Spin, Multipoles, and Massive On-Shell States

A massive on-shell state packages spin into little-group tensors, while a classical compact body is described by a representative worldline, a spin supplementary condition, and an infinite set of multipole moments. These are related dictionaries, not identical objects. The amplitude method becomes predictive only after the quantum spin limit, state normalization, centroid convention, and finite-size coefficients have all been matched to covariant body variables. For a Kerr black hole the resulting multipole series is exceptionally simple; for a generic star it is not.

Required background. Events, Intervals, and Minkowski Geometry supplies massive little-group kinematics and Lorentz invariants; Weak Metric Perturbations and Gauge Freedom supplies graviton polarizations and gauge equivalence. Helpful background. The Einstein Equation fixes the gravitational coupling and vacuum theory; Presymplectic Potentials and Currents motivates comparing invariant charges rather than coordinate representatives; Relativistic Two-Body Dynamics supplies the scattering observables to which the spin data are matched.

Covariant spin and the choice of centroid

Let a body of mass $m$ have momentum $p^\mu$, with $p^2=-m^2$. Its antisymmetric spin tensor $S^{\mu\nu}$ contains six components, three of which describe the freedom to choose a representative point inside the body. The Tulczyjew–Dixon condition removes that freedom covariantly:

$$p_\mu S^{\mu\nu}=0.$$

Using the inherited orientation $\epsilon_{0123}=+\sqrt{-g}$, one mutually consistent definition of the Pauli–Lubanski spin vector and its inverse is

$$S^\mu =- \frac{1}{2m}\epsilon^{\mu\nu\rho\sigma} p_\nu S_{\rho\sigma}, \qquad S^{\mu\nu} = \frac{1}{m}\epsilon^{\mu\nu\rho\sigma} p_\rho S_\sigma .$$

It follows that $p\cdot S=0$ and

$$S_\mu S^\mu = \frac12 S_{\mu\nu}S^{\mu\nu} \ge 0.$$

The mass-rescaled vector $a^\mu=S^\mu/m$ has dimensions of length when $c=1$. It should not be confused with the dimensionless Kerr parameter $\chi=\sqrt{S^2}/(Gm^2)$ or with a particular spatial spin vector in a canonical frame.

Changing the centroid by $z^\mu\to z^\mu+\xi^\mu$ changes the spin representative according to

$$S^{\mu\nu} \longrightarrow S^{\mu\nu}-2\xi^{[\mu}p^{\nu]},$$

so that $J^{\mu\nu}=2z^{[\mu}p^{\nu]}+S^{\mu\nu}$ is unchanged. The impact parameter changes at the same time. Thus two formulas written with different spin supplementary conditions can disagree in their raw $b^\mu$ and $S^{\mu\nu}$ while predicting the same scattering angle. A meaningful comparison translates the complete set $(z,p,S)$, not just the spin vector.

From quantum spin to a classical body

The little group is the set of Lorentz transformations that leave a chosen four-momentum fixed. In a massive particle’s rest frame these are spatial rotations; their quantum representations use $SU(2)$ and have $2s+1$ spin states. Thus a spin label specifies how a state transforms, while a classical spin vector additionally selects an orientation and magnitude. No spinor-helicity formalism is needed for the following matching argument.

For an irreducible massive quantum state of spin $s$,

$$\widehat S_\mu\widehat S^\mu = \hbar^2s(s+1).$$

A nonzero classical angular momentum therefore requires a correlated limit

$$\hbar\to0, \qquad s\to\infty, \qquad \hbar\sqrt{s(s+1)}\to\sqrt{S^2} \quad\text{fixed}.$$

Keeping $s$ fixed while sending $\hbar$ to zero gives a spinless classical source. This is the spin analogue of writing the transfer as $q^\mu=\hbar\bar q^\mu$: the quantum amplitude is an input whose scaling must be organized before it can represent a macroscopic trajectory.

There is a second subtlety. Incoming and outgoing polarization tensors live in little-group spaces attached to different momenta $p$ and $p+q$. Their components cannot be compared by silently using the same spatial axes. One may parallel them with a specified boost, use covariant wave packets, or divide a matrix element by the appropriate polarization overlap. Wigner rotations produced by this comparison contribute to the classical spin kick. The massive-state construction and the large-spin matrix elements of the Pauli–Lubanski operator are worked out in Guevara, Ochirov, and Vines 2019, §§II–III, pp. 2–5, especially Eqs. (9) and (16)–(21), PDF.

Normalization matters as well. Relativistically normalized one-particle states carry factors of $2E$; a classical wave packet supplies both the localized impact parameter and the averaging over initial spin orientation. A polarization polynomial at one momentum is not, by itself, a covariant classical spin vector.

Multipoles in the three-point amplitude

For an axisymmetric Kerr black hole, the mass moments $M_\ell$ and current moments $J_\ell$ obey

$$M_\ell+iJ_\ell=m(ia)^\ell, \qquad a=\frac{\sqrt{S^2}}{m}.$$

Consequently $M_0=m$, $J_1=ma=\sqrt{S^2}$, and $M_2=-ma^2$. This compact no-hair relation and its use in a direct 1PM two-black-hole calculation appear in Vines 2018, §I, pp. 1–3, especially Eq. (1), PDF. The scalar moments above assume an axisymmetric spin direction. With general spin orientation, one uses symmetric trace-free tensors built from $a^\mu$ and the projector orthogonal to $p^\mu$.

The same tower appears in an on-shell three-point coupling. Suppressing helicity-dependent prefactors, its spin dependence can be organized schematically as

$$\frac{\mathcal M_3^{(s)}}{\mathcal M_3^{(0)}} = \sum_{n=0}^{2s}\frac{C_n}{n!}X^n, \qquad X=-i\frac{k_\mu\varepsilon_\nu S^{\mu\nu}} {p\cdot\varepsilon}.$$

The expression is gauge invariant on the special three-point kinematics even though the displayed ratio contains the polarization vector $\varepsilon^\mu$. A finite-spin representation truncates at $n=2s$. In the correlated infinite-spin limit, minimally coupled states associated with Kerr have $C_n=1$ and produce $e^X$. Guevara, Ochirov, and Vines derive the exponential three-point form and explain both the finite-$s$ truncation and the Kerr large-spin limit in Guevara, Ochirov, and Vines 2019, §2.2, pp. 9–11, PDF.

For a generic compact body the coefficients are physical finite-size data. Even and odd powers correspond to spin-induced electric- and magnetic-type multipoles, with independent Wilson coefficients such as $C_{ES^2}$ and $C_{BS^3}$. Setting all $C_n=1$ is a black-hole specialization, not a consequence of Lorentz invariance. Tidal response introduces further operators that cannot be inferred from the permanent Kerr multipoles.

Three-point data also do not determine every four-point interaction. Factorization fixes residues on physical poles, but local Compton contact terms can remain. At sufficiently high spin those terms are required by locality, gauge invariance, or the chosen finite-size theory. A loop calculation that glues only three-point factors must establish that the omitted contact terms cannot affect the stated classical order.

From amplitude data to impulse and spin kick

The observable output is the asymptotic change of momentum and spin, not the coefficients $C_n$ or an integrand numerator. For body 1 write

$$p_1^\mu\longrightarrow p_1^\mu+\Delta p_1^\mu, \qquad S_1^\mu\longrightarrow S_1^\mu+\Delta S_1^\mu.$$

Exact preservation of the mass, spin length, and spin transversality gives

$$\begin{aligned} 2p_1\cdot\Delta p_1+(\Delta p_1)^2&=0,\\ 2S_1\cdot\Delta S_1+(\Delta S_1)^2&=0,\\ p_1\cdot\Delta S_1 +S_1\cdot\Delta p_1 +\Delta p_1\cdot\Delta S_1&=0. \end{aligned}$$

At first order in $G$ the quadratic terms drop. These three linear relations are sensitive tests of signs and of the Wigner-rotation contribution. Momentum conservation requires $\Delta p_1+\Delta p_2=0$ for conservative elastic scattering, while total angular momentum conservation balances the two spin kicks against the change in orbital angular momentum.

At 1PM order, the all-spin Kerr impulse and spin kick obtained from direct general relativity satisfy precisely these mass-shell, spin-length, and transversality checks; see Vines 2018, §II.G–H, pp. 9–11, Eqs. (54)–(59), PDF. Agreement with those covariant results is stronger evidence for an amplitude dictionary than agreement of an intermediate potential, because the latter can change under canonical transformations and SSC redefinitions.

The loop problem adds the same separation encountered for nonspinning bodies. Super-classical box terms describe iterations of lower-order spin-dependent exchange. Infrared phases cancel or exponentiate only after the complete observable is assembled. Spin-dependent triangle and contact contributions are candidates for new dynamics, but they must survive the iteration subtraction and the covariant checks above.

Common pitfalls

Identifying a spin label with a classical vector. The quantum number $s$ becomes a finite classical angular momentum only in the correlated large-$s$, small-$\hbar$ limit. Polarization transport and normalization are part of the dictionary.

Calling every minimal coupling Kerr. The exponential Kerr tower follows under a specific large-spin limit and state construction. Generic compact bodies have independent multipole and tidal coefficients.

Comparing different centroids component by component. An SSC change shifts both the spin and the worldline. Compare invariant impulses, angles, and total angular momentum after translating the full phase-space variables.

Exercise

Use the Kerr relation $M_\ell+iJ_\ell=m(ia)^\ell$ to find the monopole, spin dipole, mass quadrupole, and current octupole. Then explain why the same four values do not follow for an arbitrary rotating neutron star.

Solution

Expanding successive powers of $i$ gives

$$M_0=m, \qquad J_1=ma, \qquad M_2=-ma^2, \qquad J_3=-ma^3.$$

The complementary moment at each displayed order vanishes because equatorial reflection symmetry makes even-$\ell$ current moments and odd-$\ell$ mass moments zero. Since $S=ma$, the quadrupole may also be written $M_2=-S^2/m$.

These relations express the Kerr no-hair multipole sequence. Lorentz symmetry and the spin supplementary condition determine how moments transform but not their numerical response coefficients. A neutron star’s equation of state changes its spin-induced quadrupole and higher moments, and tidal fields induce additional multipoles. Its on-shell description therefore needs Wilson coefficients that are not all unity.

The next page tests a different dictionary: when gauge-theory numerators can be replaced by kinematic numerators to obtain gravitational observables. Continue to Double Copy and Classical Gravitational Observables.

References

• Guevara, Alfredo, Alexander Ochirov, and Justin Vines, “Black-Hole Scattering with General Spin Directions from Minimal-Coupling Amplitudes”, Physical Review D 100, 104024 (2019); Open PDF.
• Guevara, Alfredo, Alexander Ochirov, and Justin Vines, “Scattering of Spinning Black Holes from Exponentiated Soft Factors”, Journal of High Energy Physics 2019, 056 (2019); Open PDF.
• Vines, Justin, “Scattering of Two Spinning Black Holes in Post-Minkowskian Gravity, to All Orders in Spin, and Effective-One-Body Mappings”, Classical and Quantum Gravity 35, 084002 (2018); Open PDF.