geodesic equationequation
The geodesic equation is a central element of Einstein's general theory of relativity. It generalizes Newton's law of inertia to curved spacetime. A massive body bends the spacetime continuum, and a test particle moves along a line that minimizes proper time (the time measured by a clock moving with the particle). Geometrically, this is described through Christoffel symbols — quantities characterizing curvature. The equation looks like d²x^μ/dτ² + Γ^μ_{νρ} (dx^ν/dτ)(dx^ρ/dτ) = 0, where the first term is Newtonian acceleration, and the second is the curvature correction.
How it works
Satellites in orbit, planets around the Sun, a light ray passing a star — all follow geodesics in curved spacetime. Motion without engines, only under the influence of gravity, is always geodesic.
💡 Light, having no mass, also moves along geodesics, but null ones, which define the trajectories of massless particles. This leads to gravitational lensing — the distortion of images of distant galaxies.
\frac{d^2 x^\mu}{d\tau^2} + \Gamma^\mu_{\nu\rho} \frac{dx^\nu}{d\tau} \frac{dx^\rho}{d\tau} = 0
x^μ are coordinates (μ = 0,1,2,3; x^0 = ct is the time coordinate), τ is the proper time (time interval according to the clock of the moving particle), Γ^μ_{νρ} are the Christoffel symbols (connection coefficients determining spacetime curvature; dimensionless, depend on the metric g_μν), d²x^μ/dτ² is the 4-acceleration, dx^ν/dτ and dx^ρ/dτ are components of the 4-velocity.