# What am I missing about the interaction between electromagnetism and Lorentz length contraction?

I was recently reading this short article: http://galileo.phys.virginia.edu/classes/252/rel_el_mag.html It explains how magnetic and electric fields are related because in one frame of reference where there's a net magnetic force and no net electric force, there may be another frame of reference where there is no magnetic fforce but only an electric effect with the same result.

However, I'm not sure why the increase in charge density isn't symmetric between the two frames they examine. If in the frame of the lone proton, we get Lorentz length contraction that leads to an increased density of positive charge in the wire, why do we not in the frame of the wire see Lorentz contraction with the moving electrons leading to higher density of negative electric charge that would cancel out the forces we'd expect from the magnetic fields?

Why are they able to fiat that the moving electrons are uniformly distributed, but when we move to another frame of reference where the protons in the wire are the ones that appear to be moving that they are not?

Is it really just a fiat? What would happen if we started out with the wire having no current in it with the electrons initially uniformly distributed and then induced the current in the wire, would we then symmetrically see Lorentz contraction in both frames?

I'm thinking my issue is not fully understanding Lorentz contraction and its effect when acceleration is involved. For example, it also confuses me why it is both the case that the string should break between two rockets tied together with string and that if I got in a rocket and accelerated up to near the speed of light headed towards Alpha Centauri that the distance to Alpha Centauri would appear to shrink. Do empty spaces between objects get contracted, or don't they? Does it have something to do with which objects are the ones doing the accelerating to achieve the relative velocities?

• I'd find a different source. If there is a frame with a magnetic field and no electric field, it is not possible to have another frame with no magnetic field and an electric field. That is because $E^2- B^2$ (cgs) is invariant. – Rob Jeffries Dec 18 '15 at 20:57
• Maybe I misspoke, it talks about one frame with no net electrical force and another with no net magnetic force. – Shufflepants Dec 18 '15 at 21:00
• "I'm thinking my issue is not fully understanding Lorentz contraction and its effect when acceleration is involved." - You got that exactly right. Let me ask: What happens after chained accelerating rockets run out of fuel? Answer: The contraction motion continues until the rockets collide. Now let me guess what you say about that: No no, Lorentz contraction is not that kind of physical thing. Right?:) – stuffu Dec 18 '15 at 22:42
• check out this Veritasium video. At 1:15 talks about your question – Lefteris Dec 21 '15 at 7:00
• Well, thanks for the down vote with no explanation as to why... – Shufflepants Feb 10 '16 at 21:51

Let's say there's spaceship fleet A, where every spaceship driver has an order to keep a one yard's distance to the next spaceship.

Let's say there's spaceship fleet B, where every spaceship driver has an order to keep the same acceleration as the next spaceship.

Let's say there's spaceship fleet C, where every spaceship has the same electric charge, and the fleet is in a homogeneous electric field, engines off.

Let's say there's spaceship fleet D, where the fleet is in an uniform gravity field, engines off.

Let us now study those fleets.

A) When fleet A accelerates, its yardsticks shrink. As the drivers measure the distance using the contracted sticks, they notice that distances are getting too large. So they correct that somehow.

B) When fleet B accelerates, its yardsticks shrink. As the drivers measure the distance using the contracted sticks, they notice that distances are getting larger. They do not care about that, as there was no order to keep some specific distance.

C) When fleet C is accelerated by the electric field, its yardsticks shrink. As the drivers measure the distance using the contracted sticks, they notice that distances are getting larger. C is basically same as B.

D) When fleet D is accelerated by the gravity field, its yardsticks shrink. As the drivers measure the distance using the contracted sticks, they will not notice that distances are getting larger, as that would be against the principle that says that free falling in uniform gravity field is equivalent to floating in empty space.

Now we will let an inertial observer to observe those fleets. The observer says: A shrinks, B does not shrink, C does not shrink, D shrinks.

Then we use the equivalence principle and note that if a fleet falling in a gravity field shrinks according to a non-falling observer, then a fleet of toy spaceships inside an accelerating elevator falling towards the elevator floor must shrink according to somebody standing on the elevator floor. And we note that a person in an accelerating elevator is an accelerating observer according to us, and something "falling" inside that elevator is a non-accelerating thing according to us. An accelerating observer sees non-accelerating things either to accelerate and shrink, or to decelerate and expand.

Finally we note that we can model electrons inside a wire as charged particles in a homogeneous electric field. So let me copy and edit the appropriate part from above: C) When the group of electrons is accelerated by the electric field, its yardsticks would shrink, if such yardsticks existed. As the imaginary sentient electrons measure their distances using the contracted sticks, they notice that the distances are getting larger.

I don't know if this exactly answers your question, but I recently learned how the Einstein-Lorentz contraction neatly explains the magnetic attraction between two parallel wires with the current going in the same direction: With the electrons traveling near c in both wires they are in the same frame of reference and thus don't see any different relationship to each other than if they were both at rest. But, its assumed the '+' charges are at rest to the moving '-' in both wires and their frame of reference is moving near c in the opposite direction relative to them. If q is the total '+'charge in a distance s in each wire then q/s is the charge density of the '+' charges. Since the wires are moving away so fast relative to the '-' charges s seems much smaller to the '-' charges. So, q/s becomes much larger causing more attraction between them. If there were no current in either wire nothing would look any different to any charges and there would be no forces. Electrostatic fields can exist without magnetic fields. Moving charges will cause magnetic fields.

What I don't get is the magnetic B field vector is orthogonal to both velocity, v, of the charge and the electric, E, field.