Geodesic equation

The geodesic equation is the differential equation that defines the straightest possible paths — geodesics — on a curved manifold. In general relativity, geodesics are not merely geometric curves but the trajectories of freely falling bodies: a particle under no non-gravitational forces follows a geodesic of the spacetime metric. The equation takes the form d²x^μ/dτ² + Γ^μ_νρ (dx^ν/dτ)(dx^ρ/dτ) = 0, where Γ^μ_νρ are the Christoffel symbols derived from the metric and τ is proper time. The first term is the acceleration; the second is the curvature correction. Where curvature vanishes, geodesics reduce to straight lines; where curvature is strong, they bend, converge, and sometimes terminate in singularities.

The geodesic equation reveals that gravity is not a force in general relativity. A body in free fall is not accelerating; it is following the straightest path available in a curved geometry. The sensation of weight you feel while standing on Earth is not gravity pulling you down — it is the electromagnetic repulsion of the ground pushing you *up*, preventing you from following your natural geodesic into the planet's interior. The geodesic equation inverts the Newtonian intuition: falling is inertial motion; standing still is acceleration.