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Geodesic Equation

Describes the path of a freely falling object through curved spacetime, representing the shortest path between two points.
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The statement of the theorem

Let xμ(λ)x^\mu(\lambda) be the path of a freely falling particle parameterized by λ\lambda. The geodesic equation describes this path by minimizing the proper time interval and is given by:\n\nd2xμdλ2+Γνρμdxνdλdxρdλ=0\frac{d^2 x^\mu}{d\lambda^2} + \Gamma^\mu_{\nu\rho} \frac{d x^\nu}{d\lambda} \frac{d x^\rho}{d\lambda} = 0\n\nWhere Γνρμ\Gamma^\mu_{\nu\rho} are the Christoffel symbols of the second kind, defined as Γνρμ=12gμσ(∂νgρσ+∂ρgνσ−∂σgνρ)\Gamma^\mu_{\nu\rho} = \frac{1}{2} g^{\mu\sigma} (\partial_{\nu} g_{\rho\sigma} + \partial_{\rho} g_{\nu\sigma} - \partial_{\sigma} g_{\nu\rho}).
Source: Wikipedia