Spin-Orbit Coupling and Precession

Spin–orbit coupling links each body’s intrinsic angular momentum to the binary’s orbital angular momentum. In conservative dynamics it causes the spins and orbital plane to precess while preserving total angular momentum. The precession is observable through invariant frequency and waveform modulations, but individual vector components depend on the spin supplementary condition, canonical variables, and precessing frame.

Required background. Presymplectic Potentials and Currents supplies the constrained phase-space structure behind canonical spin variables.

Helpful background. The Einstein Equation fixes curvature, Weak Metric Perturbations and Gauge Freedom supplies perturbative gauge control, Black-Hole Geometry and Mechanics supplies test-body limits, and Relativistic Two-Body Dynamics supplies PN counting.

Leading spin–orbit Hamiltonian

Use canonical constant-magnitude spins in a Newton–Wigner-type spin gauge, center-of-mass orbital variables $(\mathbf r,\mathbf p)$, and coordinate time. Let

$$\mathbf L=\mathbf r\times\mathbf p, \qquad r=\lvert\mathbf r\rvert.$$

At leading 1.5PN spin–orbit order, for arbitrary mass ratio,

$$H_{\mathrm{SO}} =\frac{G}{r^3} \mathbf L\cdot \left( g_1\mathbf S_1+g_2\mathbf S_2 \right),$$ $$g_1=2+\frac{3m_2}{2m_1}, \qquad g_2=2+\frac{3m_1}{2m_2}.$$

This displayed form is specific to the stated canonical variables. Covariant spins and harmonic-coordinate positions are related by spin-dependent shifts; their Hamiltonian coefficients need not match term by term.

The spin Poisson brackets are

$$\{S_A^i,S_B^j\} =\delta_{AB}\epsilon^{ijk}S_A^k.$$

Hamilton’s equation therefore gives

$$\dot{\mathbf S}_1 =\boldsymbol\Omega_1\times\mathbf S_1, \qquad \boldsymbol\Omega_1 =\frac{Gg_1}{r^3}\mathbf L,$$

and likewise for body 2. The cross product makes

$$\frac{d}{dt}\mathbf S_A^2=0$$

at this conservative order.

Orbital-plane precession

Rotational invariance implies that the total angular momentum

$$\mathbf J =\mathbf L+\mathbf S_1+\mathbf S_2$$

is conserved when radiation reaction is omitted. Directly,

$$\dot{\mathbf L} =-\dot{\mathbf S}_1-\dot{\mathbf S}_2.$$

Thus spin torque on the orbit exactly balances orbital torque on the spins. When one spin dominates, $\mathbf L$ and that spin precess approximately about the nearly fixed $\mathbf J$; with two comparable misaligned spins, nutation is superposed on precession.

For an orbit that is circular on the orbital timescale, average $r^{-3}$ over the fast phase. At fixed, nondegenerate mass ratio, the powers of $v$ in the characteristic timescales are

$$T_{\mathrm{orb}}\sim\frac{GM}{v^3}, \qquad T_{\mathrm{prec}}\sim\frac{GM}{v^5},$$

so $T_{\mathrm{prec}}/T_{\mathrm{orb}}\sim v^{-2}\gg1$ in the PN regime. Radiation reaction is slower still, $T_{\mathrm{rr}}\sim GM/(\nu v^8)$, allowing orbit-averaged precession before precession-averaged inspiral.

The leading and higher-order spin-orbit equations, fluxes, and circular limits are developed in Kidder 1995, §§II–IV; their canonical and EFT translations are reviewed in Porto 2016, §§8.1–8.4, pp. 68–87.

Conserved combinations and radiation reaction

The leading spin–orbit Hamiltonian above is a useful first layer, but a widely used additional invariant belongs to the complete orbit-averaged 2PN black-hole spin system, including spin-spin and Kerr quadrupole–monopole couplings. With $M=m_1+m_2$, it is

$$\xi =\frac{1}{GM^2} \left[ \left(1+\frac1q\right)\mathbf S_1 +(1+q)\mathbf S_2 \right]\cdot\hat{\mathbf L}, \qquad q=\frac{m_1}{m_2}\geq1,$$

where the factor of $G$ is required for a dimensionless quantity. Writing $\boldsymbol\chi_A=\mathbf S_A/(Gm_A^2)$ reduces it to

$$\xi=\frac{m_1\boldsymbol\chi_1+m_2\boldsymbol\chi_2}{M} \cdot\hat{\mathbf L}.$$

This is the usual projected effective spin in these variables. Its conservation is specific to the stated averaging, PN order, and black-hole quadrupole coefficients; it is not an exact invariant of arbitrary spinning binaries. See Gerosa et al. 2015, § II B, Eq. (12), and § II D, Eq. (25), PDF, whose mass ratio is the reciprocal of ours and whose units have $G=1$.

One can see why the qualification matters. Set $a=1+1/q$, $b=1+q$ and keep only the displayed spin–orbit Hamiltonian. Since $L$ is constant, direct differentiation yields

$$\dot\xi_{\rm SO} =\frac{a g_2-b g_1}{M^2r^3} \hat{\mathbf L}\cdot(\mathbf S_1\times\mathbf S_2) =\frac{q^{-1}-q}{2M^2r^3} \hat{\mathbf L}\cdot(\mathbf S_1\times\mathbf S_2).$$

This quadratic-in-spin remainder generally does not vanish. It is canceled at the retained order when the complete orbit-averaged black-hole spin couplings are included. Dropping those terms while demanding exact conservation of $\xi$ would be an inconsistent check.

Radiation carries away angular momentum predominantly along the orbit-averaged $\hat{\mathbf L}$ for quasicircular motion. Then $\lvert\mathbf J\rvert$ and $\lvert\mathbf L\rvert$ drift while $\mathbf S_A^2$ remain fixed for compact bodies without absorption or accretion. Conservative precession equations should not be used to claim exact $\mathbf J$ conservation once that flux is included. If $\mathbf J$ includes the evolving black-hole spins, the secular balance is

$$\dot{\mathbf J} =-\boldsymbol{\mathcal F}_J^\infty.$$

Horizon absorption transfers angular momentum from the orbit into those spins and is internal to this total. An orbital evolution that holds the absorbing spins fixed instead includes $-\boldsymbol{\mathcal F}_J^H$ as an additional sink from its resolved variables. These two descriptions must not be combined by subtracting the horizon flux twice. Beyond the leading secular balance, near-zone angular-momentum storage can also contribute a total time derivative.

Near configurations where $\mathbf L$ and total spin nearly cancel, the direction of $\mathbf J$ can change rapidly and simple-precession approximations fail.

A numerical evolution should monitor three residuals separately: drift of each spin magnitude, drift of $\mathbf J$ in the conservative subsystem, and the mismatch between $\dot{\mathbf J}$ and the imposed angular-momentum flux after dissipation is restored. One small combined residual can conceal cancellation between two sign errors. Step refinement should reduce all three at the integrator’s expected order.

Limits and convention checks

Zero-spin limit. Setting both spins to zero removes $H_{\mathrm{SO}}$ and all spin-precession torques.

Equal masses. For $m_1=m_2$, $g_1=g_2=7/2$; the Hamiltonian is symmetric under exchange of the bodies.

Test gyroscope. When body 2 is light and carries a test spin, the $3m_1/(2m_2)$ part of $g_2$ combines with $\mathbf L\propto m_2$ to leave a finite geodetic-precession frequency. If the large body 1 spins, the coefficient $g_1\to2$ gives the leading frame-dragging torque. Recovering both pieces requires taking the mass limit before discarding the apparently divergent gyro coefficient.

Frame choice. A coprecessing triad can remove most orbital-plane rotation from the waveform modes, but it does not remove physical precession. The rotation used to define that frame and its residual angle must be supplied before mode phases from two calculations are compared.

Exercises

Prove conservation of total angular momentum and spin magnitudes for the leading conservative system.

Solution

Since $\dot{\mathbf S}_A=\boldsymbol\Omega_A\times\mathbf S_A$,

$$\frac{dS_A^2}{dt} =2\mathbf S_A\cdot (\boldsymbol\Omega_A\times\mathbf S_A)=0.$$

Rotational invariance gives the equal and opposite orbital torque $\dot{\mathbf L}=-(\dot{\mathbf S}_1+\dot{\mathbf S}_2)$, hence $\dot{\mathbf J}=0$.

Show that the leading Hamiltonian respects body exchange.

Solution

Under $1\leftrightarrow2$, $m_1\leftrightarrow m_2$, $\mathbf S_1\leftrightarrow\mathbf S_2$, and $g_1\leftrightarrow g_2$. The combination $g_1\mathbf S_1+g_2\mathbf S_2$ is unchanged, as are $r$ and the center-of-mass $\mathbf L$. Therefore $H_{\mathrm{SO}}$ is invariant.

References

• Gerosa, Davide, Michael Kesden, Ulrich Sperhake, Emanuele Berti, and Richard O’Shaughnessy. “Multi-Timescale Analysis of Phase Transitions in Precessing Black-Hole Binaries.” Physical Review D 92, 064016 (2015). DOI. Open PDF.
• Kidder, Lawrence E. “Coalescing Binary Systems of Compact Objects to (Post)$^{5/2}$-Newtonian Order. V. Spin Effects.” Physical Review D 52 (1995): 821–847. DOI.
• Porto, Rafael A. “The Effective Field Theorist’s Approach to Gravitational Dynamics.” Physics Reports 633 (2016): 1–104. DOI.