Circular orbits and spin in black-hole initial data

The construction of initial data for black-hole binaries usually involves the choice of free parameters that define the spins of the black holes and essentially the eccentricity of the orbit. Such parameters must be chosen carefully to yield initial data with the desired physical properties. In this paper, we examine these choices in detail for the quasiequilibrium method coupled to apparent-horizon/quasiequilibrium boundary conditions. First, we compare two independent criteria for choosing the orbital frequency, the “Komar-mass condition” and the “effective-potential method,” and find excellent agreement. Second, we implement quasilocal measures of the spin of the individual holes, calibrate these with corotating binaries, and revisit the construction of nonspinning black-hole binaries. Higher-order effects, beyond those considered in earlier work, turn out to be important. Without those, supposedly nonspinning black holes have appreciable quasilocal spin; furthermore, the Komar-mass condition and effective-potential method agree only when these higher-order effects are taken into account. We compute a new sequence of quasicircular orbits for nonspinning black-hole binaries, and determine the innermost stable circular orbit of this sequence.

Figure 1

Effective-potential (EP) curves $E_b/\mu$ for corotating black holes vs separation $\ell /m$. These curves are labeled by the orbital angular momentum $J/\mu m$ which is kept constant along each curve. The thick red line connecting the minima of the EP curves represents circular orbits; it terminates at the innermost stable circular orbit at the inflection point in the EP curve at $J/\mu m=3.38$. Also plotted as a dashed blue line is the sequence of circular orbits determined by the Komar-mass ansatz.

Figure 2

Violation of the Komar-mass condition when the effective-potential method is used to determine the sequence of circular orbits. Here, corotating equal-mass binaries are considered. $m{\Omega }_0$ denotes the orbital angular frequency, so that large separations correspond to small values of $m{\Omega }_0$.

Figure 3

Quasilocal spin $S/M_{\mathrm{irr}}^2$ of a black hole in a corotating equal-mass (CO) binary along the sequence of circular orbits (parametrized by the orbital angular velocity $m{\Omega }_0$). $S_z$ and $S_K$ are defined in Eq. (40). Results are given for sequences with three different lapse boundary conditions.

Figure 4

Difference between the quasilocal spin of a black hole in a corotating (CO) binary and the result of the Kerr-formula Eq. (52). For the lower set of six lines (${\Omega }_B={\Omega }_0$), the rotation rate of the black hole is simply taken to be equal to the orbital angular velocity. For the upper set of six lines (${\Omega }_B={\Omega }_T$), the rotation rate is taken to be equal to that of the tidal field of the companion hole as measured in the LARF. Each set of six lines corresponds to the cases plotted in Fig. 3.

Figure 5

Ratio of the rotational angular velocity ${\Omega }_B$ of a black hole, as determined from the quasilocal spin $S_K$, to the orbital angular velocity of the binary along the corotating (CO) sequence. The dotted line (labeled $1\mathtt{P}\mathtt{N}$) represents the leading-order correction to the analytic estimate of the ratio. The dashed line (labeled $1\mathtt{P}\mathtt{N}+$) shows the higher-order correction of Eq. (56) with $\Lambda =\Gamma =0$. Numerical results are given for three different lapse boundary conditions.

Figure 6

EP curves $E_b/\mu$ for equal-mass “leading-order” nonspinning binaries (${\Omega }_r={\Omega }_0$) plotted vs separation $\ell /m$. These curves are labeled by the value of $J/\mu m$ along each curve. Also plotted are the line connecting the minima of the EP curves, as well as the sequence of circular orbits as determined by the Komar-mass ansatz. The Komar-mass ansatz and the effective-potential method clearly disagree.

Figure 7

Quasilocal spin of the “leading-order” nonspinning (LN) binaries (as shown in Fig. 6) plotted vs orbital angular velocity. The inset displays the dimensionless spin value $S/M_{\mathrm{irr}}^2$. The main figure displays the magnitude of the spin relative to the spin of a Kerr black hole with ${\Omega }_B={\Omega }_0$. The residual spin is quite large.

Figure 8

EP curves $E_b/\mu$ for true nonspinning black holes vs separation $\ell /m$. The curves are labeled by the orbital angular momentum $J/\mu m$ which is kept constant along each curve. The thick red line connecting the minima of the EP curves represents circular orbits, and terminates at the innermost stable circular orbit at the inflection point in the EP curve at $J/\mu m=3.12$. Also plotted as a dashed blue line is the sequence of circular orbits as determined by the Komar-mass ansatz.

Figure 9

Quasilocal spin for a true nonspinning (NS) black hole in an equal-mass binary along the sequence of circular orbits, parametrized by the orbital angular velocity. The same six cases are plotted as in the inset of Fig. 7. Note that here $S_K=0$ by construction.

Figure 10

Rotation parameter $f_r={\Omega }_r/{\Omega }_0$ along the sequence of nonspinning (NS) black holes, parametrized by the $2/3$-power of the orbital angular velocity. The dotted line plots the fit to $f_r$ through the term of order $(m{\Omega }_0{)}^{2/3}$.

Figure 11

Binding energy vs total angular momentum along the sequence of nonspinning (NS) equal-mass black holes (as defined by the Komar-mass condition). For comparison, the leading-order nonspinning (LN) results from our earlier work [5] are included, as well as results from Refs. 9, 10.

Figure 12

Binding energy vs orbital frequency along the sequence of nonspinning (NS) equal-mass black holes (as defined by the Komar-mass condition). Lines are labeled as in Fig. 11.

Figure 13

Total angular momentum vs orbital frequency along the sequence of nonspinning (NS) equal-mass black holes (as defined by the Komar-mass condition). Lines are labeled as in Fig. 11.

Figure 14

Violation of the Komar-mass condition when the effective-potential method is used to determine the sequence of circular orbits. Here, nonspinning (NS) equal-mass binaries are considered, and the sequence is parametrized by the orbital angular momentum.

Figure 15

ISCO configuration for nonspinning (NS) binary black holes, computed with three different lapse boundary conditions. For comparison, the leading-order nonspinning (LN) and corotating (CO) binary black holes from our earlier work [5] are included, as well as results of Refs. 8, 9, 10, 11. For post-Newtonian calculations the size of the symbol indicates the order, the largest symbol being $3\mathtt{P}\mathtt{N}$. “$\mathtt{P}\mathtt{N}$ standard” [11] represents a $\mathtt{P}\mathtt{N}$ expansion in the standard form without use of the EOB technique (only $2\mathtt{P}\mathtt{N}$ and $3\mathtt{P}\mathtt{N}$ are plotted).

Figure 16

ISCO configuration for nonspinning (NS) binary black holes, computed with three different choices of the lapse boundary condition. Symbols as in Fig. 15.

Figure 17

ISCO configuration for nonspinning (NS) binary black holes, computed with three different choices of the lapse boundary condition. Symbols as in Fig. 15.