Lorentz Boosts and the Electromagnetic Field

In a previous note we discussed the representations of four-dimensional special orthogonal transformations (i.e., rotations) in terms of eight parameters a,b,…,h with the condition ah – bg + cf – de = 0, and we noted that such a transformation R satisfies the relation

under the more restrictive condition h = 0, which implies that af  – be + cd = 0. This has an interesting physical interpretation. Recall that the electromagnetic field in relativistic physics (see the note on Force Laws and Maxwell’s Equations) can be represented by an anti-symmetric tensor of the form

The condition that this field must satisfy in order for it to correspond to a rotation that satisfies (1) is therefore

The scalar product of the electric and magnetic field vectors is relativistically invariant, and the vanishing of this quantity signifies that those vectors are perpendicular. This condition is characteristic of a propagating electromagnetic plane wave. Thus the conditions for equation (1) to be satisfied have a physically meaningful interpretation, at least in this context.

It’s also interesting to consider the one-to-one correspondence between anti-symmetric matrices and special orthogonal (i.e., rotation) matrices established by Cayley’s transformation. Recalling that the relativistic electromagnetic field has its most natural representation as an anti-symmetric tensor, and that Lorentz transformations consist of generalized rotations, it seems natural to wonder if there could be a physically significant correspondence between these two concepts. In order to treat Lorentz transformations as ordinary rotations we use an imaginary time coordinate, and express a standard Lorentz boost in the positive x direction by

where

Letting L denote the coefficient matrix representing this Lorentz boost, the corresponding anti-symmetric matrix is given by

Inserting the L matrix into this formula, we get

If we identify this with the electromagnetic field tensor expressed in terms of the imaginary time coordinate

we find that the standard Lorentz boost with parameter b in the positive x direction corresponds to an electromagnetic field whose only non-zero component is

and hence the boost parameter corresponding to the field strength E_x is

A plot of this function is shown in the figure below.

Without knowing the scale factors, we can’t say in which regime we would expect to find ourselves. It’s possible that our experience is entirely at the very low end, where the boost is essentially just proportional to the field strength.

Conversely, given an electromagnetic field characterized by the anti-symmetric tensor P at a given point, the corresponding Lorentz boost L is given by

The observable effect of the field at a given time and place is to accelerate a charged particle located at that time and place, so we might suspect that the acceleration produced by a given field is correlated in some way with the corresponding Lorentz boost. However, since the boost given by (2) for a given field strength is not infinitesimal, we must suppose that it is applied discretely, rather than continuously, so as to give a finite acceleration. This in itself is not troubling, because the electromagnetic interaction is expected to possess some intrinsic property that can be characterized as a frequency, and we expect proportionality between frequency and energy consistent with the relation E = hn. Admittedly, it’s difficult to see how a single individual boost can have a frequency, but this is similar to the difficulty of seeing how a single discrete photon can possess a frequency. (In quantum electrodynamics the “frequency of a discrete interaction” issue is resolved – at least in some interpretations – by regarding the frequency as a property, not of the photon, but of the source.)

Perhaps a more serious objection to the overall idea of associating a certain “rotation” of spacetime with a field is that the idea would actually be more compatible with the gravitational field, because a transformation of space and time ought to affect the paths of all objects in the same way, i.e., the equivalence principle ought to be satisfied, as it is with the gravitational interaction. But gravity is represented (at least in general relativity) by a symmetric tensor, whereas the Cayley correspondence applies only to anti-symmetric tensors, such as the electromagnetic field. Electrically charged particles don’t all have the same ratio of electric charge to inertial mass, so more massive particles undergo less acceleration than less massive particles in a given field, not to mention that fact that oppositely charged particles accelerate in opposite directions. This seems to rule out any simple physical interpretation of the Cayley correspondence between Lorentz boosts and the electromagnetic field.

We might speculate that geometrical-temporal model of electromagnetism could be based on distinguishing between the trajectories of positively and negatively charged particles in some way, such as regarding them as proceeding in opposite temporal directions. On that basis a kind of equivalence principle could be restored, at least for a charged particle and its anti-particle. Indeed the idea that electrons and positrons (for example) are time-reversed versions of the same particle has often been considered, but perhaps not with the idea in mind of  extending the equivalence principle to electromagnetic interactions.

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