Geodesic equation
d²x^μ/dλ² + Γ^μ_{αβ} (dx^α/dλ)(dx^β/dλ) = 0. The trajectory of a freely-falling particle in curved spacetime; the curve whose tangent is parallel-transported along itself. Generalises the Newtonian straight line. In GR, free-fall = geodesic motion.
Definition
The geodesic equation, d²x^μ/dλ² + Γ^μ_{αβ} (dx^α/dλ)(dx^β/dλ) = 0, governs the trajectory of a freely-falling test particle in curved spacetime. It can be derived in two equivalent ways: by extremising the proper-time action S = ∫ √(g_{μν} dx^μ dx^ν) along the curve (the curve of extremal proper time between two events), or by demanding that the curve's own tangent vector be parallel-transported along itself, which is the geometric definition of "as straight as possible." The Christoffel symbols Γ^μ_{αβ} encode the gravitational acceleration relative to a coordinate-line frame; the equation says that this acceleration plus the coordinate acceleration sums to zero, which is the relativistic statement of Newton's first law in a curved spacetime.
Geodesics generalise the straight lines of Euclidean geometry to arbitrary Riemannian and pseudo-Riemannian manifolds. On a 2-sphere they are the great circles; on Minkowski spacetime they are straight lines at constant velocity; on Schwarzschild spacetime they are the timelike orbits and null light-paths around a black hole. The equivalence principle makes the identification "free-fall = geodesic" an axiom: no force acts on a body in free-fall, so by Newton's first law its world-line should be the relativistic generalisation of a straight line, and that is precisely the geodesic. A test particle's response to gravity in general relativity is fully encoded in this single second-order ordinary differential equation, parametrised by an affine parameter (proper time τ for timelike geodesics, an arbitrary affine parameter for null ones).