PN, PM, Self-Force, and EOB Regimes

Post-Newtonian (PN), post-Minkowskian (PM), gravitational self-force, and effective-one-body (EOB) calculations organize different limits of the relativistic two-body problem. Their labels are not interchangeable accuracy grades: a useful comparison must name the quantity expanded, the parameter held fixed, the observable, and the region in which the remainder is controlled.

Helpful background. Differential equations for relativity supplies the initial-value, boundary-value, and asymptotic language used below.

Four organizations of the two-body problem

Let $m_1\ge m_2$, $M=m_1+m_2$, and

$$\epsilon_m\equiv\frac{m_2}{m_1}, \qquad \nu\equiv\frac{m_1m_2}{M^2}, \qquad 0\lt\nu\le\frac14.$$

For a quasi-circular orbit, the standard invariant frequency parameter is

$$x\equiv\left(\frac{GM\Omega}{c^3}\right)^{2/3}.$$

In the weak-field circular limit, $x\sim v^2/c^2\sim GM/(rc^2)$. PN theory expands in this slow-motion, weak-field parameter while retaining arbitrary mass ratio. Its near-zone expansion must be matched to the radiative exterior; it is not obtained merely by truncating a far-zone wave solution. PM theory instead orders the field in powers of $G$ at fixed relativistic kinematics—fixed asymptotic velocities, energy, and impact parameter in a scattering formulation—so a PM order does not imply a PN order. These distinct expansion domains and their matching are set out in Blanchet 2024, § 1.1.1 and Eq. (1).

Self-force theory takes $\epsilon_m\ll1$, equivalently $\nu\ll1$ at leading order, while allowing the smaller body to move in the strong field of the larger one. The perturbative order in mass ratio and the long inspiral-time expansion are separate: a locally small force can accumulate an order-one phase change, and a resonance can require its own time scaling Barack and Pound 2019, §§ 5.2–5.3, printed pp. 36–42. The body scale, external curvature scale, buffer region, gauge, matching variables, and remainder must therefore be declared together, as emphasized by Pound and Wardell 2024, § 5.1, printed pp. 40–42.

EOB is a map and resummation framework. It rewrites two-body dynamics as an effective problem and can incorporate PN, PM, self-force, and numerical-relativity information. It is not one universal expansion with a single small parameter, so “EOB order” alone does not specify an error estimate.

Organization	Quantity ordered	Held fixed	Typical controlled domain
PN	powers of $x^{1/2}\sim v/c$	mass ratio and near-zone variables	slow motion and weak interbody field
PM	powers of $G$	relativistic scattering kinematics	weak interaction or deflection without a slow-motion assumption
Self-force	powers of $\epsilon_m$ or $\nu$	background strong-field orbit and a declared gauge class	small mass ratio, with a separate long-time treatment
EOB	mapped and resummed input	chosen potentials, waveform sector, and calibration prescription	the domain justified by those inputs and checks

Overlap requires the same invariant

The clean PN–self-force overlap is a double expansion: PN expands at small $x$ with $\nu$ fixed, whereas self-force expands at small $\nu$ with $x$ fixed. One may compare coefficients only after choosing the same invariant relation—such as binding energy versus $\Omega$, periastron advance versus frequencies, a redshift observable, scattering angle, or asymptotic flux—and matching mass normalization, boundary conditions, and admissible gauge class. Barack and Pound 2019, §§ 7.6 and 8, printed pp. 62–67 explain both the qualified meaning of self-force “invariants” and the complementary PN and small-mass-ratio limits.

The limits need not commute after truncation. Taking $x\to0$ first reorganizes a coefficient differently from taking $\nu\to0$ first; taking a slow-velocity limit of a fixed-kinematics PM result likewise requires matching before it can be identified with a PN truncation. Agreement in an overlap coefficient is a stringent mutual check, but disagreement can signal different variables, gauges, boundary data, or orders rather than a failure of either full framework.

For the detailed hierarchy of scales and accepted conventions, continue to Scales, Invariants, and Regimes of the Two-Body Problem. Current best orders, calibrations, and performance comparisons are deliberately not tabulated here because they change with the calculation and observable.

Common pitfalls

Comparing labels instead of observables. “Fourth order” has no common meaning across PN, PM, and self-force expansions. State the parameter, fixed data, invariant, and remainder before comparing coefficients.

Treating resummation as an error bound. A resummed expression may behave well beyond the visibly small-parameter region, but that behavior must be checked against independent information. The resummation name itself supplies no universal bound.

References

• Barack, Leor, and Adam Pound. “Self-Force and Radiation Reaction in General Relativity.” Reports on Progress in Physics 82 (2019): 016904. DOI
• Blanchet, Luc. “Post-Newtonian Theory for Gravitational Waves.” Living Reviews in Relativity 27 (2024): 4. DOI
• Pound, Adam, and Barry Wardell. “Black Hole Perturbation Theory and Gravitational Self-Force.” In Handbook of Gravitational Wave Astronomy. Springer, 2024. DOI