Weakfield

Weakfield refers to a regime in which a physical field is sufficiently small that nonlinear terms in the governing equations can be neglected, allowing a linear approximation. This concept appears in gravity, electromagnetism, and other field theories, and it enables analytic solutions and perturbative methods.

In general relativity, the weak-field approximation describes spacetime that is only slightly perturbed from flat Minkowski space. One writes the metric as gμν = ημν + hμν with |hμν| ≪ 1, and works in a gauge where the equations for hμν become linear. In the static, slow-motion limit, this reduces to the Newtonian Poisson equation ∇²Φ = 4πGρ, with the time-time component h00 ≈ −2Φ/c². The weak-field regime also provides a framework for gravitational waves, which are solutions to the linearized equations propagating on a nearly flat background. Relativistic corrections to Newtonian predictions are organized via post-Newtonian expansions within this regime.

The weak-field approximation is particularly applicable to gravitational phenomena in the Solar System and laboratory-scale experiments, where gravitational potentials are small and velocities are much less than the speed of light. It underpins many standard tests of general relativity, such as light deflection, time dilation, and orbital dynamics, through perturbative calculations. It also defines the domain where full nonlinear GR is unnecessary or intractable and where stronger-field methods must be employed.

Limitations include its breakdown near extremely compact objects like black holes or neutron stars, where the field strength is large and nonlinear effects become essential. In such strong-field regions, the full, nonlinear Einstein equations must be used.