Weak-Field and Slow-Motion Limits

Weak field, slow motion, and slow time dependence are independent assumptions. A small metric perturbation can describe a rapidly oscillating wave, while slow matter need not generate a weak field. This card separates the assumptions and shows which combinations give linearized gravity, Newtonian gravity, or weak radiative gravity. The complete derivations belong to Weak-Field Gravity and Classic Predictions.

Helpful background. Differential equations for relativity supplies the derivative and Poisson-equation language used to compare the quasistatic and radiative orderings.

Independent measures of smallness

Use $x^0=ct$ and the inherited signature $(-,+,+,+)$. On a stated region and in a stated gauge, a weak perturbation of Minkowski spacetime has

$$g_{\mu\nu}=\eta_{\mu\nu}+h_{\mu\nu}, \qquad \eta_{\mu\nu}=\operatorname{diag}(-1,1,1,1), \qquad \lVert h\rVert=O(\epsilon_h), \quad \epsilon_h\ll1.$$

The norm and domain matter: componentwise smallness in one chart is not a gauge-invariant statement. Linearization also requires the discarded $O(h^2)$ terms, including derivative products such as $h\,\partial^2h$ and $(\partial h)^2$, to remain smaller than the retained linear terms.

The other useful ratios are

$$\epsilon_\Phi=\frac{\lvert\Phi\rvert}{c^2}, \qquad \epsilon_v=\frac{v}{c}, \qquad \epsilon_t=\frac{L}{cT},$$

where $L$ is the spatial scale and $T$ the source-evolution time. Quasistatic ordering means

$$\frac{1}{c}\partial_t=O(\epsilon_t\partial_i), \qquad \epsilon_t\ll1,$$

not that time derivatives vanish. The weak-field and post-Newtonian bookkeeping, including the need to keep source motion, probe motion, and gauge conditions distinct, is summarized in Will 2014, §§ 3.1–3.2, printed pp. 26–33.

Which assumptions define each limit

Regime	Required control	What follows	What does not follow
Linearized gravity	$\epsilon_h\ll1$ and nonlinear residuals small in the declared norm	First-order connection, curvature, and Einstein equations	Slow motion or quasistatic fields
Weak potential	$\epsilon_\Phi\ll1$	Small gravitational potential relative to $c^2$	A Newtonian matter hierarchy or a gauge-invariant bound on every $h_{\mu\nu}$
Newtonian gravity	$\epsilon_v=O(\epsilon)$, $\epsilon_\Phi=O(\epsilon^2)$, $\epsilon_t=O(\epsilon)$, and stresses small relative to $\rho c^2$	Slow geodesic motion and Poisson’s equation at leading order	Relativistic pressure, rapid radiation, or precision post-Newtonian dynamics
Weak radiative gravity	$\epsilon_h\ll1$ with wave derivatives retained	Small-amplitude linearized waves	Quasistatic ordering or small frequency

For ordinary slow matter, the Newtonian hierarchy may be written

$$\frac{T_{0i}}{T_{00}}=O(\epsilon), \qquad \frac{T_{ij}}{T_{00}}=O(\epsilon^2), \qquad h_{00},h_{ij}=O(\epsilon^2), \qquad h_{0i}=O(\epsilon^3).$$

In a compatible weak-field gauge this gives

$$g_{00}=-\left(1+\frac{2\Phi}{c^2}\right)+O(\epsilon^4), \qquad \nabla^2\Phi=4\pi G\rho.$$

The field-equation normalization, trace reversal, and gauge reduction are derived in Reall 2013, §§ 8.1–8.2, printed pp. 77–83, PDF. See Weak Metric Perturbations and Gauge Freedom and The Newtonian Limit of the Einstein Equation for the full calculation. Distinct measures of field strength are compared in Compactness and Curvature.

Gauge and limiting checks

Pure gauge. On a flat background, $h_{\mu\nu}=-2\partial_{(\mu}\xi_{\nu)}$ can have nonzero components while its linearized Riemann tensor vanishes. A component norm must therefore be paired with a gauge choice and an invariant curvature or observable check.

Static source. For an isolated positive mass, $\Phi=-GM/r$ gives $g_{00}=-1+2GM/(rc^2)$ and $-\boldsymbol\nabla\Phi=-GM\mathbf n/r^2$, directed inward. This checks the potential sign and the Newtonian limit.

Fast weak wave. A plane perturbation $h_{\mu\nu}=\operatorname{Re}(A_{\mu\nu}e^{ik\cdot x})$ may have $\lVert A\rVert\ll1$ and $k^2=0$ while its frequency is arbitrarily large relative to the source-evolution scale. It is weak and radiative, not quasistatic.

For any claimed approximation, state the region, gauge, norm or observable, quantities held fixed, boundary data, and first omitted term. Loss of smallness, growth of the nonlinear residual, relativistic stress, or failure of the derivative hierarchy ends the corresponding claim; it does not automatically end every weaker one.

References

• Reall, Harvey S. General Relativity. Cambridge Part III lecture notes, 2013, §§ 8.1–8.2, printed pp. 77–83. PDF.
• Will, Clifford M. “The Confrontation between General Relativity and Experiment.” Living Reviews in Relativity 17 (2014): 4, §§ 3.1–3.2, printed pp. 26–33. DOI. Open PDF.