PN–PM Matching

Post-Newtonian (PN) and post-Minkowskian (PM) calculations must agree coefficient by coefficient only after they are expressed in the same invariant variables and restricted to their common asymptotic domain. PN theory expands in slow motion and weak binding; PM theory expands in powers of $G$ while retaining arbitrary velocity. Their overlap is therefore a double expansion, not an identification of a PN order with a PM order. The cleanest comparison for conservative, nonspinning scattering uses the center-of-mass energy $E$, angular momentum $J$, masses, and scattering angle $\chi(E,J)$.

Helpful background. The Einstein Equation supplies the field equations; Weak Metric Perturbations and Gauge Freedom supplies weak-field power counting; Presymplectic Potentials and Currents supplies the distinction between canonical descriptions and invariants; Computational Relativity supplies numerical-error tests; Gravitational Radiation and Waveforms supplies flux and hereditary observables; and Relativistic Two-Body Dynamics supplies the regime map used below.

The overlap as a double expansion

Let $M=m_1+m_2$, $\mu=m_1m_2/M$, and $\nu=\mu/M$. For unbound motion define the relative Lorentz factor

$$\gamma=\frac{E^2-m_1^2-m_2^2}{2m_1m_2}, \qquad p_\infty^2= \frac{\left[E^2-(m_1+m_2)^2\right] \left[E^2-(m_1-m_2)^2\right]}{4E^2},$$

and $J=b\,p_\infty$, where $b$ is the invariant asymptotic impact parameter in the center-of-mass frame. A PM expansion has the schematic form

$$\frac{\chi(E,J)}{2} =\sum_{n\geq1}\frac{\chi_n(\gamma,\nu)}{j^n}, \qquad j=\frac{J}{Gm_1m_2}.$$

At fixed $n$, expanding $\chi_n$ for $\gamma-1\ll1$ produces an infinite velocity series. Conversely, one PN order generally receives contributions from infinitely many PM orders because the Newtonian deflection parameter contains $G/v_\infty^2$. Damour makes this triangular organization explicit: inverse powers of velocity occur in the low-velocity expansion of fixed-PM coefficients, and a fixed PN coefficient is obtained by collecting terms with the same combined power of $G$ and velocity Damour 2016, §III, pp. 3–5, Eqs. (21)–(35), PDF.

It is useful to write a formal bivariate series in two small quantities,

$$\epsilon_v=v_\infty^2, \qquad \epsilon_G=\frac{GM}{b}, \qquad \chi=\sum_{n\geq1}\sum_k c_{n k}(\nu)\, \epsilon_G^n\epsilon_v^{\,k/2}.$$

Negative $k$ are allowed. The coefficient $c_{nk}$ is an overlap datum if both routes calculate that same monomial with the same definitions of $E$, $J$, and masses. Equality of coordinate accelerations, potentials, or Hamiltonian coefficients is not required: contact and canonical transformations can change those intermediate objects while leaving $\chi$ unchanged. The radial-action formula

$$\chi(E,J)=-\pi-\lim_{R\to\infty}\frac{\partial}{\partial J} \left[2\int_{r_{\min}}^R p_r(E,J;r)\,dr\right]$$

provides a representation-independent bridge. The cutoff matters because the full action has free linear and long-range logarithmic divergences, although its angular-momentum derivative has a finite limit. If the free action is subtracted first, its derivative already removes $-\pi$ and the angle is $-\partial_JW_{\rm int}$. The equivalent direct integral and the PM coefficients through the displayed orders are given by Kälin and Porto 2020, §§ 2.1–2.2, pp. 6–8, Eqs. (2.1)–(2.14), Open PDF.

A concrete 1PM-to-PN check

The exact conservative 1PM result is

$$\frac{\chi_{\mathrm{1PM}}}{2} =\frac{Gm_1m_2}{J} \frac{2\gamma^2-1}{\sqrt{\gamma^2-1}}.$$

This formula is valid at first order in $G$ without a slow-velocity assumption Damour 2016, §V, pp. 6–7, Eq. (61), PDF. For a relative speed $v_\infty$, $\gamma=(1-v_\infty^2)^{-1/2}$, so

$$\frac{2\gamma^2-1}{\sqrt{\gamma^2-1}} =\frac{1+v_\infty^2}{v_\infty\sqrt{1-v_\infty^2}} =\frac{1}{v_\infty} +\frac{3}{2}v_\infty +\frac{7}{8}v_\infty^3 +O(v_\infty^5).$$

To retain the next coefficients one must not replace $J$ by its Newtonian value too early. Writing $\Gamma=E/M=\sqrt{1+2\nu(\gamma-1)}$, the exact center-of-mass momentum is

$$p_\infty=\frac{\mu}{\Gamma}\sqrt{\gamma^2-1}.$$

Combining this relation with $J=b p_\infty$ gives

$$\chi_{\mathrm{1PM}} =\frac{2GM}{b}\, \Gamma\frac{1+v_\infty^2}{v_\infty^2} =\frac{2GM}{b} \left[ \frac{1}{v_\infty^2} +1+\frac{\nu}{2} +\frac{\nu(7-\nu)}{8}v_\infty^2 +O(v_\infty^4) \right].$$

The first term is the weak-deflection expansion of Newtonian hyperbolic scattering. The next term belongs to the relativistic low-velocity expansion at the same power of $G$. A PN calculation must reproduce both when its scattering observable is re-expanded at 1PM; a disagreement in either coefficient cannot be repaired by fitting strong-field data.

This example also shows why “1PM equals Newtonian” is false. First PM means one power of $G$, not the lowest power of velocity. The 1PM coefficient contains infinitely many PN corrections. Conversely, the exact Newtonian result

$$\chi_{\mathrm N}=2\arctan\!\left(\frac{GM}{b v_\infty^2}\right)$$

contains the odd sequence $G,G^3,G^5,\ldots$. Thus even the lowest PN dynamics samples arbitrarily high PM order when the deflection is not weak.

The common domain and noncommuting limits

The double expansion is controlled only where

$$v_\infty^2\ll1, \qquad \frac{GM}{b}\ll1, \qquad \alpha\equiv\frac{GM}{b v_\infty^2}\ll1.$$

The third inequality is independent of the first two. At fixed $G$ and $b$, taking $v_\infty\to0$ sends $\alpha\to\infty$: the orbit approaches large deflection or capture rather than the weak-scattering PM domain. Taking $G\to0$ first at fixed nonzero $v_\infty$ instead sends $\alpha\to0$. The exact Newtonian angle makes the nonuniformity visible,

$$\lim_{v_\infty\to0}\chi_{\mathrm N}=\pi, \qquad \chi_{\mathrm N}\sim2\alpha-\frac{2}{3}\alpha^3+\cdots \quad(\alpha\ll1),$$

while every finite truncation of the small-$\alpha$ series diverges when $v_\infty\to0$. The exact limits of the full function need not be algebraically noncommuting for every path; what fails is interchange of a limit with a nonuniform asymptotic truncation. This distinction prevents an apparent coefficient mismatch from being manufactured by leaving the overlap domain.

Other limits require equal care. The test-mass limit $\nu\to0$ at fixed $E/M$ is not the same scaling as fixed particle energy in a background of mass $M$. The ultrarelativistic limit $\gamma\to\infty$ can enhance a nominally higher-PM term. Bound-state continuation changes the sign of the nonrelativistic energy and assumes a conservative local radial action. Each comparison must say which dimensionless quantities remain fixed.

Local terms, tails, and what is not matched

An instantaneous conservative Hamiltonian is not the whole conservative dynamics at sufficiently high PN order. Radiation scattered from the system’s long-range gravitational field produces hereditary tail terms; in the conservative sector the 4PN dynamics includes a time-symmetric nonlocal contribution. Blanchet’s multipolar-post-Minkowskian construction displays how near-zone and exterior solutions are matched and how tails depend on the past history rather than on one instantaneous phase-space point Blanchet 2024, §2, especially §§2.2–2.4.

A naive comparison between a local PM potential and a PN Hamiltonian that includes the tail will therefore leave logarithms or scale-dependent pieces unmatched. The correct procedure is to compare like sectors: local conservative with local conservative, then include the hereditary contribution with the same boundary condition and subtraction scale. The arbitrary scale must cancel in the complete invariant. Dissipative scattering introduces radiated energy and angular momentum, so the conservative relation $\chi(E,J)$ must be replaced by a prescription that states whether $E,J$ are incoming, outgoing, or averaged. Production waveform fitting remains outside this page.

Matching also differs from the operations developed later in this chapter. Matching equates coefficients dictated by a common asymptotic expansion. Numerical calibration estimates coefficients or completions not fixed by that equality. Validation tests either result on information excluded from both procedures. Cross-regime uncertainty budgets may use the first omitted matched term as evidence, but a disagreement is not itself a calibrated probability distribution.

A solved overlap diagnostic

Suppose a PM calculation gives

$$\chi_{\mathrm{PM}} =2\epsilon_G\left(\epsilon_v^{-1} +1+\frac{\nu}{2} +\frac{\nu(7-\nu)}{8}\epsilon_v\right) -\frac{2}{3}\epsilon_G^3\epsilon_v^{-3} +O(\epsilon_G\epsilon_v^2,\epsilon_G^2,\epsilon_G^3\epsilon_v^{-2}),$$

while a PN calculation gives the first three displayed 1PM terms and the Newtonian cubic-deflection term. Which coefficients must agree, and what cannot yet be tested?

Solution

The shared monomials are $\epsilon_G\epsilon_v^{-1}$, $\epsilon_G$, $\epsilon_G\epsilon_v$, and $\epsilon_G^3\epsilon_v^{-3}$. Their coefficients must be $2$, $2+\nu$, $\nu(7-\nu)/4$, and $-2/3$, respectively, after the same definitions of $b$ and $v_\infty$ are used. The cubic term follows by expanding $2\arctan(\epsilon_G/\epsilon_v)$.

No conclusion can be drawn about an $\epsilon_G^2$ coefficient because the displayed PM remainder leaves 2PM unspecified. Nor can a local comparison settle a hereditary logarithm. Finally, the comparison is meaningful only when $\epsilon_v\ll1$ and $\epsilon_G/\epsilon_v\ll1$; choosing very small velocity at fixed $\epsilon_G$ invalidates the truncation rather than exposing a contradiction.

The broader consistency-test workflow uses this same logic to distinguish a genuine coefficient error from a variable mismatch, a missing hereditary sector, or a nonuniform limit.

References

• Blanchet, Luc. “Post-Newtonian Theory for Gravitational Waves.” Living Reviews in Relativity 27, 4 (2024). DOI.
• Damour, Thibault. “Gravitational Scattering, Post-Minkowskian Approximation and Effective One-Body Theory.” Physical Review D 94, 104015 (2016). DOI. Open PDF.
• Kälin, Gregor, and Rafael A. Porto. “From Boundary Data to Bound States.” Journal of High Energy Physics 2020, 072 (2020). DOI. Open PDF.