Geodesics and the Newtonian Limit

The equivalence principle says a freely falling particle feels no force: in its local inertial frame it moves in a straight line. Translating straight line to a curved manifold gives the geodesic, the worldline that is as straight as the geometry permits. This lesson derives the geodesic equation twice — from parallel transport and from extremal proper time — introduces the conserved quantities that symmetries supply, and shows that in the weak, slow limit the geodesic equation becomes Newton's second law with $\vec{g}=-\nabla \Phi$, identifying the metric component $g_{00}$ with the Newtonian potential.

Geodesics as straightest worldlines

In flat space a straight line is a curve whose tangent vector never changes direction. The covariant generalization is a curve that parallel-transports its own tangent vector: the tangent is carried along the curve by the connection without turning. With tangent $u^{\mu }=\mathbf{d}{x}^{\mu }/\mathbf{d}\lambda$, the condition $u^{\nu }{\nabla }_{\nu }{u}^{\mu }=0$ reads

This is the geodesic equation. The Christoffel term is the correction that distinguishes a genuinely straight worldline from one that only looks straight in a bad coordinate system: on a flat plane in Cartesian coordinates $\Gamma =0$ and the equation says ${\mathbf{d}}^2{x}^{\mu }/\mathbf{d}{\lambda }^2=0$, ordinary straight lines. In polar coordinates the same straight lines satisfy the full equation with the nonzero polar Christoffel symbols supplying the apparent curving of the coordinate description.

The parameter $\lambda$ is not arbitrary. The equation in the form above holds only for an affine parameter, one related to proper time (for a massive particle) by $\lambda =a\tau +b$ with constants $a,b$. A general reparametrization $\lambda \to \sigma (\lambda )$ adds a term proportional to $u^{\mu }$ on the right-hand side; affine parameters are precisely the ones that keep the right-hand side zero. For timelike geodesics, proper time $\tau$ is the canonical affine parameter, and the tangent is then the four-velocity $u^{\mu }=\mathbf{d}{x}^{\mu }/\mathbf{d}\tau$ with normalization $g_{\mu \nu }{u}^{\mu }{u}^{\nu }=-c^2$, preserved along the geodesic because $\nabla$ is metric-compatible.

Geodesics as extremal proper time

The second derivation makes geodesics a variational principle, the route that generalizes cleanly and supplies conserved quantities. A timelike worldline between two fixed events has a proper time

with ${\dot{x}}^{\mu }=\mathbf{d}{x}^{\mu }/\mathbf{d}\lambda$. The worldline a free particle follows extremizes this proper time (a maximum for timelike geodesics between timelike-separated events, as the twin paradox showed). Extremizing the square root is awkward, so use the equivalent action with the square root removed,

which has the same extremals when $\lambda$ is affine. The Euler–Lagrange equations $\frac{\mathbf{d}}{\mathbf{d}\lambda }\frac{\partial L}{\partial {\dot{x}}^{\sigma }}=\frac{\partial L}{\partial x^{\sigma }}$ give

Symmetrizing the second term in $\mu \nu$ and contracting with $g^{\rho \sigma }$ reconstructs exactly the Christoffel combination, returning the geodesic equation

The two derivations agree: the straightest worldline (parallel transport) is the extremal-proper-time worldline (variation). This Lagrangian $L$ is also the fastest practical way to compute Christoffel symbols, since reading the Euler–Lagrange equations off $L$ and matching to the geodesic form displays the $\Gamma$'s directly.

Conserved quantities and Killing vectors

The Lagrangian route exposes conservation laws directly. If the metric components do not depend on a coordinate $x^{\sigma }$ — the metric has a symmetry in that direction — then ${\partial }_{\sigma }L=0$ and the corresponding conjugate momentum is conserved along every geodesic,

A static metric ($t$-independent) conserves $u_t$, interpreted as energy per unit mass; an axisymmetric metric ($\varphi$-independent) conserves $u_{\varphi }$, the angular momentum per unit mass. These two conservation laws are what make the Schwarzschild orbit problem solvable, reducing it to one-dimensional motion in an effective potential.

The coordinate-free statement uses a Killing vector ${\xi }^{\mu }$, a vector field generating a symmetry of the metric, defined by Killing's equation

Whenever ${\xi }^{\mu }$ is a Killing vector, the quantity ${\xi }_{\mu }{u}^{\mu }$ is conserved along any geodesic. The proof is one line: along a geodesic $u^{\nu }{\nabla }_{\nu }({\xi }_{\mu }{u}^{\mu })=u^{\mu }{u}^{\nu }{\nabla }_{\nu }{\xi }_{\mu }+{\xi }_{\mu }{u}^{\nu }{\nabla }_{\nu }{u}^{\mu }$; the second term vanishes by the geodesic equation, and the first vanishes because $u^{\mu }{u}^{\nu }$ is symmetric while ${\nabla }_{\nu }{\xi }_{\mu }$ is antisymmetric by Killing's equation. A coordinate the metric ignores corresponds to a Killing vector $\xi ={\partial }_{\sigma }$, recovering the elementary statement above.

The Newtonian limit

General relativity must contain Newtonian gravity as the limit of weak fields and slow motion, and the geodesic equation delivers it. Three assumptions define the limit.

• Slow motion. Speeds are small, $\mathbf{d}{x}^i/\mathbf{d}t\ll c$, so on the worldline the spatial velocity components of the four-velocity are negligible next to the time component: $\mathbf{d}{x}^i/\mathbf{d}\tau \ll \mathbf{d}{x}^0/\mathbf{d}\tau$, and $\mathbf{d}\tau \approx \mathbf{d}t$.
• Weak field. The metric departs only slightly from flat, $g_{\mu \nu }={\eta }_{\mu \nu }+h_{\mu \nu }$ with $|h_{\mu \nu }|\ll 1$, and products of $h$ and its derivatives are dropped.
• Static field. The metric is time-independent, ${\partial }_0{g}_{\mu \nu }=0$.

With slow motion the geodesic equation's quadratic velocity term keeps only the $00$ piece,

For a static weak field the relevant Christoffel symbol reduces to ${\Gamma }^{\mu }{}_{00}=-\frac{1}{2}{\eta }^{\mu \lambda }{\partial }_{\lambda }{h}_{00}$, which is purely spatial (the time component vanishes because ${\partial }_0{h}_{00}=0$). Taking the spatial components and using $\mathbf{d}{x}^0/\mathbf{d}\tau \approx c\mathbf{d}t/\mathbf{d}\tau \approx c$,

Comparison with Newton's law ${\mathbf{d}}^2{x}^i/\mathbf{d}{t}^2=-{\partial }_i\Phi$ identifies

The time-time component of the metric is the Newtonian potential. Gravity, in the Newtonian regime, is entirely the bending of the time direction: clocks run at a rate set by $\sqrt{-g_{00}}\approx 1+\Phi /c^2$, and the geodesics of that warped time reproduce falling apples. For the Sun at Earth's orbit $\Phi /c^2\sim 10^{-8}$, and at the Earth's surface $\sim 10^{-9}$, confirming that the weak-field expansion is excellent everywhere in the solar system while still producing measurable effects.

The geodesic equation governs how matter moves once the metric is known, but it says nothing about tidal forces — the relative acceleration of neighbouring geodesics that the equivalence principle isolated as the true signature of gravity. Two nearby geodesics separate at a rate set by the curvature of the manifold, and quantifying that separation is the Riemann tensor, the subject of the next lesson.