---
title: Post-Minkowskian Expansion in Gravity
url: https://www.emergentmind.com/topics/post-minkowskian-expansion
type: topic
---

# Post-Minkowskian Expansion in Gravity

The post-Minkowskian (PM) expansion is a systematic perturbative approach to computing gravitational dynamics in general relativity as an expansion in Newton’s constant $G$ at fixed, arbitrary velocities. This method is distinct from the post-Newtonian (PN) expansion, which expands simultaneously in $G$ and the velocity parameter $v/c\ll1$, and is particularly well-suited for capturing relativistic effects relevant to both bound and unbound two-body dynamics. In recent work, the PM expansion has been leveraged using modern amplitude and effective-field-theory techniques, yielding results for conservative and dissipative phenomena, classical Hamiltonians, waveforms, and radiative losses at unprecedented orders.

## 1. Core Principles of the Post-Minkowskian Expansion

The PM expansion organizes the gravitational two-body problem as a series in powers of $G$, with all velocity effects retained exactly at each order. The typical ansatz for the metric is
$$
g_{\mu\nu} = \eta_{\mu\nu} + \sum_{n=1}^\infty G^n h_{\mu\nu}^{(n)},
$$
and similarly, observables (e.g., scattering angles, binding energies, radiated moments) are written as
$$
\mathcal{O} = \sum_{n=1}^\infty G^n \mathcal{O}^{(n)}(v).
$$
This methodology admits exact Lorentz (velocity) dependence at each stage and avoids the prerequisite $v/c\ll1$ expansion of the PN approach.

In the two-body context, this translates into a Hamiltonian of the form (in isotropic gauge)
$$
H(p, r) = \sqrt{p^2 + m_1^2} + \sqrt{p^2 + m_2^2} + \sum_{n\geq1} G^n c_n(p^2) / r^n.
$$
Each $n$-graviton exchange in the amplitude formalism yields $O(G^n)$ and is matched onto the corresponding term in this Hamiltonian, retaining the full velocity (momentum) dependence [2006.06665, 1911.04411].

Contrast with the PN expansion:
- **PM**: Expand only in $G$ at fixed (arbitrary) velocities, yielding results exact in $v$.
- **PN**: Simultaneously expand in $G$ and $v/c$ based on the virial assumption $v^2\sim G/r$, with each PN order corresponding to a particular scaling in both.

## 2. Amplitude and Worldline Approaches

The modern PM program draws heavily on scattering amplitude techniques and effective field theory (EFT):
- **Amplitude-based:** Relates PM Hamiltonians and dynamical invariants to the classical (long-distance) part of multi-loop gravitational scattering amplitudes of massive scalars. Techniques involve unitarity cuts, generalized gauge fixing for gravitons, and integrand-level classical truncation (e.g., scaling $\ell,q\to\epsilon\ell, \epsilon q$ to isolate classical pieces) [1906.01579, 1808.02489, 1910.09366].
- **Worldline EFT:** Integrates out metric fluctuations around Minkowski space to produce an effective action for point-particle worldlines, whose coefficients (potentials) are then computed using diagrammatic perturbation theory, matched to amplitude results [2112.11296, 2012.14224].

Key technical advances include:
- Reduction of multi-loop amplitudes to master integrals using IBP (Integration-By-Parts) identities.
- Analysis via the loop-by-loop Baikov formalism to classify the function spaces (dlog/polylog, elliptic, K3) to which PM integrals evaluate [2405.17255].
- Algorithmic recovery of all velocity dependence in PM Hamiltonians using "momentum guessing" and recurrence relations, avoiding laborious term-by-term Feynman integral expansions [1911.04411].

## 3. Conservative Dynamics: Hamiltonians, Scattering Angles, Effective One-Body

The PM expansion yields closed-form expressions for the two-body conservative Hamiltonian to high orders:
- **1PM (tree):** Recovers the exact Newtonian potential plus relativistic corrections, mapping directly to the Einstein-Infeld-Hoffmann (EIH) potential at 1PN [2006.13234].
- **2PM (one-loop/1PN):** Captures first post-Minkowskian relativistic corrections ($O(G^2)$) including full velocity dependence. Amplitude and EFT approaches exactly agree in the potential [1906.01579, 1808.02489].
- **3PM, 4PM (higher-loop):** Recent results provide explicit expressions including contributions from elliptic integrals and K3 surfaces (at 3-loop) [2405.17255, 2112.11296].

The classical scattering angle $\chi$ admits a direct representation solely in terms of the classical part of the amplitude, without constructing a potential, via an all-order formula built from the eikonal/impact-parameter transform:
$$
\chi(b) = 2b \int_{r_{\text{min}}(b)}^\infty dr \left[ r^{-2} \left(1-\frac{2V_{\text{eff}}(r)}{p_0^2}\right)^{-1/2} - r^{-1} \right].
$$
This formula is manifestly gauge-invariant and works to all PM orders [1910.09366, 2203.13024].

In the **Effective One-Body (EOB)** formalism, the PM information is encoded in an energy-dependent effective metric with coefficients $h_n(\gamma)$ systematically determined by matching to the amplitude-derived scattering angle. Up to 3PM, all contributions can be absorbed into the metric, with no need for auxiliary non-metric functions [2108.11248].

## 4. Dissipative Effects: Radiation, Memory, and Angular Momentum Loss

The PM expansion systematically treats dissipative gravitational phenomena:
- **Radiation reaction and tails**: Conservative "tail" effects (backscattering of GWs) first appear at 4PM [2112.11296], and are captured by combining potential and radiation-mode integrals.
- **Gravitational Wave Emission**: At each PM order, the radiative metric perturbation at null infinity is given as a sum over multipolar modes whose coefficients are matched to cut (on-shell) Feynman diagrams via reverse unitarity [2506.20733, 2204.06556].
- **Nonlinear Memory and Non-analyticity**: Two-loop (O($G^3$)) diagrams encode the so-called Christodoulou memory; PM methods provide closed forms for the non-analytic in frequency (1/ω poles and $\ln\omega$) parts of the waveform, with exact in-velocity predictions [2506.20733].
- **Radiated energy and angular momentum**: Calculated using amplitude-based eikonal operators, which exponentiate the elastic, radiative, and static soft (Weinberg) contributions. Static terms require careful treatment of the infrared (supertranslation) ambiguity at null infinity [2210.15689, 2308.11470, 2406.07943].
- **Spin, tidal, and multi-body corrections**: The PM scheme naturally accommodates finite-size (tidal) and spin-induced multipoles, both in conservative dynamics [2006.06665, 2204.06556, 2205.15295] and in radiative losses [2210.15689, 2308.11470].

## 5. Mathematical and Computational Structure

Higher-order PM calculations reveal intricate algebraic structures:
- **Integral Geometry**: Classification via the Baikov representation identifies which Feynman topologies yield dlog/polylog, elliptic, or K3 functions, providing analytic control and guiding computational methods [2405.17255].
- **Intersection Theory**: At 2PM and higher, master integrals can be efficiently projected using the theory of twisted cocycles and intersection numbers, streamlining the decomposition of multi-loop integrals and matching the amplitude to observables [2404.11913].
- **Algorithmic Generation**: For many quantities (up to 3PM), closed analytic forms can be algorithmically reconstructed from finite truncations in velocity expansions using symbolic summation and recurrence guessing [1911.04411].

## 6. Ambiguities, Gauge Freedom, and Null Infinity

A key subtlety in the PM framework is the **supertranslation ambiguity** in angular momentum flux at null infinity:
- **Bondi Coordinates and BMS Freedom**: The definition of radiated angular momentum possesses an ambiguity under BMS supertranslations (angle-dependent shifts of retarded time).
- **Resolution in PM**: At order $G$, Mao–Zeng [2406.07943] prescribe fixing the Bondi frame such that the self-field part of the order-$G$ metric produces vanishing next-to-leading charges, eliminating ℓ≥2 supertranslation freedom. This ensures the uniqueness of the O($G^2$) and O($G^3$) radiated angular momentum predictions, reconciling linear-response formulas with amplitude-based results. All relevant integrals and matching conditions can be stated and resolved at null infinity, without recourse to matching onto spatial infinity.
- At linear order in $G$, the Bondi 4-momentum and angular momentum match the ADM definitions of momentum and angular momentum for a system of pointlike bodies, confirming consistency between asymptotic and canonical constructions [2405.11953].

## 7. Extensions: Tidal, Spin, and Multipole Corrections

Advanced PM calculations include corrections due to tidal effects, spin-orbit coupling, and higher multipoles.
- **Tidal Contributions:** Finite-size corrections (Love numbers) enter the Hamiltonian at O($G^2$) (LO PM) and O($G^3$) (NLO), with closed expressions for mass and current-quadrupole terms in the Hamiltonian, scattering angle, and radiated fluxes, recovered by matching to the appropriate amplitude components [2006.06665, 2204.06556, 2210.15689].
- **Spin Effects:** Spin–orbit and spin–spin contributions to radiated momentum, angular momentum, and fluxes are systematically included at 3PM (O($G^3$)), with manifestly Poincaré-covariant formulas valid for arbitrary spin orientations [2205.15295, 2308.11470].
- **Three-body and Higher Multipole Interactions:** At 2PM, the three-body effective potential appears via a master triangle integral amenable to integrable (Yangian) and intersection-theory techniques [2012.14224], with a systematic pattern of PN corrections generated at higher orders.

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In summary, the post-Minkowskian expansion provides a rigorous, velocity-exact expansion for classical gravitational dynamics, unifying conservative and dissipative phenomena across relativistic and non-relativistic regimes. By integrating amplitude and EFT methods, reverse unitarity, Baikov/intersection theory, and a careful treatment of infrared and gauge ambiguities, PM results connect theoretical computational frameworks to practical applications in gravitational-wave physics, with seamless matching to post-Newtonian results and direct input for waveform models [2203.13024, 2112.11296, 2506.20733].

Source: https://www.emergentmind.com/topics/post-minkowskian-expansion