Post-Minkowskian Expansion in Gravity

• The post-Minkowskian expansion is a perturbative series in Newton’s constant that captures full relativistic effects without low-velocity approximations.
• Modern amplitude and effective field theory techniques enhance PM calculations, yielding precise conservative Hamiltonians and scattering angles for two-body gravitational dynamics.
• PM methods also account for dissipative processes by incorporating radiation reaction, waveform modeling, and corrections for tidal and spin effects at higher orders.

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 (Cheung et al., 2020, Blümlein et al., 2019).

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) (Cristofoli et al., 2019, Cheung et al., 2018, Bjerrum-Bohr et al., 2019).
• 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 (Dlapa et al., 2021, Loebbert et al., 2020).

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 (Frellesvig et al., 2024).
• Algorithmic recovery of all velocity dependence in PM Hamiltonians using "momentum guessing" and recurrence relations, avoiding laborious term-by-term Feynman integral expansions (Blümlein et al., 2019).

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 (Grignani et al., 2020).
• 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 (Cristofoli et al., 2019, Cheung et al., 2018).
• 3PM, 4PM (higher-loop): Recent results provide explicit expressions including contributions from elliptic integrals and K3 surfaces (at 3-loop) (Frellesvig et al., 2024, Dlapa et al., 2021).

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 (Bjerrum-Bohr et al., 2019, Bjerrum-Bohr et al., 2022).

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 (Damgaard et al., 2021).

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 (Dlapa et al., 2021), 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 (Georgoudis et al., 25 Jun 2025, Mougiakakos et al., 2022).
• 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 (Georgoudis et al., 25 Jun 2025).
• 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 (Heissenberg, 2022, Heissenberg, 2023, Mao et al., 2024).
• Spin, tidal, and multi-body corrections: The PM scheme naturally accommodates finite-size (tidal) and spin-induced multipoles, both in conservative dynamics (Cheung et al., 2020, Mougiakakos et al., 2022, Riva et al., 2022) and in radiative losses (Heissenberg, 2022, Heissenberg, 2023).

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 (Frellesvig et al., 2024).
• 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 (Frellesvig et al., 2024).
• 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 (Blümlein et al., 2019).

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 (Mao et al., 2024) 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 (Mao et al., 2024).

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 (Cheung et al., 2020, Mougiakakos et al., 2022, Heissenberg, 2022).
• 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 (Riva et al., 2022, Heissenberg, 2023).
• 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 (Loebbert et al., 2020), 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 (Bjerrum-Bohr et al., 2022, Dlapa et al., 2021, Georgoudis et al., 25 Jun 2025).