Relativity and gravitational theory Foundations of general relativity Fact

Newtonian limit

Builds on Einstein field equations Geodesics in general relativity Linearized gravity Poisson's equation Newton's law of universal gravitation Stress–energy tensor

Statement

Let $G$ be the gravitational constant and $c$ the speed of light. Let $(M, g)$ be a solution of the Einstein field equations $G_{\mu\nu} = \frac{8\pi G}{c^4} T_{\mu\nu}$ with zero cosmological constant. Assume that in coordinates $(x^0, x^1, x^2, x^3) = (ct, x, y, z)$ :

the field is weak: $g_{\mu\nu} = \eta_{\mu\nu} + h_{\mu\nu}$ with $\eta = \mathrm{diag}(-1, 1, 1, 1)$ and $\abs{h_{\mu\nu}} \ll 1$ , so only terms linear in $h$ are kept;
the field is quasi-static: time derivatives of $h_{\mu\nu}$ are negligible compared with spatial derivatives;
the matter moves slowly and has small pressure, so its stress–energy tensor is dominated by $T_{00} = \rho c^2$ , with $\rho$ the mass density.

Then, up to a choice of coordinates, $g_{00} = -\left(1 + 2\Phi/c^2\right)$ , where $\Phi$ solves Poisson's equation $\nabla^2 \Phi = 4\pi G \rho$ . Moreover, a test particle moving along a timelike geodesic with speed $\abs{v} \ll c$ obeys $\frac{d^2 x^i}{dt^2} = -\partial_i \Phi$ to leading order, which is Newton's law of gravitation.

Remarks

The proof uses linearized gravity. For a slow particle, $dx^0/d\tau \approx c$ dominates, so the geodesic equation becomes $\frac{d^2 x^i}{dt^2} \approx -c^2 \Gamma^i_{00} = \frac{c^2}{2} \partial_i h_{00}$ , and this is $-\partial_i \Phi$ when $h_{00} = -2\Phi/c^2$ . In the harmonic gauge the linearized $00$ field equation is $\nabla^2 \bar h_{00} = -16\pi G \rho / c^2$ for the trace-reversed perturbation, which gives $\bar h_{00} = -4\Phi/c^2$ and then $h_{00} = -2\Phi/c^2$ .

Example: outside a spherical body of mass $M$ , $\Phi = -GM/r$ , and $g_{00} = -(1 - 2GM/(rc^2))$ , which is the time-time component of the Schwarzschild metric.

The slow-motion hypothesis is needed: for light, $v = c$ , the spatial components $h_{ij} = -2\Phi \delta_{ij}/c^2$ contribute as much as $h_{00}$ . This is why the bending of light in general relativity is twice the value obtained from Newtonian gravity alone.

Going deeper

For a student

The limit is the first check that any theory of gravity must pass, and it fixes the constant $8\pi G/c^4$ in the Einstein equations. It is the leading term of the post-Newtonian expansion, whose higher orders give the relativistic corrections tested in the solar system. The derivation shows that $g_{00}$ alone controls slow motion, while $g_{ij}$ matters for fast particles and light.

Reading path

To understand this page from the ground up, read these 49 pages in order. The foundations come first.

Derivative concept
Differential equation concept
Einstein field equations concept
Euclidean space object
Euclidean vector concept
Function concept
Geodesics in general relativity concept
Interval concept
Linearized gravity concept
Ordinary differential equation concept
Poisson's equation concept
Stress–energy tensor concept
Velocity concept
Fréchet derivative definition
Ordinary differential equation definition
Cross product definition
Function of several variables concept
Vector-valued function concept
Lipschitz condition definition
Classical mechanics concept
Limit of a function of several variables concept
Momentum concept
Partial derivative concept
Solution of an ordinary differential equation definition
Inertial frame of reference concept
Continuity in R^n definition
Differentiable function of several variables definition
Partial derivative definition
Initial value problem definition
Newton's laws of motion statement
Differentiable on an open set definition
Gradient definition
Higher partial derivatives definition
Picard–Lindelöf hypotheses definition
Iterated derivative definition
Picard operator definition
C^k function characterised by continuous derivatives statement
Integral equation form of an initial value problem statement
An iterate of the Picard operator is a contraction statement
Vector field definition
Picard–Lindelöf theorem statement
Conservative vector field definition
Classical mechanics definition
Momentum definition
Potential energy definition
Angular momentum definition
Central force definition
Newton's law of universal gravitation statement
Newtonian limit statement

Tag	`X87KPFF17K`
URI	`m2://X87KPFF17K`
Kind	statement, fact
Writing	W0 stub
MSC	83C25, 83C05
Status	This copy of the site has no build data. Run `m2 status X87KPFF17K`.
About	Einstein field equations, Newton's law of universal gravitation
Uses	Einstein field equations, Geodesics in general relativity, Linearized gravity, Poisson's equation, Newton's law of universal gravitation, Stress–energy tensor
Same as	Wikidata Q7020536 Wikipedia Newtonian limit
Changed	3 November 2026