Spin-Spin and Spin-Induced Multipole Couplings

Quadratic-in-spin dynamics contains two physically distinct effects. A dipole–dipole interaction couples the spins carried by the two bodies and is present regardless of their internal structure. A spin-induced quadrupole couples each body’s rotational deformation to its companion’s tidal field and therefore carries a structure-dependent coefficient. The latter distinguishes Kerr black holes from material compact objects already at leading quadratic order.

Required background. Presymplectic Potentials and Currents supplies the reduced spin phase space.

Helpful background. The Einstein Equation fixes curvature, Weak Metric Perturbations and Gauge Freedom supplies perturbative variables, Black-Hole Geometry and Mechanics supplies Kerr multipoles, and Relativistic Two-Body Dynamics supplies PN and EFT counting.

Dipole–dipole coupling

Use canonical constant-magnitude spins, a Newton–Wigner-type spin gauge, and the center-of-mass separation

$$\mathbf r=\mathbf x_1-\mathbf x_2, \qquad \mathbf n=\frac{\mathbf r}{r}.$$

At leading 2PN quadratic-in-spin order, the cross-spin Hamiltonian is

$$H_{S_1S_2} =\frac{G}{r^3} \left[ 3(\mathbf S_1\cdot\mathbf n) (\mathbf S_2\cdot\mathbf n) -\mathbf S_1\cdot\mathbf S_2 \right].$$

It is symmetric under $1\leftrightarrow2$ because both scalar products and $r$ are unchanged when $\mathbf n\to-\mathbf n$. It vanishes if either spin vanishes and falls as $r^{-3}$, the expected dipolar tidal scaling.

For spins aligned with the orbital angular momentum, $\mathbf S_A\cdot\mathbf n=0$, so

$$H_{S_1S_2} =-\frac{G}{r^3}\mathbf S_1\cdot\mathbf S_2.$$

The sign of a Hamiltonian term alone should not be translated into a universal statement that aligned spins “attract” or “repel.” Circular binding energy at fixed frequency also includes the spin-dependent relation between radius and frequency and higher-order canonical terms.

Spin-induced quadrupoles

For body $A$, define the symmetric trace-free mass quadrupole by

$$Q_A^{ij} =-\frac{\kappa_A}{m_A} \left( S_A^iS_A^j -\frac13\delta^{ij}\mathbf S_A^2 \right).$$

This equation fixes the normalization of the dimensionless coefficient $\kappa_A$. Coupling to the companion’s leading electric tidal tensor gives

$$H_{S_A^2} =\frac{Gm_B\kappa_A}{2m_A r^3} \left[ 3(\mathbf S_A\cdot\mathbf n)^2 -\mathbf S_A^2 \right], \qquad B\ne A.$$

The full self-spin contribution is the sum of the $A=1$ and $A=2$ terms. For a four-dimensional Kerr black hole in general relativity,

$$\kappa_{\mathrm{BH}}=1$$

in this convention, reflecting the Kerr multipole relation. Neutron-star values depend on the equation of state, mass, rotation approximation, and normalization and are generally not unity. They must be matched from the object’s response or multipolar exterior; this chapter inserts them into binary dynamics but does not calculate stellar structure.

The sign of the corresponding worldline operator follows from the same quadrupole, rather than from copying an isolated coefficient. Define $E_{\mu\nu}=C_{\mu\alpha\nu\beta}u^\alpha u^\beta$ and use $L_{\rm int}=-Q^{ij}E_{ij}/2$. Since $E^i{}_i=0$, substitution gives

$$S_{ES^2} =+\int d\tau\, \frac{C_{ES^2}}{2m} E_{\mu\nu}S^\mu S^\nu.$$

Thus $C_{ES^2}=\kappa$ in this convention at leading spin-squared order. The Newtonian limit $E_{ij}=Gm_B(\delta_{ij}-3n_i n_j)/r^3$ and $H_{\rm int}=-L_{\rm int}$ recover the displayed positive coefficient of $3(\mathbf n\cdot\mathbf S)^2-\mathbf S^2$. A source with an opposite curvature convention can write a minus sign in its action with the same positive Kerr coefficient. The action, tidal tensor, and spin definitions must be translated together.

These interactions and their EFT basis are reviewed in Porto 2016, §§8.2–8.4, pp. 72–87 and Levi 2020, §§3.2–4, pp. 18–42.

Precession and orbital consequences

Spin Poisson brackets turn each Hamiltonian into a precession vector:

$$\dot{\mathbf S}_A =\frac{\partial H}{\partial\mathbf S_A} \times\mathbf S_A.$$

For the cross-spin term,

$$\boldsymbol\Omega_1^{S_1S_2} =\frac{G}{r^3} \left[ 3\mathbf n(\mathbf S_2\cdot\mathbf n) -\mathbf S_2 \right].$$

The self-spin term similarly produces a torque proportional to $\kappa_1[3\mathbf n(\mathbf S_1\cdot\mathbf n)-\mathbf S_1]$. Every torque is perpendicular to the spin, preserving its canonical magnitude. The orbital torque balances the sum, so total angular momentum is conserved in the conservative truncation.

At fixed spin magnitudes the quadratic terms enter two PN powers beyond the Newtonian orbital energy for maximally compact spins. Their importance grows in the strong field and for large, misaligned spins. Higher powers encode spin-induced octupoles and beyond; rotational symmetry, parity, and the Kerr or material matching coefficients determine their form.

Spin-induced multipoles are intrinsic and should not be confused with companion-induced tidal multipoles. The former persist for an isolated rotating body and are analytic in spin. The latter respond to an external field, can be frequency dependent and dissipative, and are developed in the tides chapter.

Independent checks

A calculation should pass:

1. exchange symmetry of the $S_1S_2$ sector;
2. the zero-spin limit body by body;
3. $\kappa_A=1$ for the Kerr matching convention;
4. $r^{-3}$ scaling at leading quadrupolar order;
5. preservation of spin magnitudes and conservative total angular momentum;
6. agreement of a gauge-invariant circular energy or scattering angle after translating spin variables.

Terms at different PN orders cannot be compared by coefficients alone. Spin-dependent centroid shifts generate quadratic terms from lower-order spin–orbit couplings, so an incomplete SSC transformation produces a false structure dependence.

The displayed Hamiltonians are valid in the weak-field, slow-motion PN regime at leading quadratic-in-spin order and for bodies whose size and spin length remain small relative to the orbital separation. They break down as isolated terms near merger, near material disruption, or when unmodeled dynamical response becomes resonant.

Exercises

Verify exchange symmetry of the displayed quadratic Hamiltonian.

Solution

Exchanging bodies sends $\mathbf S_1\leftrightarrow\mathbf S_2$, $m_1\leftrightarrow m_2$, and $\mathbf n\to-\mathbf n$. The product of the two projections is unchanged, as is $\mathbf S_1\cdot\mathbf S_2$. The two self-spin terms exchange places. Their sum and $H_{S_1S_2}$ are therefore invariant.

Compute the leading self-spin orientation factor for a spin parallel to $\mathbf n$ and for one perpendicular to it.

Solution

For $\mathbf S\parallel\mathbf n$, $3(\mathbf S\cdot\mathbf n)^2-\mathbf S^2=2\mathbf S^2$. For $\mathbf S\perp\mathbf n$, it equals $-\mathbf S^2$. The sign and magnitude of the instantaneous quadrupole coupling therefore depend on orientation even at fixed $\kappa$ and spin magnitude.

References

• Levi, Michèle. “Effective Field Theories of Post-Newtonian Gravity: A Comprehensive Review.” Reports on Progress in Physics 83 (2020): 075901. DOI.
• Porto, Rafael A. “The Effective Field Theorist’s Approach to Gravitational Dynamics.” Physics Reports 633 (2016): 1–104. DOI.