(Previous posts in this series: Part 1, Part 2, Part 3, Part 4, Part 5, Part 6, Part 7, Part 8, Part 9, Part 10, Part 11, Part 12, Part 13, Part 14)

This series started by asking a simple question, whether a charged particle and a neutral particle would fall at the same rate when dropped from the same height and reach the ground at the same time. You would think that it would have a simple answer. But no. After a fairly long journey, we arrived at the conclusion that they would. But in the process, the series had to address a whole host of related issues along the way. While many of those were seemingly resolved, there are some fundamental questions that remain murky.

We saw in part #13 that the mass of a point charge like an electron is not a simple thing, because an electric charge has an associated electric field that itself has energy and thus should be thought of as contributing to the mass, except that the field energy density goes to infinity at zero distances, which is of course awkward for point-like charges. By looking at the radiation reaction force created by an accelerating charge, we learned about something called the acceleration energy Q that increases with the speed of a charge, and the energy radiated by a linearly accelerating charge comes from this source.
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Unless you have some kind of medical condition that makes the risk of vaccination greater than the risk of getting covid-19, you should get vaccinated. But there are people who decide, against the best scientific advice, that they know better alternatives. Not all of them are rabid anti-vaxxers or are opposing it because of some ideological fixation or because they believe in some bizarre conspiracy theory that it is a device for the government to gain control over one’s body. There are those who do not get it because they have folkloric beliefs, such as that healthy people have natural immunity or that the symptoms will not be serious or that certain practices such as wellness or Ayurveda or other folk remedies will stave off the infection.
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(Previous posts in this series: Part 1, Part 2, Part 3, Part 4, Part 5, Part 6, Part 7, Part 8, Part 9, Part 10, Part 11, Part 12, Part 13)

The history of analyses of the question “Does a charge falling freely in a gravitational field radiate energy?” is quite fascinating with experts being all over the place, even to this date. The biggest disagreement is over whether the answer to the above question is yes or no. But even among those who agree on the answer, there are disagreements as to the reasons. For example, among those who say that it radiates, some argue that it leads to a loss in kinetic energy and that the Principle of Equivalence may not even hold in the case of a charge falling freely in a uniform gravitational field. Others disagree on what the source of the radiation is.
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(Previous posts in this series: Part 1, Part 2, Part 3, Part 4, Part 5, Part 6, Part 7, Part 8, Part 9, Part 10, Part 11, Part 12)

We seem to have arrived at a resolution to the paradox that this series started with that maintains our traditional expectations. Recall that it seemed like two postulates that were thought to be incontrovertible were incompatible when applied to a situation in which an electric charge and a neutral particle were dropped from the same height. It seemed like one or both had to be wrong. Those two postulates were:

Postulate #1: If we eliminate all other forces such as friction, all objects that are dropped from the same height in a gravitational field will fall at the same rate and hit the ground at the same time.

Postulate #2: An accelerating charge will radiate energy.

In the previous post, we arrived at a resolution in which the falling charge will radiate, in agreement with Postulate #2, but that this radiated energy does not result in a loss in the kinetic energy of the charge and thus it will fall at the same rate as the neutral particle, in agreement with Postulate #1.

The way this resolution was arrived at may not satisfy everyone. It involved invoking an aspect of the mass of a charged particle that we may not be familiar with. We looked at some aspects of the subtlety of mass back in Part 5. But there’s more, and this was the introduction of a mysterious source Q that provided the energy radiated by an electron that was accelerating under the influence of a uniform gravitational force. Recall that the rate of energy radiated by the charge was given by the familiar Larmor expression ℛ = 2e²g²/3c³ for a charge e having an acceleration g. Q = 2e²a^o/3c² where a^o is the zeroth component of the covariant four-acceleration of the charge given by a^μ = dv^μ/d𝜏, such that a^o = γv.a.
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An underwater volcanic eruption in the Pacific has triggered Tsunami warnings. The hardest hit is the island of Tonga and NPR reported that communications with the island had been cut off.

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I am not the kind of person who chooses to go on cruise ships. I have been on long ocean voyages as a boy three times between Sri Lanka and England but that was back in the day when travel by ocean liner was the cheapest or only way to go to distant places. The idea of being on a ship for days and even weeks on end without any specific destination in mind that I could not reach any other way does not appeal to me. This is perhaps because I am not a very sociable person and these cruises seem designed, if the many advertisements I see are any indication, to be essentially floating holiday resorts that cater to people who enjoy spending most of the day in the company of others, many of whom they have never met before, and taking part in all manner of social gatherings and organized entertainments.

I am sure it must be great fun for those who enjoy such things and can afford them. I know people who go on them every year and I have discovered since coming to Monterey and playing the game of bridge more often that bridge players seem to be big fans of them. There are cruises catering to them, and I have started getting inundated with ads for bridge cruises where experts offer lessons. The people at the bridge club exchange information about the various cruises.
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(Previous posts in this series: Part 1, Part 2, Part 3, Part 4, Part 5, Part 6, Part 7, Part 8, Part 9, Part 10, Part 11)

In the previous post we arrived at conclusions that will enable us to address the basic paradox that started this series of posts, and is restated as Issue 3 below. I will be introducing some mathematics here is order to make the line of reasoning clearer for those who know something of this topic but for those readers for whom this is unfamiliar, I would encourage you to not sweat the details but simply read the text between the equations and I think (hope!) you will get the gist.

Issue 3: If a detector D on Earth detects radiation from a falling charge (Scenario 3), that implies that the charge loses some energy in the form of radiation and thus one would expect that it will fall more slowly than a neutral particle, thus violating Postulate #1 and the Principle of Equivalence that says that all objects falling freely in a gravitational field will fall at the same rate. If it still falls at the same time as the neutral particle and thus does not violate Postulate #1, that must mean that its kinetic energy is unaffected by the emission of radiation. Does that not violate the law of conservation of energy? We seem to have arrived at two irreconcilable and unpalatable options.
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The fossilized remains of a large 10m long ichthyosaur, a sea predator, has been found in the mud of a dried out region of a reservoir in England. Here is an artist’s reconstruction of what it might have looked like.

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