Lorentz force for a relativistic charge.

The Lorentz force for a charged particle is

,

the charge of the particle, mass, the proper time, the electromagnetic tensor and the 4-velocity.

Ok... now is where the magic of the physics begins. That equation seems like a horrible monster but I'm going to try to show a beauty face of it.

Think in a particle (e.g electron) moving forward in a region where you have electric or magnetic field, the first and natural question is how do these fields change the direction of the particle? find out.

Letting

these fields are constants and uniforms with that said we can start on it.

From the Lorentz force, the 0-component is

,

the i-component is

.

When we've done with all the maths, will have this pair of equations

.

For the problem of interest the pair of equations will reduce to

recall that upper notation just means components of contravariant vector.

we can solve it just taking the derivative of respect to again and substituting

We're doing three new definitions just in case... we're calling as (because of the second derivative), and .

So the final equations for the component 2 of the 4-velocity is

.

We can check 2 cases, 1st one it leaves that and the second one which is . These both cases just are going to implicate that the solution for the equation of will be a complex exponential or a real one. In this post I'm going to begin with first one.

Assuming , we have that the solution for is

,

we're calling .

Just substituting the last result in the equations for . We have this two results

Setting the initial conditions

The world line will be,

,

I've omitted the 3 component because it's a constant(straight line).

We can notice if the component 0 of the 4-velocity and the Lorentz factor are just a constants, so the energy is a constant as well.

By another hand when the solution is very similar (setting all the constants just like the last case) but in a complex exponential form, we have

,

At the end the velocity is of course changed because of the electromagnetic field.

The next and last step is just plot how is the movement of a charged particle due to a electromagnetic field.

We're going to back to the no-relativistic limit, just putting .

The harmonic motion for , looks like.

The hyperbolic motion looks

The dominance of the electric field kills the harmonic motion.