Relativistic Spin and Supplementary Conditions

A relativistic extended body has no observer-independent center of mass. Once its stress-energy is compressed into momentum and spin about a reference curve, shifting that curve transfers angular momentum between “orbital” and “spin” pieces. A spin supplementary condition chooses the representative centroid and removes three redundant components of the antisymmetric spin tensor; it is a gauge choice within a controlled multipole expansion, not an extra force law.

Required background. Presymplectic Potentials and Currents supplies the constrained phase-space viewpoint used to identify spin-gauge freedom.

Helpful background. The Einstein Equation fixes curvature dynamics, Weak Metric Perturbations and Gauge Freedom supplies perturbative gauge reasoning, Black-Hole Geometry and Mechanics supplies the Kerr limit, and Relativistic Two-Body Dynamics supplies the orbital expansion.

Momentum, spin, and the choice of centroid

At pole–dipole order, a body is described by a worldline $z^\mu(\tau)$, four-momentum $p^\mu$, and antisymmetric spin tensor $S^{\mu\nu}$. With the site’s curvature convention, the Mathisson–Papapetrou–Dixon equations are

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

where $\tau$ is proper time, $u^\mu=dz^\mu/d\tau$, and $A^{[\mu}B^{\nu]}=(A^\mu B^\nu-A^\nu B^\mu)/2$. These ten evolution equations do not determine the thirteen components in $p^\mu$, $u^\mu$, and $S^{\mu\nu}$ after the worldline normalization is counted. Three centroid conditions are missing Costa and Natário 2015, § 3, Eqs. (14)–(16), PDF.

An SSC supplies them in the covariant form

$$S^{\mu\nu}V_\nu=0$$

for a chosen timelike vector $V^\mu$. Common choices include:

• Tulczyjew–Dixon: $V^\mu=p^\mu$, selecting the zero-momentum centroid;
• Mathisson–Pirani: $V^\mu=u^\mu$, selecting a centroid relative to the instantaneous rest frame and admitting a family of helical representatives unless additional data fix the center;
• Newton–Wigner or canonical choices: $V^\mu$ is built from $p^\mu$ and a preferred timelike frame, simplifying canonical Poisson brackets at the price of manifest covariance.

The SSC determines the relation between momentum and velocity. Even when $p^2=-m^2$, one generally has $p^\mu\not\parallel u^\mu$ because spin–curvature coupling and hidden momentum contribute.

Spin vectors and normalization

Under the Tulczyjew–Dixon condition define

$$S^\mu =-\frac{1}{2m} \epsilon^{\mu\nu\alpha\beta} p_\nu S_{\alpha\beta}, \qquad p_\mu S^\mu=0,$$

with inverse

$$S^{\mu\nu} =\frac{1}{m} \epsilon^{\mu\nu\alpha\beta} p_\alpha S_\beta$$

for the chosen orientation. The scalar magnitude

$$s^2 =\frac12 S_{\mu\nu}S^{\mu\nu} =S_\mu S^\mu$$

is conserved at pole–dipole order in this condition. A dimensionless spin is $\chi=s/(Gm^2)$. These formulas are SSC-specific: inserting a spin vector defined with $u^\mu$ into a Hamiltonian derived with $p^\mu$ silently changes the variable.

Shifting between supplementary conditions

Shift the representative curve by a small spacelike displacement $\xi^\mu=O(s/m)$. In flat spacetime the total angular momentum about a fixed origin remains unchanged under the following transformation. In curved spacetime it is the local pole–dipole transformation: identify neighboring tangent spaces by parallel transport and interpret the worldline shift with the exponential map, rather than treating separated coordinate components as vectors at one point:

$$z'^\mu=z^\mu+\xi^\mu, \qquad S'^{\mu\nu} =S^{\mu\nu}+2p^{[\mu}\xi^{\nu]}, \qquad p'^\mu=p^\mu+O(Rs\xi).$$

Contracting the new spin with a target vector $V'_\nu$ gives

$$S'^{\mu\nu}V'_\nu =S^{\mu\nu}V'_\nu +p^\mu(\xi\cdot V') -\xi^\mu(p\cdot V').$$

Choosing the displacement orthogonal to $V'^\mu$ yields

$$\xi^\mu =\frac{S^{\mu\nu}V'_\nu}{p\cdot V'}$$

through this order and enforces the new SSC. Components parallel to $p^\mu$ amount to reparameterization and must be fixed consistently.

The centroid shift changes coordinate positions and canonical spins, and it induces corresponding transformations in the orbital Hamiltonian and multipoles. It cannot change a fully transformed scattering angle, energy–frequency relation, or asymptotic waveform. The geometric meaning of the SSC and centroid shifts is explained in Costa and Natário 2015, §§2–4; the worldline-EFT implementation is reviewed in Levi 2020, §§3.2–4, pp. 18–42.

A practical comparison protocol

Before comparing two spin results, record the representative curve, SSC, spin-tensor or spin-vector definition, time variable, body frame, multipole normalization, and truncation order. Then:

1. transform both centroid and spin;
2. include induced higher-order terms from the variable change;
3. compare an invariant at a common order;
4. take the zero-spin and test-body limits.

Apparent SSC dependence usually signals an incomplete transformation or inconsistent truncation. It can become physical only when two descriptions select genuinely different bodies or internal multipoles.

The pole–dipole approximation requires body size and Møller radius $s/m$ to be small compared with curvature and orbital scales. Large shifts outside that worldtube are not gauge-equivalent centroids of the same truncated body.

Exercises

Verify that the centroid shift preserves total angular momentum in flat spacetime.

Solution

With $J^{\mu\nu}=2z^{[\mu}p^{\nu]}+S^{\mu\nu}$,

$$J'^{\mu\nu} =2(z+\xi)^{[\mu}p^{\nu]} +S^{\mu\nu}+2p^{[\mu}\xi^{\nu]}.$$

Since $2\xi^{[\mu}p^{\nu]}+2p^{[\mu}\xi^{\nu]}=0$, one obtains $J'^{\mu\nu}=J^{\mu\nu}$. The split changes, while the total does not.

Count the independent spin degrees of freedom after an SSC is imposed.

Solution

An antisymmetric tensor in four dimensions has six components. The four relations $S^{\mu\nu}V_\nu=0$ contain only three independent conditions because contraction with $V_\mu$ vanishes identically. Three components remain. Fixing the spin magnitude leaves two orientation degrees of freedom.

References

• Costa, L. Filipe O., and José Natário. “Center of Mass, Spin Supplementary Conditions, and the Momentum of Spinning Particles.” In Equations of Motion in Relativistic Gravity (2015). Open preprint.
• Levi, Michèle. “Effective Field Theories of Post-Newtonian Gravity: A Comprehensive Review.” Reports on Progress in Physics 83 (2020): 075901. DOI.