Spinning EOB Hamiltonians

A spinning EOB Hamiltonian must interpolate among three sharply constraining descriptions: the PN spin couplings of two comparable bodies, a spinning test body moving in Kerr spacetime, and the nonspinning EOB Hamiltonian. The construction uses a deformed-Kerr orbital sector, an effective particle spin, and gyrogravitomagnetic functions that carry finite-mass-ratio corrections. None of these ingredients is unique separately; the invariant content lies in the complete Hamiltonian and its observable predictions.

Required background. Presymplectic Potentials and Currents supplies canonical transformations and Hamiltonian flow on orbital and spin phase space.

Helpful background. The Einstein Equation supplies the underlying relativistic dynamics; Weak Metric Perturbations and Gauge Freedom supplies PN spin matching; Black-Hole Perturbation Theory supplies the Kerr test-body limit; and Relativistic Two-Body Dynamics supplies the mass-ratio, velocity, and spin regimes.

Spin variables and effective combinations

Let $M=m_1+m_2$, $\mu=m_1m_2/M$, and $\nu=\mu/M$. The canonical spin vectors obey

$$\{S_a^i,S_b^j\} =\delta_{ab}\epsilon^{ijk}S_a^k, \qquad a,b\in\{1,2\}.$$

They have fixed magnitudes in the conservative point-particle problem. For black holes, write

$$\boldsymbol\chi_a=\frac{\mathbf S_a}{Gm_a^2}, \qquad \lvert\boldsymbol\chi_a\rvert\leq1,$$

and define the orbital angular momentum $\mathbf L=\mathbf r\times\mathbf P$. Two mass-weighted combinations organize the PN spin-orbit interaction:

$$\mathbf S=\mathbf S_1+\mathbf S_2, \qquad \mathbf S_*=\frac{m_2}{m_1}\mathbf S_1 +\frac{m_1}{m_2}\mathbf S_2.$$

At leading order, the real spin-orbit Hamiltonian is

$$H_{\rm SO}^{\rm LO} =\frac{G}{r^3}\, \mathbf L\cdot \left(2\mathbf S+\frac{3}{2}\mathbf S_*\right).$$

Thus the leading gyrogravitomagnetic factors are $g_S=2$ and $g_{S_*}=3/2$. Higher PN orders promote them to phase-space functions,

$$H_{\rm SO} =\frac{G}{r^3}\, \mathbf L\cdot \left[ g_S(u,\mathbf p^2,p_r^2;\nu)\mathbf S +g_{S_*}(u,\mathbf p^2,p_r^2;\nu)\mathbf S_* \right].$$

Here $u=GM/r$, $\mathbf p=\mathbf P/\mu$, and $p_r=\mathbf n\cdot\mathbf p$ are dimensionless. The functions in a real Hamiltonian differ beyond leading order from the effective gyro factors inside $H_{\rm eff}$: linearizing the energy map in spin gives $H_{\rm SO}^{\rm real}=H_{\rm SO}^{\rm eff}/h_0$, where $h_0=H_{\rm real}^{\rm nonspin}/M$. Canonical and spin-gauge transformations provide further changes. The NLO mapping separates these operations in Damour, Jaranowski, and Schäfer 2008, § III, Eqs. (3.7)–(3.13), PDF.

These combinations organize the spin map; neither is automatically the literal spin of an effective particle in every representation. In the limit $m_2/m_1\to0$ at fixed black-hole spins $\chi_a$, both terms in $\mathbf S_*$ scale as $Gm_1m_2$, so its inverse mass weight is not a divergence. The physical secondary’s spin relative to the background scale satisfies $S_2/(Gm_1m_2)=(m_2/m_1)\chi_2\to0$. It is therefore a subleading test-body correction. Recovering the spin-dipole test-body Hamiltonian means keeping that order in the small-mass-ratio expansion, rather than asserting that a fixed-$\chi_2$ secondary has a leading-order spin effect as $m_2\to0$.

Deformed Kerr plus a spinning effective particle

A useful structural decomposition is

$$H_{\rm eff} =H_{\rm orb}^{\rm defKerr} +H_{\rm SO}^{\rm eff} +H_{\rm SS}^{\rm eff} +H_{\rm higher}.$$

The first term describes orbital motion in a Kerr-like metric whose radial functions are deformed by $\nu$. The second couples the effective background and particle spins to orbital angular momentum. The third contains bilinear spin-spin and spin-induced quadrupole effects. The last term stands for higher multipoles and higher powers of spin or momentum retained by a particular model. Barausse and Buonanno construct this decomposition from a canonical spinning-particle Hamiltonian and match it to PN dynamics Barausse and Buonanno 2010, §§ III–V.

The real Hamiltonian retains the nonspinning energy map,

$$H_{\rm real} =M\sqrt{1+2\nu \left(\frac{H_{\rm eff}}{\mu}-1\right)}.$$

Spin changes $H_{\rm eff}$; it does not justify silently changing the map. Any alternative spin-dependent map must be treated as a distinct canonical representation and shown to reproduce the same PN and test-body invariants.

At quadratic order, the tensor structures include

$$\mathbf S_1\cdot\mathbf S_2,\qquad (\mathbf n\cdot\mathbf S_1) (\mathbf n\cdot\mathbf S_2),\qquad \mathbf S_a^2,\qquad (\mathbf n\cdot\mathbf S_a)^2,$$

where $\mathbf n=\mathbf r/r$. The self-spin terms depend on each body’s spin-induced quadrupole coefficient. For a Kerr black hole that coefficient has its Kerr value; for a neutron star it is equation-of-state dependent. Consequently, a Hamiltonian advertised for binary black holes cannot be transferred to material bodies merely by changing the masses and spin vectors.

Spin gauge and canonical equivalence

Relativistic spin requires a representative worldline and a spin supplementary condition before reduction to canonical variables. Once a canonical spin is chosen, further spin-dependent generating functions can shift momentum-dependent pieces among the gyro factors, the deformed metric, and explicit spin couplings. Schematically, a generator linear in spin,

$$G_{\rm spin} =\frac{G}{r^2} \left[ \alpha(u,\mathbf p^2,p_r^2) \mathbf L\cdot\mathbf S +\beta(u,\mathbf p^2,p_r^2) \mathbf L\cdot\mathbf S_* \right],$$

changes the Hamiltonian by $\{H,G_{\rm spin}\}+\cdots$. Coefficients that vanish in one gauge can reappear in another. This freedom is often used to remove selected $p_r^2$ terms or to simplify the test-body limit, but it does not remove physical spin precession.

Three items must therefore accompany any quoted spinning Hamiltonian: the canonical orbital variables, the spin variables and supplementary-condition map from covariant quantities, and the spin gauge used for the gyro factors. Comparing only $g_{S_*}$ between two models may confuse a gauge choice with a physical disagreement. A clean comparison re-expands both complete Hamiltonians in one canonical gauge or directly compares invariant circular energy, spin-precession frequency, or scattering data.

For aligned or antialigned spins, reflection symmetry preserves a planar submanifold and the Hamiltonian reduces to functions of signed spin projections. Generic precession is different: the orbital plane evolves, mode frames must be transported consistently, and a circular equatorial orbit need not exist as an exact solution. Results derived for aligned circular motion should not be presented as generic-spin statements.

Hamiltonian precession and conserved spin length

Hamilton’s equation for either spin is

$$\dot{\mathbf S}_a =\frac{\partial H_{\rm real}} {\partial\mathbf S_a} \times\mathbf S_a \equiv\boldsymbol\Omega_a\times\mathbf S_a.$$

It immediately follows that

$$\frac{d}{dt}\mathbf S_a^2 =2\mathbf S_a\cdot \left(\boldsymbol\Omega_a\times\mathbf S_a\right)=0.$$

This is a valuable implementation check: a conservative Hamiltonian flow should preserve each spin magnitude to numerical tolerance. Rotational invariance also implies conservation of

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

The individual vectors precess, but their vector sum is constant until radiation reaction is added. Kidder’s PN equations give the weak-field spin precession and radiation formulae in a compatible two-body language Kidder 1995, § II, Eqs. (2.4)–(2.13).

For the leading spin-orbit Hamiltonian, differentiation gives precession axes built from $\mathbf L$. For example,

$$\boldsymbol\Omega_1^{\rm LO} =\frac{G}{r^3} \left(2+\frac{3m_2}{2m_1}\right)\mathbf L,$$

with the labels exchanged for body 2. If both spins are initially parallel to $\mathbf L$, the cross products vanish and the configuration remains aligned under conservative leading-order precession.

Independent limits

Zero-spin limit. Setting $\mathbf S_1=\mathbf S_2=0$ must remove every spin term and recover the nonspinning potentials with no residual spin gauge parameter.

Test-body-in-Kerr limit. As $\nu\to0$, the deformed background must become Kerr with the primary spin. With a nonspinning secondary, the motion becomes Kerr geodesic motion. Retaining the secondary’s spin through the proper mass scaling must reproduce the canonical spinning-test-body Hamiltonian at the included order Barausse and Buonanno 2010, § VI A, PDF.

Weak-field spin-orbit limit. Re-expansion at large $r$ must give $g_S\to2$ and $g_{S_*}\to3/2$, followed by the known PN corrections in the chosen gauge. This test catches both a faulty spin map and missing factors of mass.

Spin-length and angular-momentum conservation. In a conservative integration, $\mathbf S_a^2$, $H_{\rm real}$, and total $\mathbf J$ should remain constant. These checks are independent of comparisons to another waveform model.

Domain and breakdown

An analytic spinning EOB Hamiltonian is controlled where its PN, PM, or self-force inputs are controlled and where its chosen resummation is regular. Near-extremal spins, large anti-aligned spin at high mass ratio, and strong noncircular encounters can probe poorly constrained parts of the gyro factors or deformed metric. Truncating at a stated spin order also omits higher spin-induced multipoles. Calibration may improve agreement within a numerical-relativity domain, but it does not turn the effective metric into an exact two-body spacetime or guarantee extrapolation outside that domain. Radiation reaction further breaks conservation of orbital energy and total angular momentum and requires a compatible spin and waveform convention.

Common pitfalls

Confusing $\mathbf S_*$ with a literal sum of spins. Its mass weights are essential for the test-body limit and for PN matching.

Comparing gyro factors without a gauge map. Momentum-dependent coefficients can move under spin-dependent canonical transformations. Compare the full Hamiltonian or invariants.

Using an aligned-spin Hamiltonian for generic precession. Planar reduction discards degrees of freedom required to transport the orbital and waveform frames.

Exercise

Use the leading spin-orbit Hamiltonian to prove conservation of both spin magnitudes. Then show that total angular momentum is conserved without computing the full orbital equations component by component.

Solution

For each body, Hamiltonian spin evolution has the form

$$\dot{\mathbf S}_a =\boldsymbol\Omega_a\times\mathbf S_a.$$

Hence

$$\frac{dS_a^2}{dt} =2\mathbf S_a\cdot \left(\boldsymbol\Omega_a\times\mathbf S_a\right)=0.$$

The Hamiltonian is a rotational scalar: it depends on vectors only through scalar products such as $\mathbf L\cdot\mathbf S_a$. The generator of simultaneous rotations is $\mathbf J=\mathbf L+\mathbf S_1+\mathbf S_2$. Rotational invariance is $\{H,\mathbf J\}=0$, so $\dot{\mathbf J}=\{\mathbf J,H\}=0$. Equivalently, the torques that rotate the spins are balanced by the opposite torque on $\mathbf L$. No statement that $\mathbf L$ or either spin is separately fixed is required.

References

• Barausse, Enrico, and Alessandra Buonanno. “An Improved Effective-One-Body Hamiltonian for Spinning Black-Hole Binaries.” Physical Review D 81, 084024 (2010). DOI. Open PDF.
• Damour, Thibault, Piotr Jaranowski, and Gerhard Schäfer. “Effective One Body Approach to the Dynamics of Two Spinning Black Holes with Next-to-Leading Order Spin-Orbit Coupling.” Physical Review D 78, 024009 (2008). 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, 821–847 (1995). DOI. Open PDF.