Post-Minkowskian Iteration

Post-Minkowskian iteration constructs the nonlinear vacuum field outside an isolated source as a formal expansion in $G$, starting from the most general retarded multipolar solution of the linearized equations. Unlike a post-Newtonian expansion, it does not assume slow source motion. Its essential difficulty is that exterior multipole fields are singular at the coordinate origin, so every retarded inversion must be regularized and every harmonic constraint restored without introducing incoming radiation.

Required background. Weak-Field Gravity supplies the perturbative field and gauge freedom, while Ordinary Differential Equations supplies the order-by-order and boundary-value logic used in the iteration.

Helpful background. Einstein Equations supplies the nonlinear vacuum equation; Waves, Green Functions, and Geometric Optics supplies the retarded Green function and causal boundary condition; and Relativistic Two-Body Dynamics can later determine concrete moment functions. Detailed binary equations of motion and detector inference remain outside this construction.

Relaxed vacuum equations in harmonic coordinates

Use the gothic metric deviation

$$\mathfrak h^{\alpha\beta} =\sqrt{-g}\,g^{\alpha\beta}-\eta^{\alpha\beta}, \qquad \partial_\beta\mathfrak h^{\alpha\beta}=0,$$

with $\eta_{\alpha\beta}=\operatorname{diag}(-1,1,1,1)$. In the exterior vacuum region the Einstein equations take the relaxed form

$$\Box_\eta\mathfrak h^{\alpha\beta} =\Lambda^{\alpha\beta}[\mathfrak h;\,\partial\mathfrak h;\,\partial^2\mathfrak h],$$

where $\Lambda^{\alpha\beta}$ begins quadratically in the field. The harmonic condition and the exact Bianchi identity imply the compatibility relation $\partial_\beta\Lambda^{\alpha\beta}=0$ when the lower-order equations hold.

The multipolar post-Minkowskian (MPM) ansatz is

$$\mathfrak h^{\alpha\beta}_{\mathrm{MPM}} =\sum_{n=1}^{\infty}G^n\mathfrak h^{\alpha\beta}_{(n)}.$$

This is a formal series: the algorithm determines its coefficients but does not establish convergence of the sum. The linear seed $\mathfrak h_{(1)}^{\alpha\beta}$ is a retarded multipolar vacuum solution outside a world tube containing the source. Its two canonical STF moment families encode physical mass- and current-type content; an accompanying homogeneous gauge solution completes the general linear field. The linear STF expansion and conserved monopole and dipoles are described in Blanchet 2024, §2.1.2, pp. 38–41 and in Thorne 1980, §§II–III.

One nonlinear iteration

Suppose the coefficients through order $n-1$ are known and satisfy their harmonic constraints. Equating the coefficient of $G^n$ gives

$$\Box_\eta\mathfrak h^{\alpha\beta}_{(n)} =\Lambda^{\alpha\beta}_{(n)} [\mathfrak h_{(1)},\ldots,\mathfrak h_{(n-1)}], \qquad \partial_\beta\mathfrak h^{\alpha\beta}_{(n)}=0.$$

$\Lambda_{(n)}^{\alpha\beta}$ is divergence-free by induction. It is nevertheless singular at $r=0$, because it is made from products of exterior multipole expansions that were never meant to describe the source interior. A direct all-space retarded integral generally diverges at that artificial origin.

Introduce a complex regulator $B$ and an arbitrary length $r_0$, and define a particular solution by analytic continuation:

$$u^{\alpha\beta}_{(n)} =\underset{B=0}{\operatorname{FP}}\, \Box_{\mathrm{ret}}^{-1} \left[ \left(\frac{r}{r_0}\right)^B \Lambda^{\alpha\beta}_{(n)} \right].$$

For sufficiently large $\operatorname{Re}B$ the integral converges near the origin. This statement controls only its inner endpoint: an exterior nonlinear source also extends to infinity, where convergence can require a different range of $B$. The standard construction splits the integration into inner and outer regions, analytically continues the pieces from their respective convergence domains, and adds them before taking the finite coefficient at $B=0$. A common domain of absolute convergence for the whole integral need not exist. The resulting finite part solves the wave equation for $r\gt0$; it does not repair the fictitious multipolar singularity at the origin.

Regularization and differentiation do not commute harmlessly. The divergence

$$w^\alpha_{(n)}=\partial_\beta u^{\alpha\beta}_{(n)}$$

can be nonzero because differentiating $(r/r_0)^B$ produces $B n_i/r$; a pole in the regulated retarded integral can leave a finite remainder as $B\to0$. However, $w^\alpha_{(n)}$ obeys the homogeneous wave equation. One therefore constructs a retarded homogeneous correction $v^{\alpha\beta}_{(n)}$ such that

$$\Box_\eta v^{\alpha\beta}_{(n)}=0, \qquad \partial_\beta v^{\alpha\beta}_{(n)}=-w^\alpha_{(n)},$$

and sets $\mathfrak h_{(n)}=u_{(n)}+v_{(n)}$. The result satisfies both the nonlinear source equation and the order-$n$ harmonic constraint. This finite-part inversion and gauge-restoration step is the core MPM algorithm Blanchet 2024, §2.1.3, pp. 41–46.

Retarded boundary conditions and uniqueness

Every inverse wave operator in the construction is retarded. Together with a field that is stationary before some finite time, this enforces an isolated-source, no-incoming-radiation condition: the waveform at $(t,\boldsymbol x)$ depends on earlier moment values, never future ones. Past stationarity is a technical way to control integrals at early times; the final tail integrals can apply more broadly when their remote-past behavior is demonstrably convergent.

The boundary condition does not remove all homogeneous freedom. At higher PM order, a homogeneous retarded solution with the same multipolar structure may be absorbed into a nonlinear redefinition of the canonical moments or into a gauge transformation. A canonical convention fixes this freedom. Thus uniqueness means uniqueness after the moment definition, coordinate gauge, past behavior, and retarded condition have all been stated.

It is also important that “retarded” and “radiative coordinates” answer different questions. Retarded inversion fixes causal support. The later change from harmonic to radiative coordinates removes asymptotic logarithms and defines the waveform coefficients at null infinity.

Approximation domain and independent checks

The PM order counts powers of $G$ in the exterior field; it does not count powers of velocity. A source can have relativistic internal motion, provided an exterior weak-field region and suitable multipole moments exist. A subsequent near-zone post-Newtonian matching introduces a separate slow-motion expansion. Conflating the two loses control of which remainder is being neglected.

Four checks distinguish a valid iteration from a merely formal substitution:

1. Wave equation: applying $\Box_\eta$ to $u_{(n)}+v_{(n)}$ must return $\Lambda_{(n)}$ for $r\gt0$.
2. Constraint: $\partial_\beta(u^{\alpha\beta}_{(n)}+v^{\alpha\beta}_{(n)})$ must vanish.
3. Causality: all homogeneous additions must have retarded, not advanced, support.
4. Linear limit: dropping $\Lambda$ must leave only the prescribed $G\mathfrak h_{(1)}$.

The scale $r_0$ may remain in an intermediate PM coefficient. It must cancel against the matching or source-moment contribution in a consistently assembled radiative observable. Failure of that cancellation is an order-matching error, not new physics.

Exercise

At second PM order, suppose $\partial_\beta\Lambda_{(2)}^{\alpha\beta}=0$. Explain why the unregulated retarded inverse of a smooth compact source would satisfy the harmonic constraint automatically, but the finite-part multipolar inverse can require $v_{(2)}^{\alpha\beta}$.

Solution

For a smooth compact source and compatible boundary data, derivatives commute with the retarded inverse. Then $\partial_\beta\Box_{\mathrm{ret}}^{-1}\Lambda_{(2)}^{\alpha\beta}=\Box_{\mathrm{ret}}^{-1}\partial_\beta\Lambda_{(2)}^{\alpha\beta}=0$. An exterior multipole source is singular at $r=0$, so the MPM inverse instead acts on $(r/r_0)^B\Lambda_{(2)}$. Differentiation produces an extra term proportional to $B n_i\Lambda_{(2)}^{\alpha i}/r$. If the analytically continued integral has a $1/B$ pole, their product has a finite limit. The resulting divergence $w^\alpha_{(2)}$ is homogeneous rather than zero, and a retarded homogeneous field $v_{(2)}^{\alpha\beta}$ with opposite divergence is required. This restores the constraint without changing the sourced wave equation.

References

• Blanchet, Luc. “Post-Newtonian Theory for Gravitational Waves.” Living Reviews in Relativity 27 (2024) 4. DOI. Open PDF.
• Thorne, Kip S. “Multipole Expansions of Gravitational Radiation.” Reviews of Modern Physics 52 (1980) 299–339. DOI.