The Geodesic Equation

We now have a spacetime whose shape is curved and a metric g_{\mu\nu} that measures it. One question remains before we can do any physics: how does a free particle move? Newton said an object with no force on it travels in a straight line at constant speed. Einstein keeps the spirit and drops the word "force": a body in free fall — feeling no push, no pull — follows the straightest possible path through curved spacetime. Such a path is called a geodesic, and the equation that picks it out is the equation of motion of all of gravity.

"Straightest possible" needs care once the ground beneath you is curved. On a sphere there are no straight lines at all — the honest replacement is the path that is locally as straight as it can be, turning neither left nor right as you walk it. These are the great circles: the routes aeroplanes actually fly. This page builds the idea in three steps — what a geodesic is, why free fall follows one, and how the Christoffel symbols package the "turning of the coordinates" into the famous geodesic equation.

What a geodesic is

A geodesic is the curved-space generalisation of a straight line. Two equivalent definitions capture it:

Extremal length. A geodesic is a path whose length (or, for a worldline, whose proper time) is stationary — a small wiggle changes it only to second order. Between two points on a plane the shortest path is the straight segment; on a sphere it is the shorter great-circle arc.
No turning. A geodesic is a path that carries its own tangent vector along "parallel to itself" — you never steer. Point yourself in a direction and walk without turning your wheel, and the track you leave is a geodesic.

The great-circle route is the everyday proof. Two cities on the same line of latitude look joined by a straight east–west line on a flat map, but a plane flying the shortest route bows toward the pole. That poleward-bowing arc is the geodesic; the latitude line only looks straight because the map is a distorted chart of a curved globe.

Free fall follows a geodesic

Now the physical postulate that replaces Newton's first law:

The law of motion. A test particle in free fall — acted on by gravity alone — traces a timelike geodesic of spacetime; a light ray traces a null geodesic.
Extremal ageing. Of all worldlines between two events, the free-fall one makes the proper time stationary — in fact a maximum. Coasting ages you the most.
No force needed. There is no gravitational force in the equation; the curvature of spacetime does all the steering.

The "maximum ageing" clause is the resolution of the twin paradox in one line: the stay-at-home twin follows a geodesic and ages the most; the travelling twin, who must fire rockets to turn around, takes a non-geodesic (accelerated) path and ages less. The graph below shows that any bent worldline accumulates less proper time than the straight, geodesic one at the centre.

The geodesic equation and the Christoffel symbols

To turn "as straight as possible" into a differential equation, we need to compare a vector at one point with a vector at a neighbouring point — but in curved coordinates the basis vectors themselves twist and stretch from place to place. The bookkeeping for that twisting is the Christoffel symbols \Gamma^\mu_{\ \alpha\beta}, built from first derivatives of the metric:

\Gamma^\mu_{\ \alpha\beta} = \tfrac12\, g^{\mu\nu}\!\left(\partial_\alpha g_{\nu\beta} + \partial_\beta g_{\nu\alpha} - \partial_\nu g_{\alpha\beta}\right).

They are not a tensor — they measure how the coordinate grid bends, and in a locally inertial (free-falling) frame they vanish at a point. With them, "the tangent vector does not turn" becomes the geodesic equation:

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

Read it as Newton in disguise. The first term is the coordinate acceleration d^2x^\mu/d\tau^2. The second term, -\Gamma^\mu_{\ \alpha\beta} v^\alpha v^\beta, plays the role of the gravitational "force per unit mass" — except it is not a force, it is the curvature of the coordinates acting on the particle's velocity. Where the Christoffels vanish (flat space, Cartesian coordinates) the equation collapses to d^2x^\mu/d\tau^2 = 0 — motion in a straight line at constant velocity, exactly Newton's first law. Gravity has become geometry.

A quick count keeps the bookkeeping honest. In n dimensions the symmetry \Gamma^\mu_{\ \alpha\beta} = \Gamma^\mu_{\ \beta\alpha} (in the lower pair) leaves n \cdot \tfrac{n(n+1)}{2} = \tfrac{n^2(n+1)}{2} independent symbols — that is 40 in four-dimensional spacetime — and the geodesic equation is one equation for each of the n coordinates.

Because you are looking at the projection onto space, and throwing away time. A geodesic is straight in four-dimensional spacetime, not in three-dimensional space. Take a ball tossed across a room: its spatial path is a metre-high arc lasting maybe a second. But one second of time is, in geometric units, an enormous length — light travels 300{,}000 km in it. Plot the throw on a spacetime diagram with that true scale and the "arc" is a segment of a circle so vast its bending over a mere metre of height is imperceptibly gentle: it is, to spectacular accuracy, a straight line very slightly bent by the Earth's curvature of spacetime. The orbit of a planet is the same story stretched over a year. What looks like dramatic curving in space is a nearly straight coast through spacetime — we are simply bad at seeing the time axis at its real scale.

Two classic traps. First, the \Gamma^\mu_{\ \alpha\beta} look like tensor components but they are not a tensor — they do not transform by the tensor rule (there is an extra inhomogeneous piece). That is exactly why they can be made to vanish at any chosen point by going to a free-falling frame, whereas a genuine tensor that is zero in one frame is zero in all. Second, and following from this: Christoffel symbols being zero at a point does not mean spacetime is flat there. In plain flat space written in polar coordinates, the Christoffels are non-zero (e.g. \Gamma^r_{\ \theta\theta} = -r), yet the space is flat. Curvature lives one derivative up — in the Riemann tensor, which is built from derivatives of the Christoffels and does transform as a tensor. Non-zero \Gamma tells you the coordinates are bending; non-zero Riemann tells you spacetime is bending.