Worldline Effective Field Theory

Worldline effective field theory replaces each compact body by a trajectory carrying every local operator and causal response channel allowed by symmetry. It does not assume weak internal gravity. It assumes that external fields cannot resolve the object’s size or internal timescales. The coefficients then retain the missing information, while the bulk gravitational field transports interactions across orbital and radiation scales.

Helpful background. The Einstein Equation fixes the bulk field equations, Weak Metric Perturbations and Gauge Freedom supplies the perturbative fields, Presymplectic Potentials and Currents clarifies boundary fluxes, and Black-Hole Perturbation Theory supplies horizon response and absorption data that can be matched onto worldline coefficients.

Enter worldline effective field theory

The construction separates three scales:

$$R_A\ll r\ll\lambda_{\mathrm{rad}},$$

where $R_A$ is a body scale, $r$ the orbital separation, and $\lambda_{\mathrm{rad}}\sim\Omega^{-1}$ a radiation wavelength for a slowly moving source. The first inequality permits a multipole expansion about a worldline. The second supports a separate long-wavelength radiation theory. For relativistic encounters the two orbital scales need not be widely separated, but the body-to-external-scale hierarchy must still be declared.

At the body scale, one integrates out internal structure and matches a worldline action, retaining any low-frequency modes that must remain explicit. At the orbital scale, potential fields generate the binding interaction. At the radiation scale, composite multipoles source waves, radiation reaction, and conservative hereditary tails. Internal absorptive correlators supply an additional dissipation channel. These are successive descriptions of one system and must use the same matching and boundary conditions.

The bulk action uses the site’s metric signature $(-,+,+,+)$ and retains $G$ explicitly. Worldlines are covariant and reparameterization invariant. Every finite-size coefficient is quoted together with its operator normalization, matching scale, and response prescription. Conservative local coefficients are kept distinct from absorptive retarded correlators.

The effective action

A schematic nonspinning action is

$$S =\frac{1}{16\pi G}\int d^4x\,\sqrt{-g}\,R -\sum_A m_A\int d\tau_A +\sum_A S_{\mathrm{finite\ size},A} +S_{\mathrm{diss}}.$$

The local finite-size sector begins with curvature operators such as

$$S_{\mathrm{finite\ size}} \supset \frac12\int d\tau \left( C_E E_{\mu\nu}E^{\mu\nu} +C_B B_{\mu\nu}B^{\mu\nu} \right),$$

after operators proportional to lower-order equations of motion have been removed. Spin requires a body frame, a spin gauge or supplementary condition, and additional multipoles. Dissipation instead introduces internal worldline operators and their retarded correlation functions; a real local coefficient cannot represent irreversible heating.

Symmetry tells us which terms may occur. Power counting says how strongly they are suppressed. Matching fixes their coefficients. Renormalization transports the coefficients between separated scales. An observable is reliable only when all four steps use compatible conventions.

A guided route through the chapter

Read the six pages in order:

1. Point-Particle Actions and Reparameterization Invariance derives covariant motion and the mass-shell constraint before finite size is added.
2. Worldline Multipoles and Covariant Operators classifies spin, tidal, and derivative operators and removes redundant terms.
3. Matching Compact Objects to Wilson Coefficients equates full-object and effective observables, including the response near an internal mode.
4. Dissipative Worldline Degrees of Freedom uses retarded correlators and an in-in influence action for absorption, heating, and fluctuations.
5. Renormalization, Running, and Classical Logarithms shows how scale-dependent pieces cancel in invariant results.
6. Power Counting and EFT Breakdown compares every expansion parameter and identifies contact, resonance, and loss of scale separation.

The sequence moves from kinematics to operator content, then from coefficient determination to causal response, and finally to scale evolution and validity. The same logic will reappear in the post-Newtonian, spin, tidal, and radiation-reaction chapters.

What the coefficients do and do not mean

A Wilson coefficient is not an invariant property in isolation. It depends on the operator basis, field normalization, regulator, and subtraction scheme. For example, a field redefinition can move a term between a bulk interaction and a worldline operator. The binding energy or scattering angle remains unchanged when every induced term is transformed consistently.

Static response, dynamical response, and absorption are also distinct. A static tidal coefficient is the zero-frequency real part of a specified response problem. A dynamical mode produces frequency dependence and poles. Absorption is governed by the imaginary part of a retarded response. The vanishing of one does not establish the vanishing of the others.

For this reason, object-specific calculations remain with Black-Hole Perturbation Theory and Relativistic Matter and Compact Objects. This chapter explains how their outputs enter a two-body action and how their conventions must be translated.

The overall framework is developed in Goldberger and Rothstein 2006, §§II–V, Porto 2016, §§6–8, pp. 27–87, and Levi 2020, §§2–5, pp. 13–54.

A chapter-level consistency test

For any proposed finite-size correction, ask:

1. Which symmetry permits the operator?
2. Which ratio of scales suppresses it?
3. Which full-theory observable fixes its coefficient?
4. Is the response local and conservative, or retarded and dissipative?
5. Which scale and subtraction scheme define the coefficient?
6. Which invariant observable verifies the final result?

An answer that skips one step may still be useful as a parametrization, but it is not yet a matched prediction. Conversely, when every step is supplied, the worldline theory can describe strongly self-gravitating bodies without solving their interiors anew during every orbital calculation.

Check your understanding

Why can a vanishing static tidal coefficient coexist with nonzero absorption?

Solution

The static coefficient probes the real part of a response at zero frequency in a specified convention. Absorption probes the imaginary part of a retarded response at nonzero frequency. Analyticity relates their frequency dependence through dispersion relations, but it does not require both quantities to be nonzero at the same order. A four-dimensional nonrotating black hole provides the standard example: its usual static gravitational Love numbers vanish while its horizon absorbs waves.

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.
• Porto, Rafael A. “The Effective Field Theorist’s Approach to Gravitational Dynamics.” Physics Reports 633 (2016): 1–104. DOI.