Binary-black-hole initial data with nearly extremal spins

There is a significant possibility that astrophysical black holes with nearly extremal spins exist. Numerical simulations of such systems require suitable initial data. In this paper, we examine three methods of constructing binary-black-hole initial data, focusing on their ability to generate black holes with nearly extremal spins: (i) Bowen-York initial data, including standard puncture data (based on conformal flatness and Bowen-York extrinsic curvature), (ii) standard quasiequilibrium initial data (based on the extended-conformal-thin-sandwich equations, conformal flatness, and maximal slicing), and (iii) quasiequilibrium data based on the superposition of Kerr-Schild metrics. We find that the two conformally flat methods (i) and (ii) perform similarly, with spins up to about 0.99 obtainable at the initial time. However, in an evolution, we expect the spin to quickly relax to a significantly smaller value around 0.93 as the initial geometry relaxes. For quasiequilibrium superposed Kerr-Schild data [method (iii)], we construct initial data with initial spins as large as 0.9997. We evolve superposed Kerr-Schild data sets with spins of 0.93 and 0.97 and find that the spin drops by only a few parts in ${10}^4$ during the initial relaxation; therefore, we expect that superposed Kerr-Schild initial data will allow evolutions of binary black holes with relaxed spins above 0.99. Along the way to these conclusions, we also present several secondary results: the power-law coefficients with which the spin of puncture initial data approaches its maximal possible value; approximate analytic solutions for large spin puncture data; embedding diagrams for single spinning black holes in methods (i) and (ii); nonunique solutions for method (ii). All of the initial-data sets that we construct contain subextremal black holes, and when we are able to push the spin of the excision boundary surface into the superextremal regime, the excision surface is always enclosed by a second, subextremal apparent horizon. The quasilocal spin is measured by using approximate rotational Killing vectors, and the spin is also inferred from the extrema of the intrinsic scalar curvature of the apparent horizon. Both approaches are found to give consistent results, with the approximate-Killing-vector spin showing the least variation during the initial relaxation.

Figure 1

Convergence test for a single puncture black hole with a very large spin parameter $S/m_p^2=10000$. Plotted are results vs resolution $N$, which is the total number of basis functions. The solid lines show the relative differences of three angular momentum measures to the analytically expected value 10 000. The dashed lines show differences from the next-higher resolution of two dimensionless quantities for which no analytic answer is available.

Figure 2

Properties of single, spinning puncture black holes with spin parameter $S$ and puncture mass $m_p$. The dimensionless spin $\chi ≔S/M^2$, ADM angular momentum ${\epsilon }_J≔J_{\mathrm{ADM}}/E_{\mathrm{ADM}}^2$, and spin-extremality parameter $\zeta ≔S/(2M_{\mathrm{irr}}^2)$ are plotted against the spin parameter $S/m_p^2$. The horizon mass $M$ is related to the spin $S$ and irreducible mass $M_{\mathrm{irr}}$ in Eq. (2).

Figure 3

Properties of coordinate spheres with radius $r$ for high-spin puncture initial data. Main panel: Area of these spheres. Inset: Residual of the apparent horizon equation on these spheres. The area is almost constant over several orders of magnitude in $r$. The apparent-horizon residual vanishes at $r=R_{\mathrm{inv}}$ but is very small over a wide range of $r$.

Figure 4

Solutions of high-spin puncture initial data. Plotted are the conformal factor $\psi$ and puncture function $u$ in the equatorial plane as a function of radius $r$. Furthermore, the approximate solution $\overline{\psi }$ is included, with solid circles denoting the range of validity of this approximation; cf. Eq. (55). Three curves each are plotted, corresponding from top to bottom to $S/m_p^2=10000$, 1000, and 100.

Figure 5

Angular decomposition of the conformal factor $\psi (r,\theta ,\varphi )$ for single-black-hole puncture data.

Figure 6

Conformally flat, maximally sliced, quasiequilibrium initial-data sets with a single, spinning black hole. We plot the horizon mass $M$, irreducible mass $M_{\mathrm{irr}}$, and the (approximate-Killing-vector) spin $S$ against the rotation parameter ${\Omega }_r$ [cf. Eq. (40)]. Only ${\Omega }_r$ is varied in this figure; all other parameters are held fixed. The upper and lower points with the same ${\Omega }_r$ are obtained numerically by choosing different initial guesses. The inset shows a close-up view of the turning point, which occurs at ${\Omega }_r\approx 0.191$.

Figure 7

Conformally flat, maximally sliced quasiequilibrium initial-data sets with a single spinning black hole: The dimensionless spin $\chi$, dimensionless ADM angular momentum ${\epsilon }_J$, and spin-extremality parameter $\zeta$ plotted against ${\Omega }_r$ [cf. Eq. (40)]. Only ${\Omega }_r$ is varied in this figure; all other parameters are held fixed. The inset enlarges the area in the upper left corner; we are able to generate data sets with $\chi >0.99$, whereas the largest spin obtainable on the lower branch is $\chi \approx 0.85$.

Figure 8

Embedding diagrams for puncture and quasiequilibrium initial data. Plotted is the embedding height $Z$ as a function of the embedding radius $R$, both scaled by the mass $M$. For quasiequilibrium data (dashed lines), $Z=0$ at $r=r_{\mathrm{exc}}$; for puncture data (solid lines), $Z=0$ at $r=R_{\mathrm{inv}}$. The thin solid purple curve represents the embedding of a plane through a Schwarzschild black hole in Schwarzschild slicing.

Figure 9

Main panel: Dimensionless spin $\chi$ [Eq. (1)] and spin-extremality parameter $\zeta$ [Eq. (8)] for the family CFMS of spinning binary-black-hole initial data. Inset: Enlargement of $\chi$ toward the end of the upper branch, with circles denoting the individual initial-data sets that were constructed. Compare with Fig. 7.

Figure 10

Convergence of the spectral elliptic solver. Left panel: The residual constraint violation as a function of the total number of grid points $N$ when running the elliptic solver at several different resolutions. Right panel: Convergence of the black-hole dimensionless spin $\chi$ [Eq. (1)] with increasing resolution $L_{\mathrm{AH}}$ of the apparent horizon finder, applied to the highest-resolution initial-data set of the left panel. The three curves in each panel represent three different initial-data sets: one from the family SKS-0.99, as well as the two initial-data sets that are evolved in Sec. 5.

Figure 11

The mass $M$ (upper panel) and dimensionless spin $\chi$ (lower panel) of one of the holes for superposed Kerr-Schild, binary-black-hole initial-data sets with spins aligned with the orbital angular momentum. The mass and spin are plotted against ${\Omega }_r$ [Eq. (40)] for four different choices of the conformal spin: $\tilde{S}=0$, 0.5, 0.93, and 0.99. Also shown are the data sets SKS-0.93-E3—identical to the ${\Omega }_r=0.28\tilde{M}$, $\tilde{S}=0.93{\tilde{M}}^2$ data set on the solid curve but with lower eccentricity—and SKS-Headon; both sets are evolved in Sec. 5. The inset in the lower panel shows a close-up of the spins as they approach unity, with symbols denoting the individual data sets.

Figure 12

The irreducible mass $M_{\mathrm{irr}}$ and Euclidean coordinate radius $r$ (upper panels) and dimensionless spin $\chi ≔S/M^2$ and spin-extremality parameter $\zeta ≔S/(2M_{\mathrm{irr}}^2)$ (lower panels) for one of the black holes in the SKS-0.93 (left) and SKS-0.99 (right) initial-data-set families. These quantities are computed on two surfaces: (i) the apparent horizon (solid lines) and (ii) the excision boundary of the initial data (dashed lines). Because we enforce that the excision surface is a marginally trapped surface, typically the apparent horizon and excision boundary coincide. However, if ${\Omega }_r$ is increased beyond the values where $\chi$ approaches unity, the apparent horizon lies outside of the excision surface. The excision surface can obtain superextremal spins ($\zeta >1$) but only when it is enclosed by a subextremal horizon.

Figure 13

The time derivatives of the metric (left panel) and extrinsic curvature (right panel). In the SKS data sets, $\Vert {\partial }_t{K}_{ij}{\Vert }_{L2}$ has minima near values of ${\Omega }_r$ for which the dimensionless spin $\chi$ is approximately equal to the spin $\tilde{S}$ of the conformal metric (cf. Fig. 11). On the upper branch of the CFMS excision data, where the spin is $\chi >0.83$ (Fig. 9), the time derivatives become much larger than the SKS time derivatives. The data sets SKS-0.93-E3 (with $\chi \approx \tilde{S}=0.93$) and SKS-Headon (with $\chi \approx \tilde{S}=0.97$) are evolved in Secs. 5c, 5d; the time derivatives are significantly lower for the set SKS-Headon because of the larger coordinate separation of the holes ($d=100$ vs $d=32$).

Figure 14

Eccentricity reduction for evolutions of superposed Kerr-Schild binary-black-hole initial data. The proper separation $s$ (upper panel) and its time derivative $ds/dt$ (lower panel) are plotted for initial-data sets SKS-0.93-E0, -E1, -E2, and -E3, which have successively smaller eccentricities $e$. All evolutions are performed at resolution N1.

Figure 15

Convergence test of the evolution of the initial-data set SKS-0.93-E3. Shown are evolutions on three different resolutions N1, N2, and N3, with N3 being the highest resolution. The top panel shows the AKV spin of one of the holes as a function of time, with the top inset showing the spin’s initial relaxation; the bottom panel shows the constraint violation as a function of time.

Figure 16

A comparison of different definitions of the spin. The top panel shows the spin as a function of time for several different measures of the spin; the bottom panel shows the fractional difference between ${\chi }_{\mathrm{AKV}}$ and alternative spin definitions. Note that for $t<30E_{\mathrm{ADM}}$, the time axis has a different scaling to make the initial transients visible.

Figure 17

Convergence test of the head-on evolution SKS-Headon. Shown are evolutions at three different resolutions N1, N2, and N3, with N3 being the highest resolution. The top panel shows the AKV spin of one of the holes as a function of time; the bottom panel shows the constraint violations as a function of time.

Figure 18

A comparison of various measures of the spin for the head-on evolution of data set SKS-Headon, which is a plunge of two equal-mass black holes with parallel spins of magnitude ${\chi }_{\mathrm{AKV}}=S/M^2=0.970$ pointed normal to the equatorial plane. The top panel shows various measures of the spin as a function of time, and the bottom panel shows the fractional difference between the AKV spin ${\chi }_{\mathrm{AKV}}$ and alternative spin definitions.

Figure 19

The change $\Delta \chi$ in black-hole spin $\chi$ during the initial relaxation of black-hole initial data plotted as a function of the black-hole spin after relaxation. The SKS initial data constructed in this paper have smaller transients and allow for larger relaxed spins.

Figure 20

Error, relative to the analytic solution, of the spin on the horizon of a Kerr black hole in slightly deformed coordinates. The vertical axis represents $|{\chi }_{\mathrm{computed}}-{\chi }_{\mathrm{analytic}}|$, and data are shown for the spin computed with the standard coordinate rotation vector (in deformed coordinates, so not a true Killing vector) and with our AKVs using both the extremum norm Eq. (a21) and the integral norm Eq. (a22). The spin computed from the coordinate rotation vector quickly converges to a physically inaccurate result. The spin from approximate Killing vectors converges in resolution $L_{\mathrm{AH}}$ to the correct value $\chi =1/2$. Curves are also shown for the two spin measures defined in Appendix . These spin measures also converge exponentially to the physically correct result.