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
we get
Read more about this topic: Maxwell's Equations In Curved Spacetime
Famous quotes containing the word force:
“The whole force of the respectable circles to which I belonged, that respectable circle which knew as I did not the value of security won, the slender chance of replacing it if lost or abandoned, was against me ...”
—Ida M. Tarbell (18571944)