Beta Phase: Square45 is currently in beta testing. Expect some features or content to be incomplete or missing.
45

Geodesic Equation

Describes the path of a particle in a curved spacetime, allowing calculation of trajectories influenced by gravity, central to understanding motion near black holes.
📜

The statement of the theorem

The path xμ(λ)x^\mu(\lambda) of a test particle in a curved spacetime, parameterized by λ\lambda, follows a geodesic and is governed by the equation:\nd2xμdλ2+Γνσμdxνdλdxσdλ=0 \frac{d^2 x^\mu}{d\lambda^2} + \Gamma^\mu_{\nu\sigma} \frac{dx^\nu}{d\lambda} \frac{dx^\sigma}{d\lambda} = 0 \nwhere Γνσμ=12gμρ(∂νgρσ+∂σgρν−∂ρgνσ)\Gamma^\mu_{\nu\sigma} = \frac{1}{2} g^{\mu\rho} (\partial_\nu g_{\rho\sigma} + \partial_\sigma g_{\rho\nu} - \partial_\rho g_{\nu\sigma}) are the Christoffel symbols.
Source: Wikipedia