Power Counting and EFT Breakdown

A compact body can be replaced by a worldline only when external fields vary slowly across it and do not resolve its internal dynamics. The central hierarchy is not merely “the bodies are far apart.” Body size, orbital separation, curvature radius, wavelength, spin scale, and internal-mode frequency each generate a distinct expansion parameter. Power counting orders the resulting operators; it also says when the effective theory must stop.

Required background. The Gibbons-Hawking-York Boundary Term supplies the gravitational action, and Weak Metric Perturbations and Gauge Freedom supplies the perturbative fields being expanded.

Helpful background. The Einstein Equation fixes the bulk dynamics, Presymplectic Potentials and Currents clarifies invariant observables, and Black-Hole Perturbation Theory supplies object-scale response data.

The hierarchy behind a worldline

For body $A$, let $R_A$ be its physical or effective size, $\omega_{*,A}$ its lowest relevant internal-mode frequency, and $a_A=S_A/m_A$ its spin length. Let $r$ be a characteristic orbital separation, $\Omega$ an orbital frequency, $\omega_{\rm drive}$ a frequency actually present in the comoving tidal field, and

$$\mathcal R_A \sim \left(\frac{Gm_B}{r^3}\right)^{-1/2}$$

the curvature scale of body $B$‘s external tidal field at body $A$. A controlled local worldline expansion requires

$$\epsilon_R=\frac{R_A}{r}\ll1, \qquad \epsilon_{\mathcal R,A}=\frac{R_A}{\mathcal R_A}\ll1, \qquad \epsilon_\omega=\frac{\lvert\omega_{\rm drive}\rvert}{\omega_{*,A}}\ll1, \qquad \epsilon_S=\frac{\lvert a_A\rvert}{r}\ll1.$$

For a slowly moving bound orbit, a separate post-Newtonian parameter is

$$\epsilon_v=v^2\sim\frac{GM}{r}\ll1.$$

These parameters answer different questions. $\epsilon_v$ controls relativistic orbital corrections. $\epsilon_R$ controls spatial multipoles. $\epsilon_\omega$ controls the derivative expansion of dynamical response. $\epsilon_{\mathcal R,A}$ controls the strength of external curvature on the body scale; spatial variation is instead controlled by $R_A/r$ (or, more generally, by $R_A|\nabla E|/|E|$). A compact black hole can have strong self-gravity $Gm_A/R_A\sim1$ while all four external expansion parameters remain small. Worldline EFT does not expand in the body’s internal gravitational field.

Spatial multipole expansion and temporal derivative expansion require separate windows,

$$\frac1r\ll\mu_k\ll\frac1{R_A}, \qquad \lvert\omega_{\rm drive}\rvert\ll\mu_\omega\ll\omega_{*,A}.$$

Here $\mu_k$ and $\mu_\omega$ denote intermediate wave-number and frequency scales. A nonrelativistic internal mode need not satisfy $\omega_*\gt1/r$; a single common cutoff window would impose that unnecessary condition. If a relevant mode cannot be integrated out in time, it can remain an explicit worldline degree of freedom while the spatial expansion remains valid. For a nonrotating body in a circular binary, the oscillating quadrupole has $\omega_{\rm drive}=2\Omega$; eccentricity introduces higher harmonics, and rotation shifts the frequency measured in the body’s frame.

The systematic separation of body, orbital, and radiation scales is explained in Goldberger and Rothstein 2006, §§II–IV.

Operator size from dimensions and symmetry

The point-mass term scales as $S_{\mathrm{pp}}\sim m_A\int d\tau$. A quadrupolar electric operator has the form

$$S_{E^2} =\frac12 C_E \int d\tau\,E_{\mu\nu}E^{\mu\nu}, \qquad [C_E]=\frac{\text{length}^5}{G}.$$

For a material body, write $C_E\sim k_2R_A^5/G$, where the dimensionless response $k_2$ depends on the object’s structure and on normalization. In a weak companion field $E\sim Gm_B/r^3$, the leading quadrupolar correction relative to Newtonian orbital energy scales schematically as

$$\frac{\Delta E_{\mathrm{tidal}}}{E_{\mathrm N}} \sim k_2 \left(\frac{R_A}{r}\right)^5 \frac{m_B}{m_A},$$

up to a convention-dependent numerical factor and the contribution with $A\leftrightarrow B$. Higher multipoles add further powers of $R_A/r$; derivatives add powers of $\lvert\omega_{\rm drive}\rvert/\omega_*$ or $R_A/r$.

Spin-induced multipoles follow a different count because powers of the spin length can replace derivatives. For a Kerr black hole $\lvert a_A\rvert\le Gm_A$, so $\epsilon_S$ is bounded by the same orbital compactness scale as $Gm_A/r$. A material body may have an additional small rotation parameter or a structure-dependent spin-induced coefficient. One must not infer a neutron-star coefficient from black-hole power counting.

Operator classification, matching, and finite-size scaling are reviewed in Porto 2016, §§6.4–6.6, pp. 31–39 and Levi 2020, §§2.2–3.3, pp. 13–24.

Comparing two concrete binaries

Consider first an equal-mass black-hole binary with total mass $M$, component mass $m=M/2$, and separation $r=20GM$. Taking the body scale as $R_A\sim Gm=GM/2$,

$$\epsilon_v\simeq\frac1{20}=0.05, \qquad \epsilon_R\simeq\frac1{40}=0.025,$$

and Kepler scaling gives

$$\Omega Gm \sim \frac{1}{2\,20^{3/2}} \simeq5.6\times10^{-3}.$$

Thus the body-scale derivative expansion is substantially better than the leading orbital relativistic expansion at this separation. This does not guarantee that a truncated post-Newtonian series is accurate to $5\%$, nor that every resummation is reliable; it only compares the natural small parameters.

Now contrast an equal-mass neutron-star binary with $m_A=m_B=1.4M_\odot$, $R_A=12\,\mathrm{km}$, and $r=100\,\mathrm{km}$. Using $Gm_A\simeq2.07\,\mathrm{km}$,

$$\epsilon_v\simeq0.041, \qquad \epsilon_R=0.12.$$

A self-gravity estimate $\omega_*^2\sim Gm_A/R_A^3$ gives

$$\left(\frac{\Omega}{\omega_*}\right)^2 \sim \frac{M}{m_A} \left(\frac{R_A}{r}\right)^3 \simeq3.5\times10^{-3}.$$

For the oscillating circular quadrupole, this gives $2\Omega/\omega_*\simeq0.118$, rather than $\Omega/\omega_*\simeq0.059$. Both are small in this estimate, but the factor two matters when locating a resonance. The spatial-size parameter is larger than in the black-hole example, so finite-size effects can be relevant before the temporal derivative expansion fails. Precise neutron-star mode frequencies and Love coefficients must be matched from the material model rather than this dimensional estimate.

Where the expansion breaks

Different boundaries invalidate different approximations:

• Contact or disruption: $R_A/r$ is no longer small, or the companion’s tidal acceleration across the body becomes comparable with self-gravity, $(m_B/m_A)(R_A/r)^3\sim1$. A finite tower of local multipoles cannot describe mass shedding.
• Dynamical resonance: a tidal harmonic satisfies $\lvert\omega_{\rm drive}\rvert\simeq\omega_*$. The response develops a pole, and the resonant mode must be kept explicitly as in matching compact objects.
• Strong orbital field: $GM/r$ ceases to be small. The worldline description may still be useful, but post-Newtonian power counting must be replaced or resummed; these are logically separate limits.
• Unresolved dissipation: the driving frequency reaches a nonanalytic threshold or a narrow damped mode not represented by the matched correlator. The remedy is the causal construction of dissipative worldline degrees of freedom.
• Loss of scale separation: an external spatial or temporal scale reaches the corresponding unresolved internal scale. Coefficient running cannot restore that missing hierarchy.

The first parameter to reach unity determines the failure mechanism. There is therefore no universal “EFT breakdown radius” shared by black holes, neutron stars, eccentric encounters, and rapidly rotating bodies.

Exercises

For the neutron-star example, estimate the separation at which the simple tidal-disruption parameter $(m_B/m_A)(R_A/r)^3$ reaches unity. Explain why the result must not be used as a calibrated disruption prediction.

Solution

For equal masses the estimate gives $r=R_A=12\,\mathrm{km}$, which lies inside the two-star contact separation $2R_A=24\,\mathrm{km}$. Contact therefore intervenes first. The one-line estimate ignores relativistic gravity, stellar deformation, spin, the equation of state, and the companion’s finite size, so it identifies a scaling competition rather than a physical threshold.

At $r=20GM$, compare the curvature parameter $\epsilon_{\mathcal R,A}=R_A/\mathcal R_A$ with $\epsilon_R$ for either black hole in the equal-mass example.

Solution

Since $m_B=M/2$, $\mathcal R_A\sim[r^3/(Gm_B)]^{1/2}$, and $R_A=GM/2$,

$$\epsilon_{\mathcal R,A} \sim \frac{1}{2\sqrt{2}\,20^{3/2}} \simeq4.0\times10^{-3},$$

whereas $\epsilon_R=1/40=0.025$. The tidal field changes by a small fraction across the body, and the curvature expansion is better than the simple size-to-separation expansion in this example.

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.