If special relativity causes magnetism to arise from the electric force, how does light work?

If special relativity causes magnetism to arise from the electric force, how does light work?

My understanding of light was perturbations in the electric field that cause perturbations in the magnetic field, which caused perturbations in the electric field and so on, resulting in a packet of energy that travels through space.

If, however, magnetism can be viewed as arising from relativistic length contractions due to different reference frames of moving objects (which I just stumbled across from Veritasium's video), how do we explain electromagnetic waves under that model? As I understand it, for relativistic effects to come into play there have to be two different reference frames moving at different speeds / directions. What are the reference frames at play here? As I see it, there's only the reference frame of the photon.

Also note, from How Special Relativity causes magnetism I understand there are different ways to view the relationship between magnetism, electricity, and relativity. I'm specifically asking about how light works from the perspective of "magnetism is just charges + relativity," even if there's a better way to frame how light works in general.

• Special relativity also implies electric field given magnetic field, though. Transformation rules work both ways.
– g s
Commented Aug 10, 2023 at 4:22
• Right, regardless though, one still arises from the other. In either case this question is still relevant since it seems to take both simultaneously to make electromagnetic waves Commented Aug 10, 2023 at 4:30
• Magnetism cannot be viewed in this way. The simplest example is a light wave in a vacuum. This is an electromagnetic phenomenon. If magnetic field originated from a frame transformation, electromagnetic waves of light would not make sense. Commented Aug 10, 2023 at 4:36
• There is no frame transformation you can make which turns an electromagnetic wave into "just electric" wave or something. Maxwell's equations are invariant under Lorentz transformations but that does not mean that magnetism is a byproduct of electrical forces and special relativity. If that video says something like that, downvote it and don't watch that channel again. Commented Aug 10, 2023 at 5:10
• "As I see it, there's only the reference frame of the photon." But there isn't a frame where a photon is at rest! Commented Aug 10, 2023 at 8:42

From the point of view of Special Relativity, there are no two separate electric and magnetic fields, and charge and current densities are not independent when passing from one reference frame to another.

$$E^2-B^2$$ is a relativistic invariant (I use units such as $$c=1$$). Its value must be the same in every inertial reference frame, although the intensity of the electric and magnetic fields may vary separately. This fact puts an essential constraint on the possible changes of electric and magnetic fields from one reference frame to another. In particular, if only an electric field exists in one reference frame, a magnetic field may appear in other frames. Still, in no frame, the electric component of the electromagnetic field may vanish (that would reverse the sign of the relativistic invariant).

The invariance of $$E^2-B^2$$ helps to understand that the argument showing the magnetic field as an effect of a change of reference frame, when staring with a reference where only the electric field is present, can be reversed. It is equally possible to show that an electric field must appear in another reference frame if only a magnetic field is present in the original. In any case, Vweritasium's video is not showing that the electric field is more fundamental than the magnetic field. It just shows how the unitary description of the electromagnetic fields introduced by SR allows us to understand the deep link existing between electric and magnetic forces. Without a hierarchy among them.

In the case of an electromagnetic wave, the invariant is zero, and then it remains zero in all inertial frames. Said in another way, there is no possibility of converting an electromagnetic wave into a purely electric or magnetic wave.

• I need to re-read you answer a few more times, but I think that exactly answers my question. My takeaway from what you're saying is that it's not that one field is an "illusion", it's that by shifting reference frames you can sometimes shift the variables of the invariant so that one field or the other becomes 0. Viewed like that, light is a wave that constantly shifts back and forth between E and B in your invariant. I'll keep thinking about this. Thank you for your answer! Commented Aug 10, 2023 at 13:22

In any given frame, you have an electric field, a magnetic field, and disturbances related via Maxwell's equations. If you Lorentz transform to a different frame, you get electric and magnetic fields that are linear combinations of the old ones, and disturbances that are combinations of the old ones, and Maxwell's equations still apply.

If we trust relativity, light is pure kinetic energy. I mean, if light has no rest mass, then light is pure kinetic energy according to relativity.

So light is kinetic energy, which can become electric field energy. (Or alternatively mechanical pushing energy (radiation pressure).

• Energy is a physical property of a system. It is not a physical system. If the light is pure kinetic energy, the energy of what? Commented Aug 10, 2023 at 5:41
• @GiorgioP-DoomsdayClockIsAt-90 Electron-positron annihilation is a phenomenon where electric fields use energy to accelerate themselves. So maybe the light produced is just zero rest-energy electric fields moving at the speed of light. It is some kind of zero rest-energy moving at the speed of light. Commented Aug 10, 2023 at 6:13
• Electric fields accelerate themselves? How do you define acceleration of a field? I am afraid that you put too much confidence in Pop-Sci vague analogies. Commented Aug 10, 2023 at 6:22
• @GiorgioP-DoomsdayClockIsAt-90 Oh you doubt relativity ; ) When we have electron-positron pair at rest, we have energy at rest. Said energy accelerates itself to the speed of light, when annihilation happens. The kinetic energy of light is $pc$. And total energy of light is $pc$ Commented Aug 10, 2023 at 7:24