THE NEWTONIAN LIMIT

Every new theory of physics must pass a minimum requirement: reproduce the successes of whatever it replaces. built the field equations to describe gravity geometrically, but the equations had to recover 's inverse-square law in every regime where Newton had been tested and confirmed — planetary orbits, falling apples, ocean tides. If the Einstein field equations failed that test, they would be wrong regardless of their mathematical elegance.

The limit that enforces this test has three ingredients, taken simultaneously:

Weak field. The metric is nearly flat: $g_{\mu\nu} = \eta_{\mu\nu} + h_{\mu\nu}$ where the perturbation $h_{\mu\nu}$ is small in every component, $|h| \ll 1$. The spacetime curvature is a small ripple on flat Minkowski geometry.

Slow motion. All matter moves at speeds much less than light, $v \ll c$. This suppresses the spatial components of the four-velocity relative to the time component, so the geodesic equation simplifies drastically.

Static field. The mass distribution is not changing on the timescale of interest, $\partial_t h_{\mu\nu} \approx 0$. No gravitational waves, no time-varying potentials.

Under all three conditions the full machinery of General Relativity collapses, term by term, into a single second-order equation you have known since first-year physics. The derivation of that collapse is the subject of this section.

Write the metric as

$g_{\mu\nu} = \eta_{\mu\nu} + h_{\mu\nu}, \quad |h_{\mu\nu}| \ll 1$

and expand the Einstein field equations to first order in $h$. The Christoffel symbols are linear in $\partial h$; the Riemann and Ricci tensors are linear in $\partial^2 h$ (all quadratic terms are dropped). This is the linearised theory of gravity, valid everywhere the field is weak.

Now invoke slow motion. The geodesic equation is

$\frac{d^2 x^\mu}{d\tau^2} + \Gamma^\mu_{\alpha\beta} \frac{dx^\alpha}{d\tau}\frac{dx^\beta}{d\tau} = 0$

For slow motion, $dx^i/d\tau \ll dx^0/d\tau \approx c$, so the dominant term in the double sum is $(\alpha, \beta) = (0, 0)$. The spatial components of the geodesic equation reduce to

$\frac{d^2 x^i}{dt^2} \approx -\frac{1}{2}\,\partial^i h_{00}\,c^2$

This is Newton's second law for a particle in a potential, with the right-hand side playing the role of the gravitational acceleration $\mathbf{g} = -\nabla\Phi$. The only metric component that drives slow-motion geodesics is $h_{00}$ — the time-time perturbation. All of the spatial components of the metric are irrelevant at this order.

Comparing EQ.02 with Newton's law $d^2\mathbf{x}/dt^2 = -\nabla\Phi$ immediately gives

$h_{00} = -\frac{2\Phi}{c^2}$

where $\Phi(\mathbf{x})$ is the Newtonian gravitational potential. The identification is exact in the weak-field, slow-motion, static limit.

The dimensionless parameter $\varepsilon = |\Phi|/c^2$ measures how far from flat the spacetime is:

Newton's gravity is the leading order of a perturbation series in $\varepsilon$. It is not wrong — it is a truncation of the deeper theory, accurate to extraordinary precision wherever $\varepsilon \ll 1$.

Now take the same weak-field limit on the left-hand side of the EFE. Expand the Einstein tensor $G_{\mu\nu} = R_{\mu\nu} - \frac{1}{2}R\,g_{\mu\nu}$ to first order in $h$. The 00-component in the static case reduces to

$G_{00} \approx \nabla^2 h_{00}$

(all time-derivative terms vanish by the static assumption; the spatial Laplacian of $h_{00}$ survives).

On the matter side, the Stress-energy tensor $T_{00}$ for slow, pressureless matter is dominated by the rest-mass energy density:

$T_{00} \approx \rho c^2$

The 00-component of the full EFE, $G_{\mu\nu} = \frac{8\pi G}{c^4}\,T_{\mu\nu}$, therefore becomes

$\nabla^2 h_{00} = \frac{8\pi G}{c^4}\,\rho c^2 = \frac{8\pi G \rho}{c^2}$

Substitute EQ.03, $h_{00} = -2\Phi/c^2$:

$\nabla^2\Phi = 4\pi G\rho$

This is Poisson's equation — the exact statement of Newtonian gravity for a continuous mass distribution. The coefficient $8\pi G/c^4$ in the Einstein field equations is uniquely fixed by requiring this match. Any other coefficient would give a different coupling between geometry and matter and would fail to reproduce Newton's law.

Having shown that GR contains Newton, we can ask precisely where Newton stops being sufficient. The answer is a two-dimensional parameter space: field strength $\varepsilon = |\Phi|/c^2$ and velocity ratio $v/c$. Newton is the approximation valid for small $\varepsilon$ and small $v/c$ simultaneously.

The first quantitative departure from the Newtonian prediction is Mercury's perihelion precession. Newton's gravity predicts that a planet in a static spherical gravitational field follows a closed ellipse with no precession. Mercury's orbit is not quite closed — it precesses at a rate that cannot be fully accounted for by planetary perturbations. The residual, once all Newtonian perturbations are subtracted, is 43.0 arcseconds per century. The General Relativistic post-Newtonian correction, computed from the Schwarzschild geometry, gives exactly 43.0 arcseconds per century. This was one of the three classic tests that confirmed GR.

The second departure is the bending of light at the solar limb. Newton's corpuscular theory of light predicts a deflection of 0.87 arcseconds as a ray grazes the Sun. The correct GR value is 1.75 arcseconds — exactly twice the Newtonian prediction. The factor of 2 arises because the spatial components of the metric also contribute to photon geodesics, and Newton has no equivalent term. Eddington's 1919 eclipse expedition measured this value and made Einstein famous.

Newton was right where it counted, and the deeper theory shows him exactly where his approximation breaks. The transition from Poisson's equation to the full field equations is not a replacement but a revelation: the Newtonian potential is the leading term $h_{00}$ in a metric perturbation, and the acceleration of a falling body is the gradient of a single metric component. All of Newton's gravity lives in $g_{00}$ alone. The rest of the metric — the spatial components — only become observable when light bends, or orbits precess, or clocks drift.

The next step is to find the first exact solution to the Einstein field equations that goes beyond the Newtonian limit: the Schwarzschild metric. could derive the 43"/century from the linearised theory, but it was working with the exact Schwarzschild solution that gave the definitive calculation. The Schwarzschild metric is the unique spherically symmetric vacuum solution — no approximation, no perturbation, no linearisation. It is the geometry that Newton's inverse-square law is approximating.