We know that parallel current in two parallel wires causes them to attract. The explanation is that moving charges generate attractive magnetic fields.
https://youtube.com/watch?v=43AeuDvWc0k
If I have two charged pith balls in a space ship, then they should attract more the faster the space ship travels. But relativity says you cannot know your velocity in an inertial frame. In other words you cannot know how fast you are going unless you look outside the space ship. So that means the attraction of two moving charges should only be visible from outside the moving spaceship – which does not make sense to me.
How can a magnetic field not happen in a spaceship but be observed from outside?
There are many questions about the relativistic effects of electrons in a wire, for example
Relativistic explanation of attraction between two parallel currents
But they all hinge on the velocity difference between the moving electrons in the wire vs the static protons. In my example I am just considering the behavior of two charged pith balls through space. Moving charges should generate a magnetic field, no? Or do moving charges only generate a magnetic field when they are moving relative to protons?
Best Answer
You are right, that does not make sense. What relativity actually says is more subtle.
Relativity says that both frames will agree on all observables. They will agree on the readings of any strain gauges or accelerometers or whatever. But they will attribute those readings to different effects.
For example, suppose you attach needles and a scale to the pith balls to measure the amount of attraction. Then, regardless of the ship's speed, all frames will agree on how much the needles and scale read. In the ships frame the attraction will be due entirely to the E field, and the needles and scale will correctly measure that. In an external observer's frame, regardless of the ship's speed, the attraction will be due to both the electric field and the magnetic field and the needles and scale will be time dilated and length contracted, such that the external frame will also predict the same reading on the needles and scale. Thus regardless of the speed, the reading is the same and cannot be used as a speedometer.
In the ship's frame the charged balls have an E field and the measured forces are caused by those E fields. In the outside observer's frame there is an E field and also a B field. The measured forces are caused by both the E and B field. Furthermore, the accelerometers or strain gauges are subject to time dilation, so their values are somewhat off compared to the coordinate values.
In a coordinate-independent sense there is only the electromagnetic field. The electromagnetic field is a tensor which has the electric and the magnetic fields as different components of the tensor. Different frames will partition that into electric field and magnetic field somewhat differently. But intrinsically, physically, the electric and magnetic fields are two parts of the same overall electromagnetic field.