Electron traveling through gravitational and magnetic field

Question

When an electron travels through a magnetic field it experiences the Lorentz force. The force acting on the electron causes an acceleration and therefore Cyclotron radiation is emitted. After reading this question Energy of electron spinning in a magnetic field my understanding is that, the energy released to form the radiation must be taken from the electron's energy. The energy will be taken from the kinetic energy of the electron relative to the magnetic field. Eventually the electron will be stationary relative to the magnetic field and the Lorentz force will drop to zero.

Now lets say we have an electron traveling through a vacuum due west on the equator. Lets also assume that Earth's magnetic field is uniform in this location. Because of the left hand rule, the electron will experience an upward force.

The force due to gravity exerted on the electron is $M_e*9.81$ Newtons (kg*m/s^2 = N) in a downward direction. This is roughly $9*10^{-30}$ Newtons. Lets also assume that the electron happens to be traveling at the exact speed required so that the upward force exactly matches the downward force of $9*10^{-30}$ Newtons.

Lets say for some time period $T$ that both the gravitational field and magnetic field are perfectly uniform so that the electron travels in a straight line with constant velocity. This means there is no net acceleration and even though the electron experiences the Lorentz force it experiences no acceleration relative to the magnetic field. Does this mean that there is no cyclotron radiation, and during time period $T$, the electron will travel at this constant velocity without losing energy?

Answer 1

This is a surprisingly complicated question, and I'm not sure there is a universally accepted answer.

To see why this is turn off your magnetic field and give the electron enough velocity to keep it in orbit around the Earth. Now in the Earth frame the electron has a centripetal acceleration of $r\omega^2$ and therefore it should be emitting radiation. However in the electron's frame it is in freefall and isn't accelerating at all, so it should not be emitting radiation. And this results in a contradiction as both viewpoints cannot be correct.

But now suppose the electron is sitting stationary at the Earth's surface. In the Earth frame the electron is stationary and hence doesn't radiate. However in the electron frame it is now experiencing an acceleration (of 9.81 m/sec$^2$) and therefore should radiate. But it can't be radiating or we have the paradoxical situation where a stationary electron radiates, and violates conservation of energy in the process.

This issue is explored in some detail in Ben Crowell's answer to Does a constantly accelerating charged particle emit EM radiation or not?, though I have to confess I find myself unclear as to what exactly the answer is.

The origin of the problem is the way general relativity describes motion under gravity, and the problem exists with your question because you're considering motion under a combination of gravity and a magnetic field. A plausible argument can be made that in your experiment the electron should radiate, but an equally plausible argument can be made that it shouldn't radiate. Perhaps you can find different physicists willing to argue for both sides. Myself I have to confess that I do not know the answer. Sadly the amount of radiation we'd expect in this situation is so small that it's unlikely an experiment will ever be able to settle the issue.