Required math: algebra
Required physics: special relativity
Reference: Moore, Thomas A., A General Relativity Workbook, University Science Books (2013) – Chapter 4; Problems 4.7.
In the last post, we introduced the electromagnetic field tensor :
In terms of , the electric and magnetic (Lorentz) force laws for a charge can be combined into a single equation:
where is the four-velocity. The three spatial components give the force law, but what about the time component? The time component of the four-momentum is the relativistic energy which, for a particle of rest mass is . If we expand this in a Taylor series for small , we get
Thus the relativistic energy is the rest mass plus the Newtonian kinetic energy (plus higher order terms). In the small- limit, and the component of 2 therefore is
Since the rest mass doesn’t change, this equation is saying that the rate of change of kinetic energy is . Does this make sense?
The force on a charge in an electric field is , so the work done by this field in moving the charge through a distance is . This work accelerates the charge, thus increasing its kinetic energy. The rate at which the kinetic energy increases is therefore .
Since the magnetic force on a charge is always perpendicular to the direction of motion, magnetic forces do no work, so there is no contribution to the kinetic energy from the magnetic field.