Lorentz force
Lorentz force in general relativity

In the general theory of relativity the equation of motion for a particle with mass m {\displaystyle m} and charge e {\displaystyle e}, moving in a space with metric tensor g a b {\displaystyle g_{ab}} and electromagnetic field F a b {\displaystyle F_{ab}}, is given as m d u c d s − m 1 2 g a b, c u a u b = e F c b u b, {\displaystyle m{\frac {du_{c}}{ds}}-m{\frac {1}{2}}g_{ab,c}u^{a}u^{b}=eF_{cb}u^{b},} where u a = d x a / d s {\displaystyle u^{a}=dx^{a}/ds} (d x a {\displaystyle dx^{a}} is taken along the trajectory), g a b, c = ∂ g a b / ∂ x c {\displaystyle g_{ab,c}=\partial g_{ab}/\partial x^{c}}, and d s 2 = g a b d x a d x b {\displaystyle ds^{2}=g_{ab}dx^{a}dx^{b}}.

The equation can also be written as m d u c d s − m Γ a b c u a u b = e F c b u b, {\displaystyle m{\frac {du_{c}}{ds}}-m\Gamma _{abc}u^{a}u^{b}=eF_{cb}u^{b},} where Γ a b c {\displaystyle \Gamma _{abc}} is the Christoffel symbol (of the torsion-free metric connection in general relativity), or as m D u c d s = e F c b u b, {\displaystyle m{\frac {Du_{c}}{ds}}=eF_{cb}u^{b},} where D {\displaystyle D} is the covariant differential in general relativity (metric, torsion-free).
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