(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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As the Omicron surge continues, David Leonhardt and Ashley Wu have analyzed the data on Covid-19 cases in Seattle and New York, two cities that have the most timely data. They have produced six charts that pretty much tell the whole story about how the pandemic is affecting the vaccinated versus the unvaccinated.

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Thanks to Rob Grigjanis, I got the link to the talk I gave on Saturday on the wildly varying age of the Earth. The full day’s talks are below and mine begins soon after the 5:25:00 mark.

If you do not know how to skip to the part where my talk begins, you can go here where it has already been cued up for you.

Rob and I had a discussion about my discussion of Kelvin’s role in the comments section of my earlier post announcing the talk that those interested can go and read.

(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)

For convenience, let me summarize the results that were arrived at in the previous post of the calculations of the Poynting vector by Rohrlich for a charge Q and detector D for the following five scenarios. S represents an inertial frame (such as freely falling in a uniform gravitational field) while E is the frame of the Earth.

Scenario 1: Both Q and D are floating freely in empty space in the absence of all forces
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
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
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
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
Conclusion: D will not detect radiation and thus Q is said to not radiate.

Let us see how Rohrlich’s results affect the two postulates that began this series of posts:

Postulate #1: If we can 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 falling freely in a gravitational field will radiate energy.

The results of Scenario 3 support Postulate #2, that the falling electric charge will radiate and that radiation will be detected by a detector at rest in the frame.

But there are still three (at least) unresolved issues.
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