Post-Minkowskian Dynamics and Scattering

Post-Minkowskian dynamics organizes the relativistic two-body problem by powers of Newton’s constant while retaining exact dependence on velocity. Unbound encounters are its cleanest arena: masses, incoming momenta, and impact parameter are specified at past infinity, and impulses, scattering angles, radiated charges, and outgoing momenta are read at future infinity. These observables allow Hamiltonian, effective-one-body, worldline, self-force, and amplitude calculations to meet without identifying their gauge-dependent trajectories. This chapter develops that common language from 1PM kinematics through radiative balance and then asks, with explicit qualifications, which scattering information can be continued to bound motion.

Helpful background. The Einstein Equation identifies the nonlinear field equations being expanded; Weak Metric Perturbations and Gauge Freedom supplies the weak-field regime; Presymplectic Potentials and Currents supplies actions, canonical maps, and asymptotic charges; and Gravitational Radiation and Waveforms supplies fluxes, memory, and retarded boundary conditions.

Enter the PM scattering problem

Consider two nonspinning bodies with incoming four-momenta $p_a^{-\mu}=m_a u_a^{-\mu}$. The relative Lorentz factor and total energy are

$$\gamma=-u_1^-\!\cdot u_2^-, \qquad s=-(p_1^-+p_2^-)^2 =m_1^2+m_2^2+2m_1m_2\gamma .$$

The incoming center-of-mass momentum and orbital angular momentum are

$$p_\infty =\frac{\sqrt{\lambda(s,m_1^2,m_2^2)}}{2\sqrt{s}}, \qquad J=b p_\infty ,$$

where $b$ is the invariant magnitude of the impact vector orthogonal to both incoming velocities. A PM expansion holds $(m_1,m_2,\gamma,b)$ fixed as $G\to0$. It is therefore an expansion in weak gravitational interaction at unrestricted velocity, not an expansion in velocity.

The distinction from post-Newtonian power counting is essential. A coefficient at fixed PM order contains an infinite sequence of PN velocity corrections. In the common slow-motion domain, the leading PM result must reduce to Newtonian hyperbolic scattering; outside that domain PM retains relativistic velocity dependence that a finite PN truncation cannot. Damour’s invariant scattering formulation makes the energy, angular momentum, and PM–PN re-expansion explicit Damour 2016, §§ III–V.

The six-page route

The pages are designed to be read in order, with one observable carried through the sequence.

1. Post-Minkowskian Expansion and Relativistic Kinematics defines what is fixed at each PM order, constructs $\gamma$, $s$, $p_\infty$, $b$, and $J$, and derives the 1PM angle with both Newtonian and relativistic checks.
2. Impulse and Momentum Transfer derives the covariant endpoint momentum change. It explains why the leading impulse is transverse, why longitudinal iterations enter at higher order, and how mass-shell and momentum-balance constraints expose incomplete results.
3. Scattering Angle and Radial Action shows that the angle is the angular-momentum derivative of a regulated radial action. This establishes its canonical invariance and connects Hamiltonian, EOB, amplitude, and self-force descriptions.
4. Eikonal Methods and High-Energy Deflection moves to impact-parameter space, derives deflection by stationary phase, and separates high energy from weak deflection. It also distinguishes an elastic phase from a full scattering generator.
5. Radiative PM Scattering and Energy Loss replaces elastic momentum conservation by balance with radiated four-momentum and angular momentum. Retarded boundary conditions, recoil, hereditary tails, memory, and a linear-loss formula for the angle enter here.
6. Bound-Orbit Information from Scattering Data analytically continues the local conservative radial action and recovers periastron advance. It then compares the constructive boundary-to-bound program with the limitations caused by generic nonlocal histories, dissipative evolution, and unresolved high-order memory terms.

The first three pages form the conservative derivation spine. The fourth changes representation but not the invariant. The fifth changes boundary conditions and adds physical channels. The sixth changes the orbital class and therefore has the strongest analytic assumptions.

One invariant in four representations

At 1PM order, define

$$A(E) = \frac{2Gm_1m_2(2\gamma^2-1)} {\sqrt{\gamma^2-1}}.$$

The same coefficient appears in four statements:

$$\Delta p_{1\perp}^\mu =-A(E)\frac{b^\mu}{b^2}, \qquad \chi_{\rm 1PM} =\frac{A(E)}{J},$$ $$I_{r,{\rm 1PM}}^{\rm scatt} =-A(E)\log\!\left(\frac{J}{J_0}\right), \qquad 2\delta_1 =-A(E)\log\!\left(\frac{b}{b_0}\right).$$

The impulse is a vector endpoint observable. The angle is its elastic center-of-mass projection. The radial action is a scalar generating function whose negative $J$ derivative gives the angle. With the Fourier convention on the eikonal page, the phase derivative gives the attractive impulse: $\Delta\boldsymbol p_1=\partial_{\boldsymbol b}(2\delta_1)$. Agreement among these expressions fixes signs, flux normalizations, and factors of two. The Fourier and incoming impact parameters coincide at this order; beyond leading order their geometric conversion must be retained.

At higher PM order, the equivalence is more demanding. Lower-order trajectories must be iterated, each outgoing momentum must satisfy

$$2p_a^-\!\cdot\Delta p_a+\Delta p_a^2=0,$$

and a canonical transformation must approach an admissible transformation at the asymptotic endpoints. In amplitude calculations, classical loop terms must be separated from Born iterations; in Hamiltonian calculations, coordinate-dependent potentials must be reduced to invariant endpoint data. The wave-packet observable formalism provides a direct construction of impulse and radiated momentum from amplitudes Kosower, Maybee, and O’Connell 2019, §§ 3–6.

Conservative and retarded problems

Time-symmetric and retarded boundary conditions answer different questions. The conservative problem has

$$\Delta p_1^\mu+\Delta p_2^\mu=0.$$

The retarded problem has

$$\Delta p_1^\mu+\Delta p_2^\mu+P_{\rm rad}^\mu=0, \qquad P_{\rm rad}^0\geq0$$

in the incoming center-of-mass frame. It also has an angular-momentum balance law, whose statement requires a common origin and asymptotic Bondi frame. The radiated momentum and radiation-reaction part of the impulse begin at $O(G^3)$ for structureless nonspinning scattering, while higher orders include nonlinear and hereditary effects. A causal fourth-PM calculation demonstrates how total impulse, radiated momentum, and conservative tail pieces coexist Dlapa et al. 2023, main text around eqs. (2)–(8).

“Radiation mode” is consequently not a synonym for “dissipative.” The time-symmetric part of radiation propagation generates conservative tails; the time-antisymmetric part generates loss. Memory resides in the zero-frequency change of the asymptotic field and must not be confused with positive energy flux. These distinctions are prerequisites for using scattering data in a bound model.

Connections to other approximation schemes

PM scattering serves as a comparison surface rather than a complete theory of every regime.

• A low-velocity expansion of each PM coefficient overlaps with PN dynamics. Mismatched powers of velocity, logarithms, or regularization scales reveal a convention or region error.
• A small-mass-ratio expansion overlaps with black-hole perturbation theory and self-force calculations. The test-body limit must recover geodesic scattering before finite-mass-ratio corrections are trusted.
• The scattering angle and radial action can inform EOB potentials because they are invariant under asymptotically trivial canonical transformations. Strong-field use still requires a justified resummation and comparison with independent data.
• Amplitudes supply efficient on-shell building blocks. Restoring $\hbar$, separating classical terms from iterations, and translating signature and momentum conventions are part of the classical limit, not cosmetic steps.
• Numerical relativity can test strong-field scattering where a raw PM truncation approaches its breakdown domain. Agreement after resummation does not retroactively make the weak-deflection series convergent at capture.

The eikonal limit adds another double scaling: $J/\hbar\gg1$ and $-t/s\ll1$. At ultrarelativistic energy, weak deflection requires $b\gg G\sqrt{s}$; high energy alone does not justify an eikonal approximation.

Domain, continuation, and uncertainty

The chapter’s baseline assumes asymptotic flatness, freely separated incoming and outgoing bodies, weak interaction, and impact parameters larger than gravitational radii and relevant finite-size scales. Spin, tidal response, absorption, and additional fields require extra asymptotic data and matching coefficients. Capture, turning-point merger, and order-unity radiation loss mark genuine breakdown.

Bound continuation introduces a separate limitation. For local conservative motion,

$$\Delta\Phi(E,J) =\chi(E,J)+\chi(E,-J)$$

on the bound-energy sheet, and derivatives of the continued radial action give orbital invariants. Kälin and Porto derive this relation and reconstruct the radial action to all PM orders under the stated assumptions Kälin and Porto 2020, §§ 3–4. Generic nonlocal tails depend on the entire orbital history, however. Local pieces and universal logarithms map more broadly than finite nonlocal remainders. The final page therefore carries a dated comparison and tests that could change the conclusion rather than presenting an active-research extension as a theorem.

Review the chapter

Integrated 1PM consistency check

Starting with the leading impulse coefficient $A(E)$ above, derive the scattering angle and its Newtonian limit. Then state how the momentum-balance equation changes when radiation is included.

Solution

The transverse impulse has magnitude

$$\lvert\Delta\boldsymbol p_1\rvert=\frac{A(E)}{b}.$$

At first order in the small angle, $\lvert\Delta\boldsymbol p_1\rvert=p_\infty\chi$, and $J=bp_\infty$, so

$$\chi_{\rm 1PM} =\frac{A(E)}{J} =\frac{2Gm_1m_2}{J} \frac{2\gamma^2-1}{\sqrt{\gamma^2-1}}.$$

For $v_\infty\ll1$,

$$\sqrt{\gamma^2-1}=v_\infty+O(v_\infty^3), \qquad J=\mu b v_\infty+O(v_\infty^3),$$

with $\mu=m_1m_2/(m_1+m_2)$. Therefore

$$\chi_{\rm 1PM} =\frac{2G(m_1+m_2)}{b v_\infty^2} +O\!\left(\frac{G(m_1+m_2)}{b}\right),$$

which is Newtonian hyperbolic deflection. In a conservative encounter $\Delta p_1^\mu+\Delta p_2^\mu=0$. With retarded boundary conditions, radiation carries four-momentum and the correct equation is

$$\Delta p_1^\mu+\Delta p_2^\mu+P_{\rm rad}^\mu=0.$$

Checking only mechanical momentum would therefore misdiagnose a consistent radiative calculation as nonconserving, or accept an incomplete one if $P_{\rm rad}^\mu$ were silently omitted.

Distinguish two continuation claims

Which statement is stronger: “the local conservative scattering angle determines periastron advance” or “the full retarded scattering solution determines a generic inspiral”?

Solution

The second is much stronger. The first follows from analytic continuation of a local conservative radial action, with specified energy and angular-momentum branches. The second would also have to determine nonlocal hereditary history, radiated energy and angular momentum, recoil, the arrow of time, and cumulative phase evolution. Scattering data can constrain all of these when supplemented by fluxes and a causal model, but the conservative angle alone cannot supply them.

References

• Damour, Thibault. “Gravitational Scattering, Post-Minkowskian Approximation and Effective One-Body Theory.” Physical Review D 94, 104015 (2016). DOI. Open PDF.
• Dlapa, Christoph, Gregor Kälin, Zhengwen Liu, Jakob Neef, and Rafael A. Porto. “Radiation Reaction and Gravitational Waves at Fourth Post-Minkowskian Order.” Physical Review Letters 130, 101401 (2023). DOI. Open PDF.
• Kälin, Gregor, and Rafael A. Porto. “From Boundary Data to Bound States II: Scattering Angle to Dynamical Invariants (with Twist).” Journal of High Energy Physics 2020, 120 (2020). DOI. Open PDF.
• Kosower, David A., Ben Maybee, and Donal O’Connell. “Amplitudes, Observables, and Classical Scattering.” Journal of High Energy Physics 2019, 137 (2019). DOI. Open PDF.