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

Famous quotes containing the word force:

    “Man always made, and still makes, grotesque blunders in selecting and measuring forces, taken at random from the heap, but he never made a mistake in the value he set on the whole, which he symbolized as unity and worshipped as God. To this day, his attitude towards it has never changed, though science can no longer give to force a name.”
    —Henry Brooks Adams (1838–1918)