associated with the unperturbed analytic geodesic. As can be seen, the larger the value of the eigenvalues (i.e., geodesic deviation) the larger the effect of a $\pm 10\%$ perturbation. On the other hand, with small perturbations, we were able to find geodesics in the analytical spacetime that closely matched the dynamics (time dependence the eigenvalue) of the numerical one for geodesics farther than $r_0\sim 10M$ from the BHs.

Figure 13

A comparison of how well the curvature eigenvalues of the second-order metric and first-order metric agree with the eigenvalues of the associated numerical metrics for the $D=50M$ case. The top row shows the second-order results (which were previously shown in Fig. 8). The bottom row shows the first-order results. Note that at larger distances from the black holes the two results are comparable, while at closer distances the second-order results are qualitatively better.

Figure 14

A comparison of how well the curvature eigenvalues of the second-order metric and first-order metric agree with the eigenvalues of the associated numerical metrics for the $D=20M$ case. The top row shows the second-order results (which were previously shown in Fig. 9). The bottom row shows the first-order results. Unlike for the $D=50M$ case, there are significant differences between the analytical and numerical scalars for the first-order metric even at larger distances from the black holes.

Figure 15

A comparison of how well the curvature eigenvalues of the second-order metric and first-order metric agree with the eigenvalues of the associated numerical metrics for the $D=15M$ case. The top row shows the second-order results (which were previously shown in Fig. 10). The bottom row shows the first-order results. As with the $D=20M$ case, there are significant differences between the analytical and numerical scalars for the first-order metric even at larger distances from the black holes.

Figure 16

A comparison of how well the curvature eigenvalues of the second-order metric and first-order metric agree with the eigenvalues of the associated numerical metrics for the $D=10M$ case. The top row shows the second-order results (which were previously shown in Fig. 10). The bottom row shows the first-order results. Here, we do not see a significant improvement of the second-order metric over the first-order one.