Black hole spin axis in numerical relativity

Colliding black holes are systems of profound interest in both gravitational-wave astronomy and in gravitation theory, and a variety of methods have been developed for modeling their dynamics in detail. The features of these dynamics are determined by the masses of the holes and by the magnitudes and axes of their spins. While masses and spin magnitudes can be defined in reasonably unambiguous ways, the spin axis is a concept that, despite great physical importance, is seriously undermined by the coordinate freedom of general relativity. Despite a great wealth of detailed numerical simulations of generic spinning black hole collisions, very little attention has gone into defining or justifying the definitions of the spin axis used in the numerical relativity literature. In this paper, we summarize and contrast the various spin direction measures available in the spec code, including a comparison with a method common in other codes, we explain why these measures have shown qualitatively different nutation features than one would expect from post-Newtonian theory, and we derive and implement new measures that give much better agreement.

Figure 1

Angle, in degrees, between spin-axis measures computed in two highest-resolution simulations of the merger described in this section, for five different such measures. The top two curves show estimated error in measures based on the axis between extrema of $\mathrm{\Omega }$ or $\zeta$. The lower curves show errors in spin-axis measures based on the coordinate rotation vectors ${\overrightarrow{\varphi }}_i$ defined in Eq. (22) (blue, smoother) and those inferred by coordinate moments of $\mathrm{\Omega }$ and $\zeta$ (orange and red, respectively, and mostly overlapping on this scale).

Figure 2

Parametric plot of $x$ and $y$ components of normalized spin-axis vector computed from coordinate rotation generators defined in Eq. (22), for our two highest-resolution simulations, with the inset showing that the tracks themselves follow one another more closely than the individual data points at equal times. Analogous graphs are qualitatively similar for all of our spin-axis measures.

Figure 3

Parametric plot of $x$ and $z$ components of normalized spin-axis vector computed from coordinate rotation generators defined in Eq. (22), for our two highest-resolution simulations, with the inset showing that the tracks themselves follow one another more closely than the individual data points at equal times. Analogous graphs are qualitatively similar for all of our spin-axis measures.

Figure 4

An optimistic representation of the numerical error in the integral-based axis measures. Each time step of the highest-resolution run is compared to the best of all time-delayed counterparts in the second-highest resolution.

Figure 5

Angle (in the Euclidean background space) between spin axes defined using coordinate moments and using a line between extrema. The data for the horizon vorticity $B_{ss}$ (green curve) and the closely related curvature $\mathrm{\Omega }$ of the normal bundle (red curve) overlap so precisely that only one can be seen in this figure. For all three scalars, agreement is to within about a degree throughout the inspiral. For reference, the orbital period in the early inspiral is approximately $400m$, so the oscillations in these curves are on the timescale that one would expect from the rotation of a tidal bulge.

Figure 6

Angle (in the Euclidean background space) between spin axes defined using $B_{ss}$ and $\mathrm{\Omega }$, either using coordinate moments or a line between extrema. The measures agree remarkably well, a sign that $B_{ss}$ is very nearly equal to $\mathrm{\Omega }$, as one would expect for holes that are not undergoing strong dynamical processes. In fact, the measures agree to within a hundredth of a degree right up to the formation of the common horizon. It should be noted that we estimate the numerical truncation error of both of these measures to be roughly on the order of 0.02°, significantly greater than either of these discrepancies. Thus, even these differences could be largely due to numerical truncation.

Figure 7

Angle (in the Euclidean background space) between spin axes defined using $\zeta$ and $\mathrm{\Omega }$, either using coordinate moments or a line between extrema. The curves are visually identical if $\mathrm{\Omega }$ is replaced by $B_{ss}$. Interestingly, the two scalars agree to within our rough estimate of numerical truncation, so long as the axis is defined using coordinate moments of. The measures agree far less well when extrema are used. This is apparently a consequence of the higher-multipole structure present in $B_{ss}$ and $\mathrm{\Omega }$, already noted in Fig. 5.

Figure 8

Angle (in the Euclidean background space) between the default spec spin axis and our model of the standard spin axis in non-spec codes.

Figure 9

Nutation of the spin axes ${\widehat{\chi }}_{\zeta m}$ and ${\widehat{\chi }}_{\zeta e}$, along with the nutation expected from post-Newtonian theory. The quantity $\overrightarrow{\chi }·{\widehat{e}}_2$ is plotted on the horizontal axis and $\overrightarrow{\chi }·{\widehat{e}}_3$ on the vertical axis, where ${\widehat{e}}_i$ are defined in the text. The expectation from post-Newtonian theory is that the nutation should be represented as purely vertical oscillation on this chart, whereas the measure involving moments of $\zeta$ implies unphysical horizontal nutations greater than the vertical nutations by approximately a factor of 5. The measure involving extrema of $\zeta$ (our model of the standard non-spec axis measure) shares these unphysical horizontal nutations, as well as drastically enlarged vertical nutations.

Figure 10

Nutation of the spin axes ${\widehat{\chi }}_{\mathrm{\Omega }m}$ and ${\widehat{\chi }}_{\mathrm{\Omega }e}$, along with the nutation expected from post-Newtonian theory. The quantity $\overrightarrow{\chi }·{\widehat{e}}_2$ is plotted on the horizontal axis and $\overrightarrow{\chi }·{\widehat{e}}_3$ on the vertical axis, where ${\widehat{e}}_i$ are defined in the text. The quantity ${\widehat{\chi }}_{\mathrm{\Omega }m}$, which is the default measure of spin axis in the spec code, shares with ${\widehat{\chi }}_{\zeta m}$ the large unphysical nutations in the horizontal direction. The nutation of ${\widehat{\chi }}_{\mathrm{\Omega }e}$ is even stranger, due to the higher multipolar structure in $\mathrm{\Omega }$. We have also constructed the analogous diagram for ${\widehat{\chi }}_{B_{ss}m}$ and ${\widehat{\chi }}_{B_{ss}e}$, confirming that the nutations for that quantity are visually identical to those shown here.

Figure 11

Component of spin, on either black hole, along direction of orbital angular momentum, in a binary of equal-mass initially nonspinning black holes. When the spin is measured using the simple coordinate rotation vectors of Eq. (22), the holes spin up in the direction opposite expectations from perturbation theory. This spin-up, while small, is well resolved by the code. Moreover, this spin-up is strongly influenced by the choice of the coordinate centroid defining the rotation vectors. When the surface angular momentum density ${\omega }_A$ is boost fixed, by setting its gradient potential $\pi$ [defined in Eq. (38)] to zero, the spin (with either centroid) remains zero to well within the accuracy of numerical truncation throughout the entire inspiral (including a sharp but numerically unresolved spin-up after the formation of the common horizon).

Figure 12

Nutation in a coprecessing frame using the coordinate rotation generators ${\overrightarrow{\phi }}_i$, which are constructed from translations defined by the one form ${\mathit{\kappa }}^i=\mathbf{d}{x}^i$. This choice of rotation vector is very nicely behaved, mathematically speaking, however, it does not appear to improve upon the spurious nutations along the direction of secular precession, seen also with the spin measures using moments of $\mathrm{\Omega }$ (spec standard) or extrema of zeta (our model of the common standard outside of the spec code). The boost-fixing procedure described around Eq. (48) does not appear to provide an obvious improvement over the standard coordinate spin calculation, at least with regard to agreement with the precession track of an analogous post-Newtonian calculation, denoted by the heavy black line. All three of our ${\overrightarrow{\phi }}_i$-related coordinate spin measures seem qualitatively on par with the spec-standard measure, though intriguingly the phase lag between horizontal nutations and vertical nutations is different for the coordinate spin measures than for the moments of $\mathrm{\Omega }$.

Figure 13

Nutation in a coprecessing frame using coordinate rotation generators ${\overrightarrow{\varphi }}_i$, which are constructed from translations defined by the standard coordinate translation vectors ${\overrightarrow{\tau }}_i={\overrightarrow{\partial }}_i$. To keep the graph from getting cluttered, we have chosen a single cycle of precession near the beginning of the inspiral (after junk radiation has dissipated), though later cycles are qualitatively similar, as in other figures of the nutation track above. All four of the nutation tracks shown have dramatically better qualitative agreement with the post-Newtonian expectation (heavy black curve), with the spurious horizontal nutations reduced by an order of magnitude compared with the other measures in this paper. The remaining horizontal nutations are now roughly equal to the discrepancy of the vertical nutations with the post-Newtonian track, and hence we suspect they may all be due to remaining inaccuracies of the numerical relativity simulation such as numerical truncation error (which we have estimated to be around 0.02°), residual orbital eccentricity (the post-Newtonian calculation assumes a noneccentric orbit), or possibly genuine nonlinear behavior in the numerical relativity simulation.

Figure 14

Angle between post-Newtonian spin axis and four possible numerical-relativity spin axes. Fitting the post-Newtonian initial data to minimize the quantity $\mathcal{S}$ defined in the text, where the integration window of the fit extends from $t=4000m$ to $t=5500m$. The uppermost curve (light, solid) shows the discrepancy with ${\widehat{\chi }}_{\zeta e}$, our model of a standard axis measure defined by a line between poles of the approximate horizon symmetry. The intermediate curve (light, dashed) represents the standard axis measure in the spec code, ${\widehat{\chi }}_{\mathrm{\Omega }m}$, which also differs from the best-fit post-Newtonian spin axis by approximately a half degree throughout the inspiral, due to the wild horizontal oscillations visible in Fig. 10. The bottom curves show the best fit to spin measures based on the quasilocal angular momentum formula using ${\overrightarrow{\varphi }}_i$ as rotation generators. The heavy, solid curve employs the fixing of boost gauge described in the text, while the heavy dashed curve calculates the spin in the default boost gauge of the simulation, adapted to the spacetime slicing. These measures also use the flat-space centroid condition defined in Eq. (33), but the curved-space centroid gives similar results. As in the nutation tracks of Fig. 13, agreement with post-Newtonian expectations is better for the angular-momentum-based measures by approximately an order of magnitude.