Dixon-Quadrupole Term in MPD Equations

• Dixon-quadrupole term is the explicit quadrupolar contribution in Dixon’s MPD equations, coupling an extended body's quadrupole tensor to spacetime curvature.
• It extends the pole–dipole approximation by incorporating spin-induced and general quadrupole effects, enabling the capture of equation-of-state information for compact objects.
• The term adds key force and torque corrections that modify observable dynamics, including ISCO shifts and tidal interactions in relativistic orbital motion.

The Dixon-quadrupole term is the explicit quadrupolar contribution in Dixon’s covariant multipole expansion of an extended body’s stress-energy tensor, as it appears in the Mathisson–Papapetrou–Dixon (MPD) equations of motion. In the relativistic extended-body literature, the term denotes the coupling of a reduced quadrupole tensor $J^{\alpha\beta\gamma\delta}$ to spacetime curvature and its derivatives, beyond the pole–dipole approximation that keeps only mass monopole and spin dipole. In its force sector it has the characteristic structure $$-\tfrac16 J^{\alpha\beta\gamma\delta}\nabla^\mu R_{\alpha\beta\gamma\delta}$$, while in the spin-evolution sector it appears as $$\tfrac43 J^{\alpha\beta\gamma[\mu}R^{\nu]}{}_{\gamma\alpha\beta}$$; together these terms encode how internal quadrupolar structure modifies the translational and rotational dynamics of compact bodies in curved spacetime (Shahzadi et al., 22 May 2025).

1. Covariant definition within the MPD hierarchy

Dixon’s multipolar construction expands the conserved stress-energy tensor $T^{\mu\nu}$ around a chosen center-of-mass worldline. The lowest moments are the four-momentum $p^\mu$ and the antisymmetric spin tensor $S^{\mu\nu}$; the next nontrivial moment is the quadrupole tensor $J^{\alpha\beta\gamma\delta}$. Truncated at quadrupole order, the MPD system takes the form

$\dot x^\alpha=u^\alpha,$

$$\dot p^\alpha = -\frac12 R^\alpha{}_{\mu\nu\rho}\,S^{\nu\rho}u^\mu -\frac16 J^{\mu\nu\rho\sigma}\nabla^\alpha R_{\mu\nu\rho\sigma},$$

$$\dot S^{\alpha\beta} = p^\alpha u^\beta-u^\alpha p^\beta +\frac43 J^{\mu\nu\rho[\alpha}R^{\beta]}{}_{\rho\mu\nu},$$

with overdots denoting covariant derivatives along the worldline. The first right-hand term in $$-\tfrac16 J^{\alpha\beta\gamma\delta}\nabla^\mu R_{\alpha\beta\gamma\delta}$$0 is the Papapetrou spin-curvature force; the second is the Dixon-quadrupole force. The first term in $$-\tfrac16 J^{\alpha\beta\gamma\delta}\nabla^\mu R_{\alpha\beta\gamma\delta}$$1 is the pole–dipole spin transport term; the second is the Dixon-quadrupole torque (Bini et al., 2015).

The quadrupole tensor has the algebraic symmetries of the Riemann tensor,

$$-\tfrac16 J^{\alpha\beta\gamma\delta}\nabla^\mu R_{\alpha\beta\gamma\delta}$$2

These symmetries are not ancillary: they are what make $$-\tfrac16 J^{\alpha\beta\gamma\delta}\nabla^\mu R_{\alpha\beta\gamma\delta}$$3 the appropriate reduced quadrupole moment in Dixon’s formalism and guarantee that only the curvature-coupled combinations relevant to force and torque appear at this order (Shahzadi et al., 22 May 2025).

Setting $$-\tfrac16 J^{\alpha\beta\gamma\delta}\nabla^\mu R_{\alpha\beta\gamma\delta}$$4 recovers the pole–dipole MPD system. Setting also $$-\tfrac16 J^{\alpha\beta\gamma\delta}\nabla^\mu R_{\alpha\beta\gamma\delta}$$5 reduces the motion to geodesic motion. The Dixon-quadrupole term is therefore the first correction that depends explicitly on the body’s internal structure beyond spin.

2. Quadrupole tensor structure and spin-induced constitutive models

A general Dixon quadrupole can be decomposed relative to a timelike vector $$-\tfrac16 J^{\alpha\beta\gamma\delta}\nabla^\mu R_{\alpha\beta\gamma\delta}$$6 into a stress quadrupole $$-\tfrac16 J^{\alpha\beta\gamma\delta}\nabla^\mu R_{\alpha\beta\gamma\delta}$$7, a mass quadrupole $$-\tfrac16 J^{\alpha\beta\gamma\delta}\nabla^\mu R_{\alpha\beta\gamma\delta}$$8, and a flow quadrupole $$-\tfrac16 J^{\alpha\beta\gamma\delta}\nabla^\mu R_{\alpha\beta\gamma\delta}$$9: $$\tfrac43 J^{\alpha\beta\gamma[\mu}R^{\nu]}{}_{\gamma\alpha\beta}$$0 This decomposition is useful because it separates electric-type and current-type quadrupolar content before one imposes a constitutive model (Shahzadi et al., 22 May 2025).

A widely used specialization is the spin-induced quadrupole model. In one formulation, the stress and flow quadrupoles are set to zero and the mass quadrupole is taken to be

$$\tfrac43 J^{\alpha\beta\gamma[\mu}R^{\nu]}{}_{\gamma\alpha\beta}$$1

with $$\tfrac43 J^{\alpha\beta\gamma[\mu}R^{\nu]}{}_{\gamma\alpha\beta}$$2 a polarizability constant. In the notation of another formulation, the reduced quadrupole tensor is written directly as

$$\tfrac43 J^{\alpha\beta\gamma[\mu}R^{\nu]}{}_{\gamma\alpha\beta}$$3

These are notationally distinct presentations of the same physical idea: the quadrupole is algebraically determined by the spin tensor, so the quadrupole sector carries no new independent dynamical degree of freedom beyond a response parameter (Quyet, 21 Mar 2026).

In these models the quadrupole is quadratic in spin, hence of order $$\tfrac43 J^{\alpha\beta\gamma[\mu}R^{\nu]}{}_{\gamma\alpha\beta}$$4. The response parameter depends on the nature of the compact body. The cited literature gives $$\tfrac43 J^{\alpha\beta\gamma[\mu}R^{\nu]}{}_{\gamma\alpha\beta}$$5 for a Kerr black hole and $$\tfrac43 J^{\alpha\beta\gamma[\mu}R^{\nu]}{}_{\gamma\alpha\beta}$$6 for neutron stars depending on the equation of state; in the $$\tfrac43 J^{\alpha\beta\gamma[\mu}R^{\nu]}{}_{\gamma\alpha\beta}$$7-based notation it gives $$\tfrac43 J^{\alpha\beta\gamma[\mu}R^{\nu]}{}_{\gamma\alpha\beta}$$8 for black holes and $$\tfrac43 J^{\alpha\beta\gamma[\mu}R^{\nu]}{}_{\gamma\alpha\beta}$$9 for neutron stars (Quyet, 21 Mar 2026). This suggests that the Dixon-quadrupole term is the first MPD contribution sensitive to equation-of-state information in a clean, covariant way.

A more general treatment need not be purely spin-induced. In Kerr spacetime, one analysis retains a fully general effective quadrupole tensor and parametrizes it by electric-type STF tensors $T^{\mu\nu}$0 and magnetic-type STF tensors $T^{\mu\nu}$1, with the former interpreted as mass quadrupole components and the latter as current quadrupole components (Bini et al., 2014). In that broader setting, the Dixon-quadrupole term is not reducible to $T^{\mu\nu}$2 alone.

3. Supplementary conditions, centroid dependence, and circular-orbit expansions

The MPD system is not closed until one specifies a spin supplementary condition (SSC), written generically as

$T^{\mu\nu}$3

The two choices emphasized in the recent Kerr circular-orbit literature are the Tulczyjew–Dixon SSC,

$T^{\mu\nu}$4

and the Mathisson–Pirani SSC,

$T^{\mu\nu}$5

These conditions select different centroid worldlines. Because the multipole moments are defined relative to the centroid, the quadrupole sector is SSC-dependent at the level of representation even when one is describing the same physical body (Shahzadi et al., 22 May 2025).

The centroid shift under a change of SSC is represented by

$T^{\mu\nu}$6

This is the mechanism through which SSC changes induce changes in the spin measure and in the decomposition of the Dixon quadrupole into $T^{\mu\nu}$7, $T^{\mu\nu}$8, and related objects (Shahzadi et al., 22 May 2025).

For circular equatorial orbits of extended bodies with spin-induced quadrupole around Kerr, the orbital frequency is expanded as

$T^{\mu\nu}$9

The geodesic term $p^\mu$0 and the linear spin term $p^\mu$1 are pole–dipole contributions. The quadrupole first enters at order $p^\mu$2, through $p^\mu$3. An explicit example given in the cited work shows $p^\mu$4 splitting into a pure pole–dipole piece and a $p^\mu$5-dependent Dixon-quadrupole piece proportional to $p^\mu$6 (Shahzadi et al., 22 May 2025).

The same study shows that, before any centroid corrections, the TD and MP descriptions agree through $p^\mu$7, including the quadrupole contribution, and start differing at $p^\mu$8. After applying the radial centroid shift, the agreement extends to $p^\mu$9 generally; for the black-hole value $S^{\mu\nu}$0, the agreement extends to $S^{\mu\nu}$1 in the zero-order radial-shift scheme, but not to all orders (Shahzadi et al., 22 May 2025). The common misconception that SSC choices become exactly equivalent once one introduces the proper centroid transformation is therefore not supported at quadrupole order.

4. Force, torque, mass evolution, and generic-orbit dynamics

The physical content of the Dixon-quadrupole term is twofold. First, the force term

$S^{\mu\nu}$2

couples the body’s quadrupole to curvature gradients. Second, the torque term

$S^{\mu\nu}$3

couples the same internal structure directly to curvature. The force is the covariant expression of a tidal-gradient interaction; the torque modifies spin transport and, in general, the momentum–velocity relation (Bini et al., 2015).

In a small-spin expansion for a spin-induced quadrupole in Kerr, one convenient quantity is the effective mass shift

$S^{\mu\nu}$4

so that

$S^{\mu\nu}$5

The same analysis shows that the worldline tangent $S^{\mu\nu}$6 and the momentum direction $S^{\mu\nu}$7 are not generally parallel once quadrupole effects are retained. For circular equatorial motion, the misalignment is quadratic in spin and proportional to $S^{\mu\nu}$8; for $S^{\mu\nu}$9, corresponding to a black hole, $J^{\alpha\beta\gamma\delta}$0 and $J^{\alpha\beta\gamma\delta}$1 align at this order (Bini et al., 2015).

Allowing a fully general effective quadrupole in Kerr reveals a sharp difference between electric-type and magnetic-type components. With only diagonal electric quadrupole components, the orbit oscillates around the reference circular geodesic. When off-diagonal or magnetic-type quadrupole components are switched on, the motion can spiral inward or outward and can display spin-flip-like behavior, meaning that the signed spin variable crosses zero and changes sign while remaining axis-aligned in the chosen model (Bini et al., 2014). In that formulation the spin magnitude evolves according to

$J^{\alpha\beta\gamma\delta}$2

so nonconservation of spin is itself a quadrupole effect.

For generic Kerr orbits with a spin-induced quadrupole, numerical integrations show further consequences. The dynamical mass $J^{\alpha\beta\gamma\delta}$3 varies with the radial period, reaching a minimum near perihelion and a maximum near aphelion; the effective spin parameter $J^{\alpha\beta\gamma\delta}$4 oscillates correspondingly. The same study concludes that spin-induced quadrupoles need not be included in extreme-mass-ratio inspirals for spatial gravitational-wave detectors, and that no chaotic orbits were found for extended bodies with physical spins and spin-induced quadrupoles; more specifically, it states that no chaos was found for $J^{\alpha\beta\gamma\delta}$5 even for $J^{\alpha\beta\gamma\delta}$6 (Han et al., 2016). A plausible implication is that the dynamical relevance of the Dixon-quadrupole term is highly regime-dependent: it is structurally indispensable in the theory, but often parametrically subleading in realistic EMRIs.

5. Observable consequences: ISCO shifts and non-geodesic lensing

The quadrupole sector enters strong-field observables at order $J^{\alpha\beta\gamma\delta}$7. For circular equatorial motion in Kerr, the radial effective potential acquires explicit $J^{\alpha\beta\gamma\delta}$8-dependent terms, and the ISCO radius can be expanded as

$J^{\alpha\beta\gamma\delta}$9

with

$\dot x^\alpha=u^\alpha,$0

The ISCO frequency likewise receives a quadratic-in-spin quadrupole correction,

$\dot x^\alpha=u^\alpha,$1

These formulas make the internal-structure parameter $\dot x^\alpha=u^\alpha,$2 observable in principle through strong-field orbital spectroscopy (Bini et al., 2015).

A distinct recent development places the Dixon-quadrupole term inside a Jacobi-metric/Gauss–Bonnet lensing framework for massive spinning particles. In that approach the physical ray $\dot x^\alpha=u^\alpha,$3 is not a geodesic of the Jacobi manifold because the MPD force contains a quadrupole term. The spatial acceleration is split as

$\dot x^\alpha=u^\alpha,$4

and, for the spin-induced model,

$\dot x^\alpha=u^\alpha,$5

The associated geodesic curvature contribution is

$\dot x^\alpha=u^\alpha,$6

so the Dixon-quadrupole term appears directly as a line-integral contribution in the Gauss–Bonnet deflection formula (Quyet, 21 Mar 2026).

In Schwarzschild spacetime, after weak-field and high-energy expansions and straight-line trajectory approximation, the quadrupole correction to the deflection angle becomes

$\dot x^\alpha=u^\alpha,$7

The full leading hierarchy is then

$\dot x^\alpha=u^\alpha,$8

One paper proposes interpreting the resulting $\dot x^\alpha=u^\alpha,$9-dependent splitting of trajectories as a form of gravitational birefringence for spinning test bodies (Quyet, 21 Mar 2026). This suggests a direct conceptual link between the Dixon-quadrupole term and internal-structure diagnostics in strong-field scattering.

6. Scope, approximations, and distinct usages

The presence of the phrase “Dixon-quadrupole term” in a paper using MPD methods does not imply that the explicit quadrupole force and torque are actually retained. In an EMRB study of a millisecond pulsar around a quasi-Kerr massive black hole, the pulsar is treated strictly in the pole–dipole approximation,

$$\dot p^\alpha = -\frac12 R^\alpha{}_{\mu\nu\rho}\,S^{\nu\rho}u^\mu -\frac16 J^{\mu\nu\rho\sigma}\nabla^\alpha R_{\mu\nu\rho\sigma},$$0

with the black-hole quadrupole incorporated through the background metric $$\dot p^\alpha = -\frac12 R^\alpha{}_{\mu\nu\rho}\,S^{\nu\rho}u^\mu -\frac16 J^{\mu\nu\rho\sigma}\nabla^\alpha R_{\mu\nu\rho\sigma},$$1, not through a pulsar quadrupole tensor $$\dot p^\alpha = -\frac12 R^\alpha{}_{\mu\nu\rho}\,S^{\nu\rho}u^\mu -\frac16 J^{\mu\nu\rho\sigma}\nabla^\alpha R_{\mu\nu\rho\sigma},$$2 (Kimpson et al., 2020). In that setting the quadrupole is a property of the spacetime, not an explicit Dixon quadrupole of the test body.

A closely related S-star study around Sgr A$$\dot p^\alpha = -\frac12 R^\alpha{}_{\mu\nu\rho}\,S^{\nu\rho}u^\mu -\frac16 J^{\mu\nu\rho\sigma}\nabla^\alpha R_{\mu\nu\rho\sigma},$$3 likewise treats spin-curvature coupling as the leading-order manifestation of MPD dynamics and models central or cluster quadrupoles through metric multipoles or Newtonian potentials, not through a relativistic test-star quadrupole tensor. Its “quadrupole” observables are therefore structurally analogous to, but not identical with, the Dixon-quadrupole term in the strict MPD sense (Alush et al., 2022).

There is also a separate nonrelativistic usage in the electrostatic finite-size literature. There, “Dixon–quadrupole term” refers to the quadrupolar contribution in the Makov–Payne correction for charged periodic cells,

$$\dot p^\alpha = -\frac12 R^\alpha{}_{\mu\nu\rho}\,S^{\nu\rho}u^\mu -\frac16 J^{\mu\nu\rho\sigma}\nabla^\alpha R_{\mu\nu\rho\sigma},$$4

rather than to the curvature-gradient force $$\dot p^\alpha = -\frac12 R^\alpha{}_{\mu\nu\rho}\,S^{\nu\rho}u^\mu -\frac16 J^{\mu\nu\rho\sigma}\nabla^\alpha R_{\mu\nu\rho\sigma},$$5 of relativistic extended-body dynamics (Rutter, 2023). The two usages share a historical association with multipole expansions, but they concern different tensors, different background fields, and different physical theories.

Within relativistic mechanics, the most precise meaning of the term remains the explicit quadrupole-force and quadrupole-torque sector of the MPD equations. In that sense, the Dixon-quadrupole term marks the transition from spin-curvature dynamics to genuinely structure-dependent motion.