postMinkowskian

Post-Minkowskian (PM) expansion is a perturbative method in general relativity that expands the spacetime metric around flat Minkowski space. It treats the gravitational field as a small perturbation and organizes the solution as a series in the gravitational coupling, effectively in powers of Newton’s constant G. The zeroth term is the Minkowski metric, the first-order term is linear in G (1PM), the second-order term is quadratic in G (2PM), and so on. Unlike the post-Newtonian approach, the PM expansion does not require slow motion; velocities can be relativistic as long as the gravitational field remains weak.

Formally, one writes g_{μν} = η_{μν} + h_{μν}^{(1)} + h_{μν}^{(2)} + ..., with each h^{(n)} ~ G^n. Einstein’s equations are then solved order by order, typically using a convenient gauge such as the harmonic (de Donder) gauge. The PM framework is particularly suited to weak-field regions and to problems where the gravitational field is small but velocities may be large, such as certain scattering problems and radiation calculations.

Applications include determining the gravitational field and dynamics of isolated systems in the weak-field regime, computing gravitational wave generation, and providing a bridge to numerical relativity and post-Newtonian results. In modern gravity research, the PM expansion also appears in effective field theory approaches to gravity, where gravity is treated perturbatively in G to study quantum and classical corrections.

Relation to PN: PM expands in G, while PN expands in v/c; the two can be combined to describe different regimes. Limitations include its unsuitability for strong-field regions near compact objects where g_{μν} deviates significantly from η_{μν}.