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Geodesic Equation

Describes the path of a particle moving under the influence of gravity in a curved spacetime: \ddot{x} u + eta u( rac{ ext{D}g_{ u ho}}{ ext{D}x^ ho}) = 0
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The statement of the theorem

The path xμ(λ)x^\mu(\lambda) of a particle moving freely in curved spacetime (a geodesic) is parameterized by λ\lambda and satisfies the equation:\nd2xμdλ2+Γνρμdxνdλdxρdλ=0\frac{d^2x^{\mu}}{d\lambda^2} + \Gamma^{\mu}_{\nu\rho} \frac{dx^{\nu}}{d\lambda} \frac{dx^{\rho}}{d\lambda} = 0 \nWhere Γνρμ\Gamma^{\mu}_{\nu\rho} are the Christoffel symbols of the second kind, defined by the metric tensor gμνg_{\mu\nu} as:\Γνρμ=12gμσ(∂νgσρ+∂ρgσν−∂σgνρ)\Gamma^{\mu}_{\nu\rho} = \frac{1}{2} g^{\mu\sigma} (\partial_{\nu} g_{\sigma\rho} + \partial_{\rho} g_{\sigma\nu} - \partial_{\sigma} g_{\nu\rho})\n
Source: Wikipedia