Regimes and Approximations

An approximation is controlled only after its small parameter, fixed data, domain, error measure, and stopping condition have all been named. This directory separates several limits that are often conflated and routes each one to the Library treatment that supplies its derivation.

Helpful background. Differential equations for relativity supplies the asymptotic, initial-value, and boundary-value language used to state a controlled regime.

Choose the controlled approximation

Approximation	Expansion or hierarchy	Assumptions to check	First sign of breakdown
Weak-Field and Slow-Motion Limits	Metric amplitude, $\lvert\Phi\rvert/c^2$, $v/c$, and time-versus-space derivatives	Gauge choice, matter scaling, curvature or an invariant observable	Strong potential, relativistic matter, rapid source evolution, or nonlinear field terms
PN, PM, Self-Force, and EOB Regimes	Orbital velocity, powers of $G$, small-body ratio, or a mapped and resummed description	Variables held fixed, invariant comparison quantity, matching region, and accumulated time	Loss of scale separation, a nonuniform long-time term, or disagreement in an overlap region
Geometric Optics and WKB	Wavelength divided by the background-variation scale	Wave system, branch, ray family, transport order, and local remainder	Turning point, caustic, mode conversion, sharp boundary, or Stokes transition
Black-Hole Perturbation Validity	Perturbation amplitude together with derivative, duration, detuning, and near-extremal scales	Background family, gauge-invariant observable, norm, boundary data, and perturbative order	Secular growth, resonance, background drift, nonlinear coupling, or throat-scale enhancement
Curved-Spacetime QFT and Semiclassical Validity	Curvature, proper frequency, adiabaticity, backreaction, fluctuation, and species-counting ratios	State, renormalized couplings, smearing scale, causal response, and observable tolerance	Cutoff-scale physics, inadmissible states, large response, unsuppressed fluctuations, or secular failure

These are independent coordinates, not labels on a single ladder. A metric can carry a small-amplitude wave with rapid time dependence; a system can have a small mass ratio in a strong field; and a low-curvature geometry can support a state with cutoff-scale proper frequencies.

Read the small parameter as a contract

For a quantity $Q$ approximated through order $N$, the useful statement is not merely $\epsilon\ll1$ but

$$Q=\sum_{n=0}^{N}\epsilon^n Q^{(n)}+R_N, \qquad \lVert R_N\rVert_{D}\leq C_D\epsilon^{N+1}.$$

The record must therefore say what $\epsilon$ measures, which variables are held fixed as it tends to zero, the domain $D$ and norm or observable used for the remainder, and how the coefficient $C_D$ behaves. A local series may be excellent while a phase error accumulates over long times or while the bound loses uniformity near a boundary. An overlap calculation, conservation law, invariant comparison, or direct numerical solution supplies the independent check.

Move from selection to derivation

These cards answer whether a regime is applicable and where it fails. The linked Library chapters derive the equations and explain the physics; the Methods directory covers matching, asymptotic repair, numerical verification, and other execution techniques. Accuracy claims that depend on a current model, calibration set, observation, or disputed frontier belong in a dated Research assessment, not in a timeless regime card.