Why do these particle trajectories appear to be different in different inertial frames?

Question

This question is based on Example 15.12 in John R. Taylor's Classical Mechanics. It concerns the trajectory of a particle experiencing electromagnetic forces as observed in two different inertial reference frames.

In the noted example, Taylor compares the electromagnetic fields of an infinite line charge in two reference frames related by a Lorentz boost along the axis of the line charge (i.e. the z-axis in cylindrical coordinates). In one frame, the line charge is at rest; in the other, it moves uniformly along its length thereby producing a current. In the first frame, there is only a radial electric field, but in the second, there is also a magnetic field circulating around the line charge in the $\phi$-direction.

The example does not analyze this system further, but I noticed something unusual about the behavior of a point charge in these fields. Here's the issue:

A point charge starting at rest in the frame with both an electric and magnetic field will experience acceleration mainly in the radial direction, but it will also experience acceleration along the axis of the line charge (in the z-direction) due to its radial velocity coupling to the magnetic field. Additionally, its radial acceleration will degrade over time due to its z-velocity also coupling to the magnetic field.

Contrast this with the same particle in the other frame. It will have an initial velocity in the negative z-direction, but because there is no magnetic field, it will only experience an acceleration in the radial direction. In this frame, the component of velocity along the z-direction should remain constant and the radial acceleration should not degrade.

These differences should not be possible because the trajectory of the particle needs to be consistent between frames. If it hits a bullseye in one frame, it won't in the other if the trajectories are inconsistent. For example, consider a target at rest with respect to the particle that begins at rest in the frame with both an electric and magnetic field. The target will be missed in that frame but will be hit in the frame where it and the particle move with a constant z-velocity. Therefore, there must be an aspect of this particle's motion that is not being accounted for and which would rectify this discrepancy between frames.

Question: How can this analysis be corrected to guarantee consistent trajectories between frames of reference?

Answer 1

Electromagnetism is a relativistic theory. If you're going to talk about EM and Lorentz transforms, remember to use relativistic kinematics, too.

In the frame where the line charge has no current, the radial acceleration of the point charge is reduced with time, because the faster a particle is, the harder it is to make it go even faster. It's not $\mathbf F=q(\mathbf E+\mathbf v\times\mathbf B)$ that fails. It is $\mathbf F=m\mathbf a!$ (Thought experiment/mnemonic device: if $1\;\mathrm{kg}$ is moving at $0.9999c$ and you apply a steady $1\;\mathrm N$ of force in the direction of motion, is it possible that the mass accelerates at $1\;\mathrm{m}\,\mathrm{s}^{-2}$ steadily? No!) In fact, $\mathbf F=\mathbf{\dot p}=\frac{d(\gamma m\mathbf v)}{dt}$ doesn't even have to be parallel to $\mathbf a=\mathbf{\dot v}.$ The radial force applied to the particle actually does cause it to slow down in the $z$ direction. (Mnemonic: the light speed limit suggests that, at some point, if you push something, its velocity in other directions must reduce in order to "make room" for more velocity in the direction of the force.)

Note that $F^\mu=ma^\mu$ is true for the four-force $F^\mu$ and four-acceleration $a^\mu.$ These are different from the "Newtonian" 3-force and 3-acceleration $\mathbf F,\mathbf a$ (and $\mathbf F,\mathbf a$ aren't even identical to any components of $F^\mu,a^\mu$), and $\mathbf F,\mathbf a$ are the quantities you are worried about.

So, qualitatively, both frames predict that the radial acceleration of the particle will drop with time and that the $z$-velocity will change. (NB: the frames agree on what "radial" and "$z$-direction" mean, but not on what "acceleration" and "velocity" mean. Be careful making comparisons!)