---
title: Geodesics and the Newtonian Limit
module: Curved Spacetime
moduleNumber: 5
lessonNumber: 4
order: 504
summary: >
  Free fall is geodesic motion: a freely falling particle follows the straightest
  possible worldline, obtained either by parallel-transporting its own tangent
  vector or by extremizing proper time. Both routes give the geodesic equation.
  Affine parameters, and conserved quantities from symmetries via Killing vectors,
  make it solvable. In the weak-field slow-motion limit the geodesic equation
  reproduces Newton's law of gravity, fixing the time-time metric component as
  the Newtonian potential.
topics: [Curved Spacetime]
draft: false
sources:
  - book: Hartle
    ref: "Gravity, Ch. 8 — Geodesics"
  - book: Carroll
    ref: "Lecture Notes on General Relativity, §3 — Curvature"
---

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 $u^\mu = \d x^\mu/\d\lambda$, the condition
$u^\nu\nabla_\nu u^\mu = 0$ reads

$$
\frac{\d^2 x^\mu}{\d\lambda^2}
+ \Gamma^\mu{}_{\nu\rho}\,\frac{\d x^\nu}{\d\lambda}\frac{\d x^\rho}{\d\lambda} = 0.
$$

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 $\d^2 x^\mu/\d\lambda^2 = 0$, 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 $u^\mu = \d
x^\mu/\d\tau$ with normalization $g_{\mu\nu}u^\mu u^\nu = -c^2$, preserved along
the geodesic because $\nabla$ is metric-compatible.

$$
% caption: A geodesic parallel-transports its own tangent vector: the velocity
% arrow is carried along the worldline without turning relative to the local
% geometry. The Christoffel term supplies the correction that makes "no turning"
% coordinate-independent.
\begin{tikzpicture}[>=stealth, font=\footnotesize, scale=1.0]
  \definecolor{acc}{HTML}{4A6FA5}
  % curved worldline
  \draw[acc, very thick] (0,0) .. controls (1.5,0.6) and (3.0,1.6) .. (4.5,3.0);
  \node[acc, anchor=north west] at (0.1,0.1) {geodesic};
  % tangent arrows along it, each aligned with the curve
  \draw[->, black, thick] (0.75,0.28) -- ++(0.75,0.35);
  \draw[->, black, thick] (2.25,1.0) -- ++(0.7,0.55);
  \draw[->, black, thick] (3.6,2.1) -- ++(0.55,0.7);
  \node[black, anchor=west] at (1.3,0.35) {tangent};
  \node[black, anchor=west] at (3.7,2.35) {carried along};
\end{tikzpicture}
$$

## 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

$$
\tau = \int \d\tau = \frac{1}{c}\int \sqrt{-g_{\mu\nu}\,\d x^\mu\,\d x^\nu}
= \frac{1}{c}\int \sqrt{-g_{\mu\nu}\,\dot x^\mu \dot x^\nu}\;\d\lambda,
$$

with $\dot x^\mu = \d x^\mu/\d\lambda$. 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,

$$
S = \int L\,\d\lambda, \qquad L = \tfrac{1}{2}\,g_{\mu\nu}\,\dot x^\mu \dot x^\nu,
$$

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

$$
g_{\sigma\nu}\ddot x^\nu + \partial_\mu g_{\sigma\nu}\,\dot x^\mu \dot x^\nu
= \tfrac{1}{2}\,\partial_\sigma g_{\mu\nu}\,\dot x^\mu \dot x^\nu.
$$

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

$$
\ddot x^\rho + \Gamma^\rho{}_{\mu\nu}\,\dot x^\mu \dot x^\nu = 0.
$$

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.

$$
% caption: Among timelike worldlines joining two events, the geodesic extremizes
% proper time; neighbouring paths accumulate less. On the sphere the analogous
% extremal curves are great circles, the shortest paths between two points.
\begin{tikzpicture}[>=stealth, font=\footnotesize, scale=1.0]
  \definecolor{acc}{HTML}{4A6FA5}
  % left: spacetime extremal
  \draw[->, black] (-0.2,-0.2) -- (-0.2,3.4) node[above] {ct};
  \draw[->, black] (-0.4,0) -- (2.6,0) node[right] {x};
  \fill[black!70] (0.2,0.2) circle (1.7pt);
  \fill[black!70] (0.2,3.0) circle (1.7pt);
  \draw[acc, very thick] (0.2,0.2) -- (0.2,3.0);
  \node[acc, anchor=west] at (0.3,2.4) {geodesic};
  \draw[black, dashed] (0.2,0.2) .. controls (1.5,1.0) and (1.4,2.2) .. (0.2,3.0);
  \node[black, anchor=west] at (0.95,0.85) {less proper time};
  % right: great circle on sphere
  \draw[black, very thick] (5.0,1.4) circle (1.6);
  \draw[black, dashed] (5.0,1.4) ellipse (1.6 and 0.5);
  \draw[acc, very thick] (3.9,1.9) .. controls (4.6,2.3) and (5.4,2.3) .. (6.1,1.9);
  \node[acc, anchor=south] at (5.0,2.35) {great circle};
  \fill[black!70] (3.9,1.9) circle (1.6pt);
  \fill[black!70] (6.1,1.9) circle (1.6pt);
\end{tikzpicture}
$$

## 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,

$$
p_\sigma = \frac{\partial L}{\partial \dot x^\sigma} = g_{\sigma\nu}\dot x^\nu
= u_\sigma = \text{const}.
$$

A static metric ($t$-independent) conserves $u_t$, interpreted as energy per unit
mass; an axisymmetric metric ($\phi$-independent) conserves $u_\phi$, 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

$$
\nabla_\mu \xi_\nu + \nabla_\nu \xi_\mu = 0.
$$

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.

$$
% caption: A symmetry direction of the metric is a Killing vector; the projection
% of a particle's four-velocity onto it stays constant along the geodesic. Time
% translation gives conserved energy, rotation gives conserved angular momentum.
\begin{tikzpicture}[>=stealth, font=\footnotesize, scale=1.0]
  \definecolor{acc}{HTML}{4A6FA5}
  % worldline
  \draw[acc, very thick] (0,0.1) .. controls (1.6,0.7) and (3.0,1.1) .. (4.6,1.8);
  \node[acc, anchor=north west] at (0.05,0.1) {geodesic};
  % Killing field arrows (uniform, e.g. horizontal symmetry)
  \foreach \x/\y in {1.0/0.35,2.4/0.72,3.7/1.2} {
    \draw[->, black] (\x,\y) -- ++(0.75,0);
  }
  \node[black, anchor=west] at (3.85,0.95) {Killing vectors};
  % projection stays constant label
  \node[black, anchor=west] at (0.3,2.5) {projection stays constant};
\end{tikzpicture}
$$

## 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, $\d x^i/\d t \ll c$, so on the worldline the
  spatial velocity components of the four-velocity are negligible next to the time
  component: $\d x^i/\d\tau \ll \d x^0/\d\tau$, and $\d\tau \approx \d t$.
- **Weak field.** The metric departs only slightly from flat,
  $g_{\mu\nu} = \eta_{\mu\nu} + h_{\mu\nu}$ with $\lvert h_{\mu\nu}\rvert \ll 1$,
  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,

$$
\frac{\d^2 x^\mu}{\d\tau^2} + \Gamma^\mu{}_{00}\Big(\frac{\d x^0}{\d\tau}\Big)^2 = 0.
$$

For a static weak field the relevant Christoffel symbol reduces to
$\Gamma^\mu{}_{00} = -\tfrac{1}{2}\eta^{\mu\lambda}\partial_\lambda h_{00}$, which
is purely spatial (the time component vanishes because $\partial_0 h_{00} = 0$).
Taking the spatial components and using $\d x^0/\d\tau \approx c\,\d t/\d\tau
\approx c$,

$$
\frac{\d^2 x^i}{\d t^2} = -\tfrac{1}{2}c^2\,\partial_i h_{00}.
$$

Comparison with Newton's law $\d^2 x^i/\d t^2 = -\partial_i \Phi$ identifies

$$
h_{00} = -\frac{2\Phi}{c^2},
\qquad
g_{00} = -\Big(1 + \frac{2\Phi}{c^2}\Big).
$$

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.

$$
% caption: In the weak, slow, static limit the geodesic equation reduces to
% Newton's law of gravity, and matching fixes the time-time metric component as one
% plus twice the Newtonian potential over c squared. Gravity is the warping of the
% time direction.
\begin{tikzpicture}[>=stealth, font=\footnotesize, scale=1.0]
  \definecolor{acc}{HTML}{4A6FA5}
  \draw[acc, thick, fill=acc!10] (0,0.9) rectangle (3.0,2.1);
  \node[acc, anchor=center] at (1.5,1.5) {geodesic equation};
  \draw[->, black] (3.05,1.5) -- (4.35,1.5);
  \node[black, anchor=south] at (3.7,1.6) {weak, slow};
  \draw[black, thick] (4.45,0.9) rectangle (7.7,2.1);
  \node[black, anchor=center] at (6.05,1.5) {Newton g = grad potential};
  \node[black, anchor=north] at (3.55,0.35) {match sets time-time metric};
\end{tikzpicture}
$$

> **Worked example.** A clock at rest at radius $r$ in the static field of a mass
> $M$ has $\Phi = -GM/r$, so its proper time runs at rate
> $\d\tau/\d t = \sqrt{-g_{00}} \approx 1 - GM/(rc^2)$ relative to a clock at
> infinity. A clock deeper in the well ($r$ smaller) ticks slower. For two clocks
> at heights differing by $H$ near the Earth's surface, $\Phi = gH$ gives a
> fractional rate difference $gH/c^2$; over $H = 22.5\text{ m}$ this is
> $\approx 2.5\times 10^{-15}$, the shift the Pound–Rebka experiment measured and
> the effect the GPS lesson quantifies. The prediction here rests only on the
> geodesic Newtonian limit, before any exact solution is known.

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.

[^hartle-geo]: **Hartle**, _Gravity_, Ch. 8 — geodesics as extremal-proper-time worldlines, the geodesic equation from the variational principle, conserved quantities from metric symmetries, and the Newtonian limit fixing $g_{00}$.

[^carroll-geo]: **Carroll**, _Lecture Notes on General Relativity_, §3 — the geodesic equation from parallel transport and from the action, affine parameters, Killing vectors and conserved quantities, and the weak-field correspondence with Newtonian gravity.