Post-Minkowskian Expansion and Relativistic Kinematics

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The post-Minkowskian (PM) expansion solves a weak-interaction problem without first assuming slow motion. Its natural data are the masses and asymptotic four-momenta of two bodies, together with an impact parameter; the relative Lorentz factor is held fixed while the answer is expanded in Newton’s constant. This organization makes relativistic scattering especially clean: incoming and outgoing momenta are observable at infinity, while gauge-dependent trajectories remain intermediate constructions. The leading result already contains every power of velocity and reduces to Newtonian hyperbolic scattering only after a separate low-velocity expansion.

Required background. Events, Intervals, and Minkowski Geometry supplies invariant four-momentum kinematics, and Weak Metric Perturbations and Gauge Freedom supplies the weak-field expansion and its gauge freedom.

Helpful background. The Einstein Equation fixes the nonlinear field equations being expanded; Presymplectic Potentials and Currents clarifies why asymptotic charges are preferable to coordinate trajectories; and Gravitational Radiation and Waveforms provides the radiative charges needed once retarded boundary conditions are imposed.

Invariant scattering data

Let the incoming momenta be

p_a^{-\mu}=m_a u_a^{-\mu}, \qquad u_a^-\!\cdot u_a^-=-1, \qquad \gamma\equiv-u_1^-\!\cdot u_2^-\geq 1 .

The invariant center-of-mass energy and the magnitude of either incoming spatial momentum in that frame are

s\equiv-(p_1^-+p_2^-)^2 =m_1^2+m_2^2+2m_1m_2\gamma ,

p_\infty =\frac{\sqrt{\lambda(s,m_1^2,m_2^2)}}{2\sqrt{s}}, \qquad \lambda(x,y,z)=x^2+y^2+z^2-2xy-2xz-2yz .

Choose the impact vector $b^\mu$ to point from body 2’s incoming line to body 1’s incoming line at closest approach, with

b\!\cdot u_1^-=b\!\cdot u_2^-=0, \qquad b=\sqrt{b^\mu b_\mu}, \qquad J=b\,p_\infty .

Translations shift both incoming lines together and leave $b^\mu$ unchanged. Reparametrizations of either line also leave it unchanged. Thus $(m_1,m_2,\gamma,b)$ , or equivalently $(s,J)$ with the masses, are a complete set for planar, nonspinning, structureless scattering. Spin orientations and finite-size response coefficients become additional asymptotic data when those effects are retained.

The outgoing momenta define the impulses

\Delta p_a^\mu=p_a^{+\mu}-p_a^{-\mu}.

For conservative scattering, $\Delta p_1^\mu+\Delta p_2^\mu=0$ . With radiation, that equation must include the four-momentum carried to null infinity; the precise balance is developed on Radiative PM Scattering and Energy Loss. The covariant wave-packet construction of the impulse and radiated momentum is given in Kosower, Maybee, and O’Connell 2019, §§ 3.1–3.5, pp. 6–16.

What PM order means

For an invariant observable $\mathcal O$ , PM power counting has the schematic form

\mathcal O(m_a,\gamma,b;G) =\sum_{n\geq 0}G^n\mathcal O_n(m_a,\gamma,b),

where $m_a$ , $\gamma$ , and $b$ are fixed as $G\to0$ . The coefficient $\mathcal O_n$ retains its full dependence on velocity. Calling a result “ $n$ PM” means retaining terms through $G^n$ ; it does not mean that $G$ itself is dimensionless. At moderate relativistic energy the useful weak-field ratio is of order $GM/b$ ; at very large energy the relevant scale is $G\sqrt{s}/b$ . In the slow-motion corner, small deflection instead requires the stronger condition

\frac{GM}{b\,v_\infty^2}\ll1, \qquad M=m_1+m_2,

because a slowly passing body interacts for a longer time. A formally small $GM/b$ therefore does not guarantee a uniformly small scattering angle as $v_\infty\to0$ .

The post-Newtonian (PN) expansion makes a different cut through the same dynamics: it expands in small velocities and weak binding, with the virial relation $v^2\sim GM/r$ for a bound system. One PM coefficient contains infinitely many PN terms; conversely, one PN order can receive several powers of $G$ . The two expansions can be compared only in their common weak-field, slow-motion domain. Damour makes this distinction explicit by expanding the invariant scattering function in $1/J$ at fixed energy and then re-expanding each coefficient at low velocity Damour 2016, § IV and eqs. (26)–(33).

Leading deflection at fixed velocity

At first PM order, straight incoming worldlines may be used inside the interaction. A convenient reduced interaction function is

W_1(b) = \frac{2Gm_1m_2(2\gamma^2-1)} {\sqrt{\gamma^2-1}} \log\!\left(\frac{b}{b_0}\right),

where $b_0$ is an irrelevant infrared length. Differentiating with respect to the transverse separation gives the leading impulse,

\Delta p_1^{\mu(1)} =-\frac{\partial W_1}{\partial b_\mu} =-\frac{2Gm_1m_2(2\gamma^2-1)} {\sqrt{\gamma^2-1}} \frac{b^\mu}{b^2}, \qquad \Delta p_2^{\mu(1)}=-\Delta p_1^{\mu(1)} .

The sign is physical with our definition of $b^\mu$ : body 1 is pulled toward body 2. The arbitrary $b_0$ disappears after differentiation. Since $b^\mu$ is orthogonal to both incoming velocities, $p_a^-\!\cdot\Delta p_a^{(1)}=0$ , which is exactly the order- $G$ part of the outgoing mass-shell condition. Longitudinal components enter at higher order so that

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

continues to hold.

For small deflection, $|\Delta\boldsymbol p_1|=2p_\infty\sin(\chi/2)$ , and hence

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

This agrees with the direct linearized-gravity calculation and the EOB scattering dictionary Damour 2016, § V and eqs. (54)–(61). It also previews the logic of the next two pages: the same invariant may be obtained from a force integral, an amplitude, an eikonal phase, or a radial action.

Using $p_\infty=m_1m_2\sqrt{\gamma^2-1}/\sqrt{s}$ makes the weak-deflection requirement explicit for $\gamma\gt1$ :

\chi_{\rm 1PM} =\frac{2G\sqrt{s}}b\frac{2\gamma^2-1}{\gamma^2-1}\ll1.

This one expression exhibits both the slow-encounter enhancement and the high-energy gravitational radius. It is a necessary leading-order check; near capture or a finite-size resonance it does not replace the other validity conditions.

Boundary conditions and physical sectors

“Conservative PM dynamics” normally means that radiation modes are treated with time-symmetric boundary conditions, or that the conservative part is separated from a retarded calculation. “Radiative PM dynamics” uses retarded boundary conditions and retains fluxes and radiation reaction. This distinction is not identical to “potential modes versus radiation modes.” Beginning at sufficiently high order, radiation modes generate conservative hereditary tails as well as dissipative loss; a local instantaneous Hamiltonian alone cannot represent the full history dependence. Blanchet’s multipolar-post-Minkowskian construction exhibits how no-incoming-radiation boundary conditions, near-zone matching, and hereditary integrals enter as distinct ingredients Blanchet 2024, §§ 2.1–2.3.

The point-particle PM expansion additionally assumes $b$ is large compared with each body’s gravitational radius and any material size relevant to the interaction. It breaks down near capture, at a strong-field separatrix, or when tidal excitation and absorption cease to be perturbative. Large $\gamma$ can also compensate for large $b$ : the relevant gravitational radius grows with center-of-mass energy. Finally, an asymptotic scattering expansion does not directly describe a bound orbit, because bound motion has no free incoming and outgoing states. The controlled continuation of conservative information is the subject of Bound-Orbit Information from Scattering Data.

Exercises

Recover Newtonian hyperbolic deflection

Show that the 1PM angle reduces to $2GM/(b v_\infty^2)$ when the relative speed is small.

Solution

For $v_\infty\ll1$ ,

\gamma=1+\frac12v_\infty^2+O(v_\infty^4), \qquad \sqrt{\gamma^2-1}=v_\infty+O(v_\infty^3).

Also $p_\infty=\mu v_\infty+O(v_\infty^3)$ and $J=\mu b v_\infty+O(v_\infty^3)$ , with $\mu=m_1m_2/M$ . Since $2\gamma^2-1=1+O(v_\infty^2)$ ,

\chi_{\mathrm{1PM}} =\frac{2Gm_1m_2}{\mu b v_\infty^2} +O\!\left(\frac{GM}{b}\right) =\frac{2GM}{b v_\infty^2} +O\!\left(\frac{GM}{b}\right).

This is the weak-deflection limit of Newtonian Rutherford scattering for an attractive $-Gm_1m_2/r$ potential. The exact Newtonian angle is $2\arctan[GM/(bv_\infty^2)]$ ; the 1PM result keeps its first term. The divergence of that first term as $v_\infty\to0$ signals loss of uniform PM control, rather than an unbounded exact Newtonian angle.

Check the invariant center-of-mass momentum

Starting from $s=m_1^2+m_2^2+2m_1m_2\gamma$ , show that

p_\infty=\frac{m_1m_2\sqrt{\gamma^2-1}}{\sqrt{s}}.