(Previous posts in this series: Part 1, Part 2, Part 3, Part 4, Part 5, Part 6, Part 7, Part 8, Part 9)

In the previous post, I quoted Fritz Rohrlich who said that asking whether an accelerated charge really radiates is meaningless and that we need to also look at the value of the Poynting vector at the location of the detector. In discussing the radiation from an electric charge that is freely falling in a gravitational field, the state of motion of the charge and the detector both have to be taken into account. and this requires transformations between frames and will depend on whether the detector is an inertial observer or not. The transformations to accelerating frames requires the use of the mathematical machinery of general relativity.

With that. in mind, let us consider a charge Q and a radiation detector D and see what happens under various scenarios that describe different states of motion of each. There are five possible scenarios of interest to consider. It should be borne in mind that these scenarios are hard to test experimentally and, as far as I am aware, have not been tested.

Scenario 1: Both Q and D are floating freely in empty space in the absence of all forces

Suppose that Q is freely floating in outer space in the absence of all forces, gravitational and otherwise. Thus, by the law of inertia, Q will be at rest or moving with constant velocity. If D observes Q as being at rest or moving with constant velocity, that means that D is an inertial frame. All frames that move with a uniform velocity (i.e., with a constant speed that does not change direction) with respect to D are also inertial frames. Without loss of generality, we can choose D to be an inertial frame that has the same velocity as Q so that Q is at rest in this inertial frame, that we will call S to signify empty space. Maxwell’s equations predict that in inertial frames, a charge Q at rest will only produce an electric field, which is the Coulomb field. The magnetic field is zero everywhere and hence the Poynting vector is also zero, which means that Q will not radiate in S and hence D should not detect any radiation. This is not controversial, in that there seems to be general agreement on this.

Conclusion: D will not detect any radiation and and thus Q is said to not radiate.

Scenario 2: Both Q and D are both falling freely in a uniform gravitational field

Suppose that we now enclose Q and D in a rectangular room that accelerates ‘upwards’ (i.e., in the direction from what we label the ‘floor’ of the room to the ‘ceiling’) with a value g. We call this frame E to signify that this is something like an elevator. In this frame, both Q and D will appear to have an acceleration g ‘downwards’. According to the Principle of Equivalence (PoE), this scenario is equivalent to Q and D falling freely in a frame that consists of a uniform gravitational field g acting downwards, so the label E also signifies Earth. (Although the Earth’s gravitational field is not uniform since the Earth is a sphere, in any small region it is approximately uniform. Hence E can be taken to represent the Earth.)

But the status of Q and D has not materially changed from Scenario 1. It is just that ‘we’ have introduced ourselves as a third party watching the situation while stationary in the frame E. As far as the freely falling D is concerned, there is still no magnetic field in its presence and hence D will not detect anything and thus we infer that the freely falling Q does not radiate, which makes sense since it is still at rest in the inertial frame S. Thus the conclusion is that a detector that is falling freely along a uniform gravitational field (which is approximately the Earth’s gravitational field in a small region) will not detect any radiation from a charge that is also freely falling in that gravitational field.

The electric field as seen by D is just the Coulomb field of the charge, just as it was in Scenario #1. Hence we see that the Principle of Equivalence is preserved in that the results of experiments to detect the fields will not enable an observer to distinguish whether they are in the frame S or E.

Conclusion: D will not detect any radiation and and thus Q is said to not radiate.

Scenario 3: Q is freely falling while D is on the floor in E

Now consider the detector D at rest on the ‘floor’ of the frame E while the charge Q is freely falling. This is the scenario that started this whole series of posts because it describes the situation of an electric charge dropped from a height on Earth, and we tacitly assumed then that we were measuring the radiation using a detector that was stationary in this frame. Note that in the frame S, Q will be at rest while D has a constant acceleration g upwards in its own frame. Since Q is at rest in the inertial frame S, one might think that Q will not radiate and thus D will not detect any radiation, since according to our understanding of Maxwell’s equations, a charge only radiates when it is accelerating in an inertial frame. In part Part 8 of this series, this was the argument used to arrive at the conclusion that the detector D would not detect any radiation. But we need to look more closely.

Recall the comment by Rohrlich that “Since radiation is not a generally covariant concept the question whether the charge really radiates is meaningless unless it is referred to a particular coordinate system.” We need to see what is happening in the frame of the detector D. In this case, the detector D is not an inertial frame and, according to Rohrlich, it will detect radiation in this situation. How does he arrive at this conclusion, given that Q is at rest in the inertial frame S? He says that we have to look at the electric and magnetic fields as seen by D in the frame E. Since D is accelerating, we have to calculate the Poynting vector at the location of the detector in its accelerating frame E and this requires transforming the electric and magnetic fields into that required frame.

In the frame S in which Q is at rest, the detector D which is also at rest in S (Scenario 1) will see just the Coulomb electric field and no magnetic field and thus no radiation. To get the fields in the frame E, we have to transform the Coulomb field to that frame. We cannot use Lorentz transformations because the frame E is accelerating with respect to S. Since E is equivalent to a frame that consists of a uniform gravitational field, in order to do the required transformations we have to use general relativity and calculate the curvature metric for a space that consists of a uniform gravitational field.

Rohrlich does so and finds that both electric and magnetic fields are non-zero in the frame E and thus the Poynting vector is non-zero and hence D will detect radiation, thus contradicting the conclusion arrived at in Part 8 by an insufficiently thought-out application of the Principle of Equivalence.

Conclusion: D will detect radiation and thus Q is said to radiate.

Scenario 4: Q is on the floor in E while D is freely falling

Now consider the charge Q at rest on the ‘floor’ of the frame E while the detector D is freely falling. Hence Q will experience a constant acceleration g upwards in its own frame.

Since Q is accelerating in an inertial frame S, then according to Maxwell’s equations, it should radiate. Rohrlich calculates the electric and magnetic fields in the frame S due to Q accelerating and finds that they are both non-zero everywhere, thus resulting in a non-zero Poynting vector at the detector D. Since the freely falling D is static in the inertial frame S, it is an inertial observer and thus will detect this radiation, confirming that it will be inferred that Q is radiating.

Conclusion: D will detect radiation and thus Q is said to radiate.

Scenario 5: Both Q and D are at rest on the floor in E

In the frame S, both Q and D will have a constant acceleration g upwards in their own frames. To find out if D will detect any radiation, Rohrlich starts with the electric and magnetic fields he calculated in Scenario 4 where Q (which was at rest on the floor of E) was accelerating in the frame S and the detector was at rest in that same inertial frame S. He then transforms those fields into the frame E to see what a detector at rest in the frame E would see. Since we are transforming between accelerating frames, we have to use general relativity again. Using the transformations, Rohrlich finds that the magnetic fields vanish in E. Hence the Poynting vector is zero and hence we can infer that D will not detect any radiation and thus it will be inferred that Q is not radiating.

Conclusion: D will not detect radiation and thus Q is said to not radiate.

That concludes the main results, at least as according to Rohrlich.

So far, so good. We have arrived at concrete results based on actual detailed calculations of the Poynting vector in the various situations, rather than using heuristic arguments that can lead us astray when we are dealing with subtleties like this.

These results agree with two of our common expectations. One is that a detector at rest on Earth will detect radiation from a freely falling charge (Scenario 3). The other is that a detector at rest on Earth will not detect any radiation from a charge that is also at rest on Earth (Scenario 5). The latter in particular is a relief because otherwise it would imply that we could have an unlimited source of electromagnetic energy in the form of radiation. While that would solve the world’s energy problems, it would violate the laws of physics in a major way.

But as I warned right at the beginning of this series, resolving some issues raises new issues which arise when we try to find meaning in the results that can be reconciled with our intuitive sense of what should happen. The problem is that our intuition is not a reliable guide in many situations that are far removed from our actual sensed experience. This is true in the case of special relativity because our everyday experience is with speeds are nowhere close to the speed of light. It is true for quantum mechanics where our everyday experience is with objects that are massive. And it is true here because our everyday experience is conditioned by living in inertial (or almost inertial) frames, and by masses and electromagnetic phenomena that cause the curvature of space to be small enough to be ignored. So in these situations, one has to tread warily and depend upon the results of calculations to tell us what to expect, rather than using our intuition to decide what we should see.

So we are still not the end of the story. Some readers may already have noticed some seeming new paradoxes in the results of the five scenarios. The next post will deal with those.

(To be continued.)

1. Rob Grigjanis says

Scenario 2:

The electric field in the frame E is just the Coulomb field of the charge

Do you mean in the frame S? We don’t know the electric field in the frame E yet; that’s calculated in scenario 3.

2. Rob Grigjanis says

CD @2: Wait until he gets to the Killing vector (actually, he probably won’t in this seres).

3. Mano Singham says

Rob @#1,

I have changed the wording to make it clearer what I meant.

4. Rob Grigjanis says

Mano @4: I still have a quibble. I’ll just give my wording for scenario 2.

The frame S is the deep space frame in which Q and D are at rest. E’ is the elevator frame; the one containing Q and D, but accelerating with respect to them.

For the gravitational field, E is the Earth frame; the pov of someone standing on the surface. T is the free-falling frame in the gravitational field, in which Q and D are stationary.

Then the PoE is a statement of the equivalence of E and E’, and, as a corollary, the equivalence of S and T.