Map of Approximation Overlaps

Approximation schemes become most powerful where two of them are simultaneously controlled. An overlap supplies more than reassurance: it identifies shared invariants, fixes otherwise undetermined coefficients, and reveals whether a disagreement is physical, representational, or caused by an inconsistent truncation. This page gives a working method for choosing such comparisons without assuming that every formal double limit commutes.

Required background. The Einstein Equation identifies the common underlying theory, and Weak Metric Perturbations and Gauge Freedom explains why intermediate fields and coordinates cannot be compared directly.

Helpful background. Presymplectic Potentials and Currents supplies invariant charges and canonical transformations, while Black-Hole Perturbation Theory supplies the strong-field small-mass-ratio anchor.

The principal expansion directions

Let $x=(GM\Omega)^{2/3}$ label quasi-circular weak-field motion, $\nu=m_1m_2/M^2$ label mass ratio, and $\alpha=GM/b$ label encounter coupling at fixed asymptotic velocity. Small $\alpha$ alone does not ensure weak deflection when the velocity is also small: the Newtonian angle is controlled by $GM/(bv_\infty^2)$. The main analytic organizations expand along different directions:

Method	Small quantity	Held fixed	Natural outputs
Post-Newtonian	orbital velocity or $x^{1/2}$	mass ratio and scaled orbital data	binding energy, precession, flux, phase
Post-Minkowskian	$G$ or $GM/b$	relativistic asymptotic energy	impulse, scattering angle, radiated momentum
Self-force	small mass ratio	background strong-field orbit	redshift, precession, local force, flux
Worldline EFT	body size over external scale	resolved orbital and radiation scales	Wilson coefficients, forces, response
Effective one body	not one expansion by itself	chosen invariant information and resummation	global effective dynamics and waveform inputs
Numerical relativity	resolution and domain errors	finite physical parameters	nonlinear benchmark observables

The EOB entry is deliberately different. EOB is a map and resummation framework constrained by PN, PM, self-force, test-mass, tidal, and numerical information; calling “EOB” a small-parameter expansion hides which input controls a particular statement.

For each method, the approximation metadata must include the order, variables held fixed, boundary conditions, and observable. “3PM” at fixed Lorentz factor is not the same limit as a “3PN” coefficient, although expanding the former at low velocity can constrain the latter.

PN and self-force

For a nonspinning circular invariant such as reduced binding energy $e=E_b/\mu$, suppressing logarithms for the moment,

$$e(x,\nu)=\sum_{n\geq1}e_n(\nu)x^n.$$

Each coefficient can in turn be expanded around $\nu=0$:

$$e_n(\nu) =e_n^{(0)}+\nu e_n^{(1)} +\nu^2 e_n^{(2)}+\cdots.$$

At higher PN orders terms such as $x^n\ln x$ also occur. Their coefficient functions must be matched separately; a pure power series is not the full PN ansatz.

PN theory at general mass ratio and self-force theory at strong field overlap when both $x\ll1$ and $\nu\ll1$. The test-mass term $e_n^{(0)}$ must agree with geodesic motion. First-order self-force determines $e_n^{(1)}$ and, when known as a function of $x$, constrains strong-field EOB potentials beyond a finite PN series.

The comparison must use a gauge invariant such as $E(\Omega)$, periastron advance, redshift, or spin precession. A coordinate self-force cannot be equated to a PN coordinate acceleration without constructing the gauge map. The role of self-force information across mass ratios is reviewed in Barack and Pound 2019, §8.

A useful consistency check is exchange symmetry. A nonspinning, exchange-symmetric dimensionless observable should be expressible in $\nu$ after fixing its mass normalization, but its re-expansion in $\epsilon=m_2/m_1$ contains higher powers generated kinematically by $\nu=\epsilon/(1+\epsilon)^2$. Those terms must be separated from genuinely higher-order self-force contributions. Body-specific redshifts and spinning observables also require their body labels or spin combinations.

PN and PM

A conservative scattering angle can be organized as

$$\frac{\chi(E,J)}2 =\sum_{n\geq1} \left(\frac{GM\mu}{J}\right)^n \chi_n(\gamma,\nu).$$

This is a PM expansion: the coefficient functions retain arbitrary relative Lorentz factor $\gamma$. Expanding $\gamma$ near unity generates a velocity expansion that must reproduce PN information in the scattering sector. Conversely, a sufficiently high-PN Hamiltonian can be evaluated on a hyperbolic orbit and reorganized in powers of $G$.

The overlap check proceeds in a fixed order:

1. Express both results in the same asymptotic $E$ and $J$.
2. Remove lower-order iterations consistently.
3. Expand the PM coefficient at low velocity or the PN result at fixed PM order.
4. Compare only terms controlled by both truncations.

Scattering-angle and EOB dictionaries through several PM orders are developed in Damour 2020, §§II–IV. That analysis also illustrates why high-energy and weak-coupling limits can fail to commute. A low-velocity match does not validate an extrapolation to $\gamma\to\infty$.

Amplitudes, Hamiltonians, and EOB

Quantum scattering amplitudes provide on-shell information from which classical impulse or scattering data can be extracted. A potential or Hamiltonian obtained in the process is not unique. Born iterations, infrared phases, canonical transformations, and radiation modes must be treated consistently.

The clean chain is

$$\text{amplitude} \longrightarrow \text{classical impulse or }\chi(E,J) \longleftrightarrow \text{radial action} \longrightarrow \text{Hamiltonian or EOB potentials}.$$

The angle is invariant in the specified asymptotic frame. The radial action also needs a long-range subtraction and endpoint convention, and recovering it from the angle leaves a function of energy undetermined. The end representations are not unique. The amplitude–PM relation and the separation of classical terms are reviewed in Bjerrum-Bohr et al. 2022, §§ 2–5. Agreement of an amplitude-derived potential coefficient with an ADM coefficient is unnecessary until a canonical dictionary is fixed.

Double copy can simplify construction of gravitational integrands or classical fields, but it does not automatically preserve every off-shell observable, finite-size coefficient, or global boundary condition. Its validation belongs at the level of a specified gravitational observable.

Self-force, EOB, and numerical relativity

EOB potentials are functions of radius, momentum, spin, and $\nu$. Their small-$\nu$ coefficients can be constrained by self-force invariants. Their finite-$\nu$ strong-field completion can be compared with numerical-relativity binding energy, scattering angle, periastron advance, or waveforms.

Calibration and validation are distinct. If numerical data determine adjustable EOB coefficients, those same data cannot provide an independent accuracy test. A defensible workflow withholds configurations, varies mass ratio, spin, eccentricity, and encounter energy, and propagates numerical extraction and truncation errors into the comparison.

Numerical agreement also cannot cure an analytic inconsistency. Before calibration, the model should reproduce:

• the exact Newtonian limit;
• known PN and PM coefficients;
• the geodesic limit;
• available first-order self-force information;
• symmetry under exchange of bodies;
• the correct spinless and zero-tide limits.

Each condition probes a different direction in parameter space. Passing all of them sharply reduces the chance that a flexible fit is compensating one error with another.

Noncommuting limits and overlap failure

Consider a function

$$F(\alpha,\gamma)=\frac{\alpha\gamma}{1+\alpha\gamma}.$$

At fixed $\gamma$, expansion in $\alpha$ gives $F=\alpha\gamma-\alpha^2\gamma^2+\cdots$. The expansion is not uniform as $\gamma\to\infty$ because the true control parameter is $\alpha\gamma$. This toy example captures a real danger in ultrarelativistic PM scattering: a formally higher-PM term enhanced by energy can compete with lower orders.

Other overlap failures include a PN series near its strong-field singularity, self-force expansion when the smaller body tidally disrupts, adiabatic averaging at resonance, and worldline EFT when body size is comparable to curvature radius. The correct response is to shrink the claimed overlap or introduce a new inner scaling, not to average incompatible answers.

A reproducible overlap test

For any pair of methods, record:

1. the common theory, object model, asymptotics, and conventions;
2. the two small parameters and the region where both are controlled;
3. the common invariant inputs and output;
4. the exact double truncation;
5. source translations and iteration subtractions;
6. at least one limiting check not used to construct the answer;
7. the numerical or analytic uncertainty on each side.

For example, a PN–self-force test might compare the coefficient of $\nu x^5$ in $E(\Omega)$, after expanding both results in the same $\nu$ and $x$. A PM–numerical test might compare $\chi(E,J)$ after subtracting radiated energy and angular momentum consistently. A statement that two “potentials agree” without this metadata is not yet a cross-regime test.

Exercises

Suppose a quantity is known including $x^4$, with remainder $O(x^5)$, at arbitrary $\nu$ from PN theory, and including the term linear in $\nu$, with remainder $O(\nu^2)$, at arbitrary $x$ from self-force theory. For a power-series example without logarithms, which coefficients of

$$F(x,\nu)=\sum_{m,n\geq0}c_{mn}x^m\nu^n$$

are independently available from both methods?

Solution

PN theory supplies $c_{mn}$ for $m\leq4$ at all mass-ratio powers present in its exact coefficient functions. First-order self-force supplies $c_{m0}$ and $c_{m1}$ for all weak-field orders that can be extracted from its strong-field function. The genuine overlap is therefore $c_{m0}$ and $c_{m1}$ with $m\leq4$. Coefficients with $m\gt 4$ come only from self-force, while $n\geq2$ comes only from the finite-mass-ratio PN result in this example.

References

• Barack, Leor, and Adam Pound. “Self-Force and Radiation Reaction in General Relativity.” Reports on Progress in Physics 82 (2019): 016904. DOI.
• Bjerrum-Bohr, N. E. J., Poul H. Damgaard, Ludovic Planté, and Pierre Vanhove. “The SAGEX Review on Scattering Amplitudes, Chapter 13: Post-Minkowskian Expansion from Scattering Amplitudes.” Journal of Physics A 55 (2022): 443014. DOI.
• Damour, Thibault. “Classical and Quantum Scattering in Post-Minkowskian Gravity.” Physical Review D 102 (2020): 024060. DOI.

Dated Research assessment. For open questions and recent results, follow the link below. Each assessment states the date through which it considers results and the limits of its conclusions.

• Consistency of PN, PM, Amplitudes, and Self-Force