Newtonian limit

The Newtonian limit is the regime in which Einstein's field equations reduce continuously to Newton's law of gravity. Three conditions define the limit: gravity is weak, so the metric departs only slightly from Minkowski (g_{μν} = η_{μν} + h_{μν} with |h_{μν}| ≪ 1); the sources are slow-moving (v ≪ c) and the configuration is static or nearly so; and the stress-energy tensor is dominated by rest-mass density (T_{00} ≈ ρc² ≫ p, momentum-flux components negligible). Under these conditions one identifies the gravitational potential Φ with the metric perturbation through h_{00} = −2Φ/c² — the time-time component of the metric carries the Newtonian potential — and works out, after some index-juggling, that G_{00} ≈ −2∇²Φ/c². The 00-component of the field equations then reads ∇²Φ = 4πGρ, which is exactly Poisson's equation for the Newtonian gravitational potential.

The recovery of Newton in this limit is what legitimises general relativity. Newtonian gravity, with all its centuries of experimental verification, must reappear as the slow-moving weak-field reduction of any acceptable relativistic theory of gravity — and the Einstein field equations satisfy this requirement structurally, with the constant 8πG/c⁴ on the right-hand side fixed precisely by demanding agreement. The same limit also reproduces the geodesic equation's reduction to Newton's second law with gravitational force −∇Φ, since in the Newtonian limit the geodesic equation's spatial components yield d²x^i/dt² ≈ −∂^i Φ. Beyond Newton, the next-order corrections in the post-Newtonian expansion encode the perihelion precession of Mercury, light deflection by the Sun, and the Shapiro time delay — the classical solar-system tests of GR.