Editing Lorentz force (section)

=== Lorentz force in general relativity ===
In the [[general theory of relativity]] the equation of motion for a particle with mass <math>m</math> and charge <math>e</math>, moving in a space with metric tensor <math>g_{ab}</math> and electromagnetic field <math>F_{ab}</math>, is given as

<math display="block">m\frac{du_c}{ds} - m \frac{1}{2} g_{ab,c} u^a u^b = e F_{cb}u^b ,
</math>

where <math>u^a= dx^a/ds</math> (<math>dx^a</math> is taken along the trajectory), <math>g_{ab,c} = \partial g_{ab}/\partial x^c</math>, and <math>ds^2 = g_{ab} dx^a dx^b</math>.

The equation can also be written as

<math display="block">m\frac{du_c}{ds}-m\Gamma_{abc}u^a u^b = eF_{cb}u^b ,</math>

where <math>\Gamma_{abc}</math> is the [[Levi-Civita connection#Christoffel symbols|Christoffel symbol]] (of the torsion-free metric connection in general relativity), or as

<math display="block">m\frac{Du_c}{ds} = e F_{cb}u^b ,</math>

where <math>D</math> is the [[covariant differential]] in general relativity.