We've looked at how forces can be described in relativistic mechanics, but not yet discussed how the fields producing them change. It's time to correct that oversight by examining the quintessential physics scenario.
No, not the harmonic oscillator, the other one, the case of a charged particle in an electromagnetic field.
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| Plot of particle trajectories in EM fields |
Let us imagine an electron moving at some relativistic velocity through a combination of electric and magnetic fields. We need a way to express the fields in both our, laboratory, frame of reference and in the electron's frame.
Electromagnetic Fields
Let's start with what we already know in classical electromechanics, consider the force on a charged particle in a combination of electric and magnetic fields:
Rewriting this in purely matrix notation, where the ~ indicates a skew-symmetric matrix representation of a vector, which allows us to represent cross products in matrix notation:
Now, let's go back to how we expressed four-force and four-velocity:
We can assume that the four-force acting on the particle carries the same form as in the three dimensional case, by inspection we see that:
The middle term here is termed the Electromagnetic Tensor, or Faraday Tensor, and expresses electromagnetic fields in relativistic terms. Like before, we see that it is skew-symmetric, which properly fits all six degrees of freedom, three components of each the electromagnetic and electric fields.
But what in the world is that matrix full of ones and zeroes doing there? It looks like a bit like, η, the metric that we introduced before. In fact, it is another way of expressing the flat spacetime metric.
We'll also notice that we can very easily combine the charge and four-velocity into a single term, the four-current. Where we know that I = qv for a single moving charge:
Which may, in fact, prove more general for some arbitrary distribution of moving charges, as it does in conventional cases, where we look at the motion of streams of electricity rather than individual electrons.
Lorentz Boosted Electromagnetism
Thus far, we've only tried applying boosts to four-vectors, rank-one tensors, represented as (4x1) matrices, but the Faraday Tensor is not a vector it's a rank-two tensor, which we've represented as a (4x4) matrix. When a charged particle moves through an EM field, it experiences forces deriving from the electric and magnetic fields. We can measure these fields and particle's velocity in our frame, the laboratory frame, and wish to calculate what they look like in the particle's reference frame.
Consider the equation above, which describes the force upon the particle in one frame. We've well established that both position and force are subject to Lorentz transformations:
Four-Potential and Fields
Let's again return to the non-relativistic case. We typically like to describe the source of fields in terms of potentials, specifically the electric scalar potential and magnetic vector potential, which we want to combine into a single electromagnetic four-vector potential.
Electric potential reflects the energy per unit charge required to move a hypothetical test charge from infinitely far away to a given location. Where the quantity ρdV is some differential electric charge:
Electric field lines point from locations of high to low potential. A stationary point charge creates a potential at some distance r from the charge: φ=
Magnetic potential can be described as the potential energy per unit current, when considering the magnetic field acting upon some current. Because current has three directional components, so too does the magnetic potential:
Time and space derivatives of the magnetic and electric potentials both contribute to the electric and magnetic fields at some point in space through the relations:
Well, that already has most of the term we expect for a four-dimensional equation, so let's just write out the Faraday tensor using potentials and see if something pops out:
Where we notice that the diagonal terms could be anything, so long as they are equal, and still give us the intended zeroes. We've got two square matrices, with very similar looking terms, which we can represent as two distinct vectors using the outer product:
Once again, the metric appears in our equation, and we've been playing pretty fast and loose with where it goes. It's time to discuss what it does and what it means. In order to properly discuss the metric, we'll need to dig into notions of covariance and contravariance, as we'll get to next.
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