Multipolar Equations of Motion

Stress-energy conservation turns a compact body’s extended distribution into a hierarchy of covariant multipole moments about a representative worldline. At monopole order the curve is geodesic. At dipole order spin couples to curvature. At quadrupole and higher orders, curvature gradients exert force and tidal fields exert torque. The hierarchy is controlled only while the body and spin lengths are small compared with the external curvature scale.

Required background. Presymplectic Potentials and Currents supplies the constrained covariant phase-space setting.

Helpful background. The Einstein Equation gives $\nabla_\nu T^{\mu\nu}=0$, Weak Metric Perturbations and Gauge Freedom supplies multipolar gauge control, Black-Hole Geometry and Mechanics supplies curved backgrounds, and Relativistic Two-Body Dynamics supplies matching and power counting.

From conservation to moments

Choose a timelike reference curve $z^\mu(\tau)$ inside the body’s worldtube and spatial hypersurfaces intersecting it. Covariant moments of $T^{\mu\nu}$ are defined using separation bitensors so that tensors at nearby points are compared at $z^\mu$. Their precise normalization depends on the moment convention; the equations below use Dixon’s quadrupole tensor $J^{\alpha\beta\gamma\delta}$ with the algebraic symmetries of the Riemann tensor. In particular, $J^{\alpha\beta\gamma\delta}$ is antisymmetric in each index pair, symmetric under pair exchange, and obeys $J^{\alpha[\beta\gamma\delta]}=0$; it has 20 independent components in four dimensions. This is a stress-energy quadrupole, not just the five components of a spatial STF mass quadrupole Dixon 1970, §§ 4–5.

Truncating stress-energy conservation at quadrupole order gives

$$\frac{Dp^\mu}{d\tau} =-\frac12 R^\mu{}_{\nu\alpha\beta} u^\nu S^{\alpha\beta} -\frac16 J^{\alpha\beta\gamma\delta} \nabla^\mu R_{\alpha\beta\gamma\delta} +O(\text{octupole}),$$ $$\frac{DS^{\mu\nu}}{d\tau} =2p^{[\mu}u^{\nu]} +\frac43 J^{\alpha\beta\gamma[\mu} R^{\nu]}{}_{\gamma\alpha\beta} +O(\text{octupole}).$$

The pole–dipole terms are the Mathisson–Papapetrou–Dixon system. The quadrupole force samples a curvature gradient; the quadrupole torque samples curvature itself. Numerical factors change if $J$ is rescaled, so a moment must never be imported without its defining action or stress-energy decomposition.

Here $u^\mu=dz^\mu/d\tau$, $u^2=-1$, and antisymmetrization has weight $1/2$. The signs and moment normalization above agree with Chen et al. 2019, § 2.1, footnote 4, and § 2.5, Eqs. (2.19)–(2.20), PDF, which explicitly uses the site’s Riemann convention. The multipolar derivation and translation to Dixon’s moments are given in Steinhoff and Puetzfeld 2010, §§ VI–VII.

Closing the hierarchy

The equations do not predict $J^{\alpha\beta\gamma\delta}$ from kinematics. A constitutive or matching model closes the system. Examples include:

• a spin-induced quadrupole constructed from $S^\mu S^\nu$ with coefficient $\kappa$;
• an adiabatic tidal quadrupole proportional to the electric curvature $E_{\mu\nu}$;
• a dynamical quadrupole obeying an internal oscillator equation;
• a retarded response with an imaginary part for absorption.

In a local worldline action,

$$S_{\mathrm{body}} =-m\int d\tau +S_{\mathrm{spin}} +\int d\tau \left[ +\frac{\kappa}{2m}E_{\mu\nu}S^\mu S^\nu +\frac{\lambda_E}{4}E_{\mu\nu}E^{\mu\nu} +\cdots \right],$$

we use $E_{\mu\nu}=C_{\mu\alpha\nu\beta}u^\alpha u^\beta$, $Q_{\rm spin}^{ij}=-\kappa S^{\langle i}S^{j\rangle}/m$, and the coupling $-Q^{ij}E_{ij}/2$. These fix the positive sign of the spin-induced operator. The tidal term follows by eliminating $Q_{\rm tidal}^{ij}=-\lambda_E E^{ij}$ including its elastic energy; its coefficient is $\lambda_E/4$ in the tidal chapter’s convention. Variation supplies compatible force and torque. A generic quadrupole also needs the supplementary condition and a momentum–velocity relation; the displayed action is an example of closure, not a replacement for those constraints.

Reduction checks

Pole–dipole limit. Setting $J^{\alpha\beta\gamma\delta}=0$ recovers

$$\frac{Dp^\mu}{d\tau} =-\frac12 R^\mu{}_{\nu\alpha\beta}u^\nu S^{\alpha\beta}, \qquad \frac{DS^{\mu\nu}}{d\tau}=2p^{[\mu}u^{\nu]}.$$

An SSC is still required to select the centroid and determine the momentum–velocity relation.

Spinless monopole limit. Set $S^{\mu\nu}=0$ and all higher moments to zero. The torque equation implies $p^{[\mu}u^{\nu]}=0$, so $p^\mu=mu^\mu$. If $m$ is constant, the force equation becomes

$$u^\nu\nabla_\nu u^\mu=0.$$

Thus geodesic motion follows as the final consistent truncation.

Flat-spacetime limit. With $R_{\alpha\beta\gamma\delta}=0$, the force and quadrupole torque vanish. Momentum is constant and total angular momentum is conserved, though a poorly fixed Mathisson–Pirani centroid can still trace a helical representative inside the same worldtube.

These reductions detect unwanted forces and missing terms, but cannot test the coefficients of terms that vanish in every chosen limit. A nonzero curvature-gradient check and a symmetry charge are needed as well.

Power counting and validity

Let $R_{\mathrm b}$ be a body scale, $\mathcal R^{-2}$ a typical curvature component in the body frame, and $\mathcal L_\nabla$ the curvature-variation scale, so that $|\nabla R_{\alpha\beta\gamma\delta}|\sim 1/(\mathcal R^2\mathcal L_\nabla)$. Natural mass moments scale as $I_\ell\sim mR_{\mathrm b}^\ell$. The force per unit mass scales as

$$\frac{|F_{\rm quad}|}{m} \sim\frac{R_{\mathrm b}^2}{\mathcal R^2\mathcal L_\nabla}, \qquad \frac{|F_{\rm spin}|}{m} \sim\frac{s/m}{\mathcal R^2}.$$

There is no covariant ratio to a nonzero “monopole acceleration”: a geodesic has zero proper acceleration. In a Newtonian companion field one may instead compare coordinate accelerations in the same frame. There $\mathcal R^{-2}\sim GM_c/r^3$, $\mathcal L_\nabla\sim r$, and $a_{\rm N}\sim GM_c/r^2$, giving

$$\frac{a_{\rm quad}}{a_{\rm N}} \sim\left(\frac{R_{\mathrm b}}{r}\right)^2.$$

Successive multipoles add powers of body size over the scale on which curvature changes. The hierarchy breaks near material disruption, when an internal mode is resonantly excited but not retained, or when a claimed centroid shift is comparable with the curvature scale.

A pole–quadrupole truncation must also evolve or prescribe the quadrupole in a way compatible with its symmetries. Arbitrarily freezing its coordinate components can inject spurious work and torque. A matched body-frame constitutive law should recover energy and angular-momentum balance in a stationary or axisymmetric background.

Conserved quantities in symmetric backgrounds

For a Killing vector $\xi^\mu$, Dixon’s momentum and spin give the charge

$$\mathcal C_\xi =p_\mu\xi^\mu +\frac12S^{\mu\nu}\nabla_\mu\xi_\nu.$$

It remains conserved at quadrupole order; no extra explicit quadrupole term is needed in this convention. Differentiate it, insert both MPD equations, and use $\nabla_{(\mu}\xi_{\nu)}=0$ and the Killing identity. The pole–dipole terms cancel. The remaining terms combine into

$$\frac{d\mathcal C_\xi}{d\tau} =F_\mu\xi^\mu +\frac12N^{\mu\nu}\nabla_\mu\xi_\nu =-\frac16J^{\alpha\beta\gamma\delta} \mathcal L_\xi R_{\alpha\beta\gamma\delta}=0,$$

where $F$ and $N$ are the quadrupole force and torque. A Killing symmetry preserves curvature, so its Lie derivative vanishes. This conclusion does not require a rigid or constant quadrupole; it requires evolving the matching force and torque consistently. In fact the same Dixon charge is conserved with arbitrary higher moments Chen et al. 2019, § 2.3, Eq. (2.15), PDF. Energy $-\mathcal C_{\partial_t}$ and axial angular momentum $\mathcal C_{\partial_\phi}$ are therefore strong checks for motion in a stationary axisymmetric background. They are distinct from the generally variable mass $\sqrt{-p^2}$.

Exercises

Derive geodesic motion from the spinless monopole truncation.

Solution

With $S=J=0$, the torque equation gives $p^\mu=mu^\mu$. The force equation gives

$$0=\frac{D(mu^\mu)}{d\tau} =\dot m\,u^\mu+m\,\frac{Du^\mu}{d\tau}.$$

Contracting with $u_\mu$ and using $u^2=-1$ gives $\dot m=0$. The remaining orthogonal part is $Du^\mu/d\tau=0$.

Why is the quadrupole force sensitive to spatial variation of curvature while the torque is not?

Solution

A force compares how the interaction energy changes as the body’s reference point is displaced, introducing a gradient of the tidal field. A torque compares how the quadrupole orientation changes at a fixed point and therefore couples directly to curvature. The displayed equations implement these translation and rotation variations.

References

• Chen, Bin, Geoffrey Compère, Yan Liu, Jiang Long, and Xuao Zhang. “Spin and Quadrupole Couplings for High Spin Equatorial Intermediate Mass-Ratio Coalescences.” Classical and Quantum Gravity 36, 245011 (2019). DOI. Open PDF.
• Dixon, W. G. “Dynamics of Extended Bodies in General Relativity. I. Momentum and Angular Momentum.” Proceedings of the Royal Society A 314 (1970): 499–527. DOI.
• Steinhoff, Jan, and Dirk Puetzfeld. “Multipolar Equations of Motion for Extended Test Bodies in General Relativity.” Physical Review D 81 (2010): 044019. DOI.