Maxwell's Equations in Curved Spacetime - Lorentz Force Density

Lorentz Force Density

The density of the Lorentz force is a covariant vector density given by

The force on a test particle subject only to gravity and electromagnetism is

where pα is the linear 4-momentum of the particle, t is any time coordinate parameterizing the world line of the particle, Γβαγ is the Christoffel symbol (gravitational force field), and q is the electric charge of the particle.

This equation is invariant under a change in the time coordinate; just multiply by and use the chain rule. It is also invariant under a change in the x coordinate system.

Using the transformation law for the Christoffel symbol

\bar{\Gamma}^{\beta}_{\alpha \gamma} \, = \, \frac{\partial \bar{x}^{\beta}}{\partial x^{\epsilon}} \, \frac{\partial x^{\delta}}{\partial \bar{x}^{\alpha}} \, \frac{\partial x^{\zeta}}{\partial \bar{x}^{\gamma}} \, \Gamma^{\epsilon}_{\delta \zeta} \, + \frac{\partial \bar{x}^{\beta}}{\partial x^{\eta}}\, \frac{\partial^2 x^{\eta}}{\partial \bar{x}^{\alpha} \partial \bar{x}^{\gamma}} \,

we get

\begin{align} & \frac{d \bar{p}_{\alpha}}{d t} \, - \, \bar{\Gamma}^{\beta}_{\alpha \gamma} \, \bar{p}_{\beta} \, \frac{d \bar{x}^{\gamma}}{d t} \, - \, q \, \bar{F}_{\alpha \gamma} \, \frac{d \bar{x}^{\gamma}}{d t} \\ & = \, \frac{d}{d t} \left( \frac{\partial x^{\delta}}{\partial \bar{x}^{\alpha}} \, p_{\delta} \right) \, - \, \left( \frac{\partial \bar{x}^{\beta}}{\partial x^{\theta}} \, \frac{\partial x^{\delta}}{\partial \bar{x}^{\alpha}} \, \frac{\partial x^{\iota}}{\partial \bar{x}^{\gamma}} \, \Gamma^{\theta}_{\delta \iota} + \, \frac{\partial \bar{x}^{\beta}}{\partial x^{\eta}}\, \frac{\partial^2 x^{\eta}}{\partial \bar{x}^{\alpha} \partial \bar{x}^{\gamma}} \right) \, \frac{\partial x^{\epsilon}}{\partial \bar{x}^{\beta}} \, p_{\epsilon} \, \frac{\partial \bar{x}^{\gamma}}{\partial x^{\zeta}} \, \frac{d x^{\zeta}}{d t} \, - \, q \, \frac{\partial x^{\delta}}{\partial \bar{x}^{\alpha}} \, F_{\delta \zeta} \, \frac{d x^{\zeta}}{d t} \\ & = \, \frac{\partial x^{\delta}}{\partial \bar{x}^{\alpha}} \, \left( \frac{d p_{\delta}}{d t} \, - \, \Gamma^{\epsilon}_{\delta \zeta} \, p_{\epsilon} \, \frac{d x^{\zeta}}{d t} \, - \, q \, F_{\delta \zeta} \, \frac{d x^{\zeta}}{d t} \right) + \frac{d}{d t} \left( \frac{\partial x^{\delta}}{\partial \bar{x}^{\alpha}} \right) \, p_{\delta} \, - \, \left( \frac{\partial \bar{x}^{\beta}}{\partial x^{\eta}}\, \frac{\partial^2 x^{\eta}}{\partial \bar{x}^{\alpha} \partial \bar{x}^{\gamma}} \right) \, \frac{\partial x^{\epsilon}}{\partial \bar{x}^{\beta}} \, p_{\epsilon} \, \frac{\partial \bar{x}^{\gamma}}{\partial x^{\zeta}} \, \frac{d x^{\zeta}}{d t} \\ & = \, 0 \, + \, \frac{d}{d t} \left( \frac{\partial x^{\delta}}{\partial \bar{x}^{\alpha}} \right) \, p_{\delta} \, - \, \frac{\partial^2 x^{\epsilon}}{\partial \bar{x}^{\alpha} \partial \bar{x}^{\gamma}} p_{\epsilon} \, \frac{d \bar{x}^{\gamma}}{d t} \, = \, 0 \ . \end{align}

Read more about this topic:  Maxwell's Equations In Curved Spacetime

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