Worldline Multipoles and Covariant Operators

A worldline represents an extended compact body only after its unresolved structure is encoded in a systematic operator expansion. Covariant multipoles couple to curvature and its derivatives; response coefficients distinguish black holes, neutron stars, and other objects. The operator basis must respect reparameterization, diffeomorphism, spin-gauge, parity, and internal symmetries, while redundant terms are removed only through transformations that preserve observables.

Required background. The Gibbons-Hawking-York Boundary Term supplies gravitational action methods, and Weak Metric Perturbations and Gauge Freedom supplies perturbative curvature and gauge translation.

Helpful background. The Einstein Equation fixes the bulk dynamics, Presymplectic Potentials and Currents clarifies variational generators, and Black-Hole Perturbation Theory computes response data that later become Wilson coefficients.

Curvature in the comoving frame

Let $u^\mu$ be the unit tangent to the representative worldline and define the spatial projector

$$P^\mu{}_\nu=\delta^\mu{}_\nu+u^\mu u_\nu.$$

In vacuum, the Weyl tensor carries the local tidal field. Its electric and magnetic parts are

$$E_{\mu\nu} =C_{\mu\alpha\nu\beta}u^\alpha u^\beta,$$ $$B_{\mu\nu} =\frac12 \epsilon_{\mu\alpha\rho\sigma} C^{\rho\sigma}{}_{\nu\beta} u^\alpha u^\beta.$$

With the site’s orientation, both are spatial, symmetric, and trace free:

$$E_{\mu\nu}u^\nu=B_{\mu\nu}u^\nu=0, \qquad E^\mu{}_\mu=B^\mu{}_\mu=0.$$

These properties provide an immediate check on signs and projections. A source with the opposite metric signature or dual convention must be translated before its magnetic-type coupling is used.

For a nonspinning parity-even body, the leading local response action is

$$S_{\mathrm{resp}} =\frac12\int d\tau \left[ C_E E_{\mu\nu}E^{\mu\nu} +C_B B_{\mu\nu}B^{\mu\nu} \right]+\cdots.$$

Since curvature has dimension $L^{-2}$, $d\tau$ has dimension $L$, and the retained-$G$ action has dimension $L^2/G$, the coefficients obey

$$[C_E]=[C_B]=\frac{L^5}{G},$$

or simply $L^5$ after setting $G=1$. Convention-dependent numerical normalizations relate $C_E$ to the tidal deformability $\lambda$; they do not change this power of $G$. Dimensional analysis prevents confusing either coefficient with a Love number or a dimensionless $\Lambda$.

Multipolar couplings

A general extended body carries mass and current multipoles. Schematically,

$$S_{\mathrm{mult}} =-\frac12\int d\tau\,Q^{\mu\nu}E_{\mu\nu} -\frac16\int d\tau\,Q^{\mu\nu\rho} \nabla_\rho E_{\mu\nu} +\cdots.$$

Each multipole is spatial with respect to $u^\mu$ and decomposed into symmetric trace-free components. The spatial derivative expansion is controlled by

$$\frac{R}{\mathcal L_{\nabla E}}\ll1, \qquad \mathcal L_{\nabla E} \sim\frac{|E|}{|\nabla E|},$$

where $R$ is the body scale and $\mathcal L_{\nabla E}$ is the length over which the external curvature varies. A separate curvature-strength scale is $\mathcal L_E\sim(E_{\mu\nu}E^{\mu\nu})^{-1/4}$; the local geometry is weak across the body when $R/\mathcal L_E\ll1$. In a Newtonian companion field, $\mathcal L_{\nabla E}\sim r$, whereas $\mathcal L_E\sim[r^3/(Gm_B)]^{1/2}$. Keeping these scales distinct prevents a weak tidal amplitude from being mistaken for a slowly varying field. An $\ell$-pole coupling is suppressed by the appropriate spatial-gradient ratio, unless an internal resonance enhances its response.

One may treat $Q^{\mu\nu}$ as an algebraic induced multipole,

$$Q^{\mu\nu}=-\lambda E^{\mu\nu},$$

or as a dynamical variable with its own kinetic term and mode frequency. The first is an adiabatic low-frequency limit; the second is required for dynamical tides. Integrating out a dynamical oscillator reproduces a frequency-dependent, generally nonlocal response. Thus a local $E^2$ operator has a domain of validity rather than being an exact description of every tide.

The displayed normalizations also fix the map between the two common local coefficients. Adding $-Q_{\mu\nu}Q^{\mu\nu}/(4\lambda)$ to the $-Q^{\mu\nu}E_{\mu\nu}/2$ coupling and eliminating $Q^{\mu\nu}$ gives $+\lambda E_{\mu\nu}E^{\mu\nu}/4$. Comparison with the earlier $+C_EE_{\mu\nu}E^{\mu\nu}/2$ operator therefore gives

$$C_E=\frac{\lambda}{2}$$

in this page’s convention. A different coefficient is not comparable until both the multipole coupling and the $E^2$ operator normalization are translated.

The covariant operator hierarchy and matching logic are developed in Goldberger and Rothstein 2006, §§II–IV and Levi 2020, §§2.2–3.3, pp. 13–24.

Spin as a worldline degree of freedom

A spinning body carries an antisymmetric tensor $S^{\mu\nu}$ or a Lorentz-frame angular momentum. A spin supplementary condition selects the representative center. In a covariant operator basis, spin-induced multipoles include terms such as

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

where $S^\mu$ is a spatial spin vector defined from $S^{\mu\nu}$ after the supplementary condition and dual convention are fixed. The sign follows from this page’s $E_{\mu\nu}=C_{\mu\alpha\nu\beta}u^\alpha u^\beta$ and its multipole coupling $-Q^{\mu\nu}E_{\mu\nu}/2$: a spin-induced quadrupole $Q^{ij}=-(C_{ES^2}/m)S^{\langle i}S^{j\rangle}$ gives the displayed positive action term. Angle brackets mean symmetric trace-free projection in the rest frame. For Kerr, $C_{ES^2}=1$; material bodies generally differ.

The Newtonian limit checks the convention without relying on an imported action sign. In the companion potential $\Phi_B=-Gm_B/r$,

$$E_{ij}=\partial_i\partial_j\Phi_B =\frac{Gm_B}{r^3}(\delta_{ij}-3n_in_j),$$

so $H_{ES^2}=-L_{ES^2}$ at this leading, velocity-independent order becomes

$$H_{ES^2} =\frac{Gm_BC_{ES^2}}{2mr^3} \left[3(\mathbf n\cdot\mathbf S)^2-\mathbf S^2\right].$$

This matches Spin–Spin and Spin-Induced Multipole Couplings. An opposite convention for the electric curvature reverses the action sign and must be translated along with the quadrupole coupling.

Higher operators describe current quadrupoles and cubic or quartic spin-induced moments. Their coefficients are physical only after the operator and spin normalization are stated. A change of spin variable or center can reshuffle lower multipoles and orbital terms while preserving total charges and scattering observables.

The leading minimal spin coupling can be written through a body-fixed tetrad and angular velocity. Local Lorentz invariance introduces a spin-gauge freedom analogous to the freedom in choosing the centroid. Fixing it too early can obscure constraints or create apparent disagreement between EFT and canonical Hamiltonians.

Redundant and essential operators

Not every covariant scalar belongs to an independent basis. Operators containing the Ricci tensor can often be removed in vacuum using field redefinitions and the lowest-order equations of motion. Acceleration terms can be shifted by a worldline redefinition. Integration by parts relates derivative operators.

Suppose an action changes under

$$z^\mu\mapsto z^\mu+\Delta z^\mu, \qquad g_{\mu\nu}\mapsto g_{\mu\nu}+\Delta g_{\mu\nu}.$$

To first order, the action changes by equations-of-motion terms plus boundary terms. On-shell scattering, binding-energy relations, and other observables remain unchanged when all induced operators and observable definitions are transformed. Off-shell potentials need not.

An operator is essential when it changes an on-shell response that cannot be removed by such a transformation. The coefficient of $E_{\mu\nu}E^{\mu\nu}$, for example, is fixed by a low-frequency quadrupolar response after conventions are matched. The absence of a coefficient in a particular basis is not evidence that the object has no finite-size effect.

Power counting and a sample check

For body $A$ in a binary of separation $r$, the companion’s tidal curvature scales as

$$E_{\mu\nu}\sim\frac{Gm_B}{r^3}.$$

The adiabatic quadrupole interaction of body $A$ is

$$L_{\mathrm{tidal},A} \sim \lambda_A E_{\mu\nu}E^{\mu\nu} \sim \lambda_A\frac{G^2m_B^2}{r^6}.$$

Compared with Newtonian binding energy $L_{\mathrm N}\sim Gm_1m_2/r$, the relative size is

$$\frac{L_{\mathrm{tidal},A}}{L_{\mathrm N}} \sim \frac{\lambda_A Gm_B}{m_A\,r^5}.$$

For $\lambda_A\sim R_A^5/G$, this is $(m_B/m_A)(R_A/r)^5$; the companion contributes the term with $A\leftrightarrow B$. For comparable masses, both scale as $(R/r)^5$. That fifth power independently checks why leading quadrupolar tides enter late in a weak-field expansion even when their accumulated phase can be measurable.

The estimate fails near a mode resonance, when the response coefficient becomes frequency dependent, and near disruption, when a finite number of worldline multipoles no longer captures the body.

Exercises

Verify from Weyl symmetries that $E_{\mu\nu}u^\nu=0$ and $E^\mu{}_\mu=0$ in vacuum.

Solution

Contracting with $u^\nu$ gives

$$E_{\mu\nu}u^\nu =C_{\mu\alpha\nu\beta} u^\alpha u^\beta u^\nu=0,$$

because the last two velocities are symmetric while the Weyl tensor is antisymmetric in $\nu,\beta$. The trace is

$$E^\mu{}_\mu =C^\mu{}_{\alpha\mu\beta}u^\alpha u^\beta=0$$

because the Weyl tensor is trace free.

Using $E_A\sim Gm_B/r^3$ and $\lambda_A\sim R_A^5/G$, recover the $(R/r)^5$ scaling for comparable masses.

Solution

The tidal term of body $A$ scales as

$$L_{\mathrm{tidal},A} \sim\frac{R_A^5}{G}\frac{G^2m_B^2}{r^6} =\frac{Gm_A m_B}{r}\frac{m_B}{m_A} \left(\frac{R_A}{r}\right)^5.$$

Dividing by $L_{\mathrm N}\sim Gm_A m_B/r$ gives $(m_B/m_A)(R_A/r)^5$. For comparable masses, this is $(R/r)^5$ up to normalization factors; the second body’s term has the exchanged labels.

References

• Goldberger, Walter D., and Ira Z. Rothstein. “An Effective Field Theory of Gravity for Extended Objects.” Physical Review D 73 (2006): 104029. DOI.
• Levi, Michèle. “Effective Field Theories of Post-Newtonian Gravity: A Comprehensive Review.” Reports on Progress in Physics 83 (2020): 075901. DOI.