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Already I’ve had a few people ask me for clarification of a key point in the book, having to do with a certain type of unusual “standing wave.” It’s so central to the story that I’ve decided to address it right away.

The point that there are two quite different types of standing waves; the familiar ones you may know from musical instruments or from physics class, and less familiar ones that play a key role in the book. You can jump right to my new webpage comparing these two types of standing waves, or you can read the post below, which provides more context.

Note: Going forward, you’ll see a lot of posts and new webpages like this one. One of the great things about 21st century books is that they aren’t contained within their covers. I’ve always planned to continue the book into this website, allowing me to expand here upon key issues that I knew would will raise questions from readers. So even though the book in printed form is done and published, it will continue to live and grow on this website.

The Stationary Electron as a Standing Wave

In the book’s chapter 17, I suggest a sort of mental image of a stationary electron. In particular, electrons should be visualized as waves, not as little dots, and a stationary electron is a standing wave — a wave that vibrates in place. [I focus on a stationary electron because it’s the best context in which to understand an electron’s “rest mass” (see chapters 5 and 8.)]

If you know something about standing waves already, perhaps from music classes or from a first-year physics class, this statement is potentially confusing. An electron could be stationary out in the middle of nowhere, light-years from the nearest star. But the standing waves of music and physics classes are never found in the middle of nowhere; they are always found inside or upon objects of finite size, perhaps upon a guitar string, inside a room or in the Sun. So how could a free-floating, isolated electron out in the open be a standing wave?

This mismatch is naturally puzzling, and indeed it has already raised questions among listeners of my recent podcast appearances. [Here’s the conversation on Sean Carroll’s podcast, and here are the first half and second half of the conversation with Daniel Whiteson on his podcast.]

The point, which I didn’t have time to address in the podcasts and which is discussed in the book only in examples (see chapter 20.2), is that there is a type of standing wave that is not covered in first-year physics classes, and that appears in no human musical instruments that I’m aquainted with. Unlike familiar standing waves, it needs no walls.

Familiar Standing Waves

A string or other extended vibrating object may have many types of standing waves. For a string, the four simplest standing waves are shown below (Figure 25 from the book chapter 11; illustration by C. Cesarotti).

But we can just focus our attention on the simplest of all such waves, the one at upper left, which has a single crest that over time becomes a single trough and back again, over and over.

The simplest standing wave on a string (in green) of finite length; the dashed line shows where the string would be located in the absence of a wave.

Classic examples of these simplest standing waves are found on the strings of guitars and pianos; somewhat similar waves are found in organ pipes and flutes. As budding musicians quickly learn, it’s generally the case that the longer the string or pipe or bell, the lower the frequency of the wave and the lower the musical note. An organ’s lowest notes come from its longest pipes, and shortening a guitar string with your finger causes the instrument to create a higher note.

[It’s not quite as simple as that because, as covered in chapter 10, there are other ways to change frequency; tightening a string raises it, while replacing air with another gas can raise or lower it. But for a fixed material with fixed properties, what I’ve said is true.]

A simple version of this basic idea is illustrated by taking a box whose sides are of length , filling it with some sort of material, and considering that material’s simplest standing wave. For most familiar materials, the frequency of the standing wave decreases as the length of the box increases; specifically, if you double the length of the box, the frequency drops in half. Thus frequency is inversely proportional to length, as it is on many musical instruments, and as the box’s size becomes infinite, no standing wave remains — its frequency becomes zero, meaning that it no longer vibrates at all.

[In math, we would write

• ,

where is the speed with which traveling waves can move across the substance.]

Unfamiliar Standing Waves

However, there are other standing waves whose frequency of vibration does not decrease in this way. For standing waves of this unfamiliar sort, doubling the length of the box does not cause the frequency to drop in half. In fact, if the box is big enough, doubling its size barely has any effect on the frequency at all! (This can happen in unfamiliar materials, or in familiar materials treated in unusual ways; see chapter 20.2 for a couple of examples.)

If you put one of these unfamiliar standing waves in a box and make the box larger and larger, its frequency won’t drop all the way to zero. Instead it will settle down to a steady frequency, which I’ll label and refer to as the “resonance frequency”. No matter how big the box, the smallest the wave’s frequency can possibly be is the resonance frequency . Said another way, if is sufficiently large, the difference between and will be too small to notice, or even to measure.

[In math, the wave’s frequency is related to the resonance frequency and to the length of the box by an equation similar to

The precise form of the expression depends on details of the box and the vibrating material; but the details are not important here.]

Summarizing the Two Waves

The size and shape of a large box thus affects these two types of standing waves differently,

• impacting the shape and the frequency of familiar standing waves, while
  
  
• impacting the shape of unfamiliar standing waves, but not their frequency

This difference is illustrated in animations found here, and in the graph below, which shows how the frequencies of familiar and unfamiliar standing waves depend on the length of the box.

The frequencies of familiar standing waves (blue) decrease down to zero as L goes to infinity, but those of unfamiliar standing waves (orange) never go below a non-zero minimum frequency f0. A Wave that Stands Without Support

Suppose, then, that we take an unfamiliar standing wave and dismantle its box, or let its box grow to infinite size. Do we then have a standing wave that stretches all across the entire universe?! Well, mathematically speaking, this would indeed be true. But in realistic situations, the standing wave will always run into some obstructions around its edges. Perhaps your hand is close by, as is the ground and a nearby wall; trees and mountains block the wave in other directions, and so forth. Even out in deep space, the space is not completely empty; there are always stray particles moving by.

These objects will affect the shape of the standing wave. But if they are far enough away from the core of the standing wave, they do not affect its frequency — at least, not by an amount that anyone could readily notice or measure.

Thus even a realistic standing wave, with a diameter far smaller than the size of the universe, will vibrate at its resonance frequency if it is sufficiently isolated. That’s more than enough for the purposes of the universe.

The Importance of These Waves

And so an isolated, stationary electron corresponds to an unfamiliar standing wave whose shape is determined by its local environment, but whose frequency is not. It will vibrate at the same rate whether it is in deep space far from any stars, or whether it is in an empty, airless box the size of a sugar cube.

Why should we care? Because there is a direct connection between the frequency of this standing wave and the electron’s rest mass. Explaining that connection is the most important goal of the first two-thirds of the book (through chapter 18). In addition, there’s a link between this issue and the Higgs field — a topic of the book’s remainder (especially chapter 20.)

I hope I’ve managed to write this post in a way which is useful even if you aren’t reading the book. For those reading it, it may well be helpful in clarifying chapters 17-20, where these issues take center stage. If you have found this post confusing, please leave comments, or ask a question on my new Book Questions page. As always, your questions and suggestions will help me improve this website.

In response to a recent surge in interest in alien phenomena and claims that the US government is hiding what it knows about extraterrestrials, the Pentagon established a committee to investigate the question – the All-Domain Anomaly Resolution Office (AARO). They have recently released volume I of their official report – their conclusion:

“To date, AARO has not discovered any empirical evidence that any sighting of a UAP represented off-world technology or the existence a classified program that had not been properly reported to Congress.”

They reviewed evidence from 1945 to 2023, including interviews, reports, classified and unclassified archives, spanning all “official USG investigatory efforts” regarding possible alien activity. They found nothing – nada, zip, goose egg, zero. They did not find a single credible report or any physical evidence. They followed up on all the fantastic claims by UFO believers (they now use the term UAP for unidentified anomalous phenomena), including individual sightings, claims of secret US government programs, claims of reverse engineering alien technology or possessing alien biological material.

They found that all eyewitness accounts were either misidentified mundane phenomena (military aircraft, drones, etc), or simply lacked enough evidence to resolve. Eyewitness accounts of secret government programs were all misunderstood conversations or hearsay, often referring to known and legitimate military or intelligence programs. Their findings are familiar to any experience skeptic – people misinterpret what they see and hear, fitting their misidentified perception into an existing narrative. This is what people do. This is why we need objective evidence to know what is real and what isn’t.

I know – this is a government report saying the government is not hiding evidence of aliens. This is likely to convince no hard-core believer. Anyone using conspiracy arguments to prop up their claims of aliens will simply incorporate this into their conspiracy narrative. Grand conspiracy theories are immune to evidence and logic, because the conspiracy can be used to explain away anything – any lack of evidence, or any disconfirming evidence. It is a magic box in which any narrative can be true without the burden of evidence or even internal consistency.

But the report is devastating to those who claim the government has known for a long time that aliens exist and are in possession of alien tech. It also means that in order to maintain such a belief, you have to enlarge the conspiracy, give it more power and scope. You have to believe the secret program is secret even from Congress and the executive branch, and that it is either secret from the defense and intelligence communities or they are fully involved at every level. At some point, it’s not really even a government program, but a rogue program somehow existing secretly within the government.

This is how grand conspiracy theories fail. In order to be maintained against negative evidence, they have to be enlarged and deepened. They then quickly collapse under their own weight. Imagine what it would take to fund and maintain such a program over decades, over multiple administrations and generations. How total would their control need to be to keep something this huge secret for so long? There have been no leaks to the mainstream press, like the Pentagon papers, or the Snowden leaks, or even the Discord fiasco. And yet, some rando UFO researchers know all about it. There is no way to make this story make sense.

I also don’t buy the alleged motivation. Why would such an agency keep the existence of aliens secret for so long? I can see keeping it a secret for a short time, until they had a chance to wrap their head around what was going on – but half a century? The notion that the public is “not ready” for the revelation is just silly. We’ve been ready for decades. If they want to keep the tech secret, they can do that without keeping the very existence of aliens secret. Besides, wouldn’t the principle of deterrence mean that we would want our enemies to know – hey, we have reverse-engineered alien technology, so don’t mess with us?

Also, the conspiracy theories often ignore the fact that the US is not the only government in the world. So do all countries in the world who might come into possession of alien artifacts have similarly powerful and long-lived secret organizations within their government? Some conspiracy theorists solve this contradiction by, again, widening the conspiracy. This leads to “secret world government” territory. Perhaps the lizard aliens are really in charge, and they are trying to keep their own existence secret.

I’ll be interested to see what the effect of the report will be (especially in our social-media post truth world). Interests in UFOs wax and wane over the years. It seems each generation has a flirtation with the idea then quickly grows bored, leaving the hard core believers to keep the flame alive until a new generation comes up. This creates a UFO boom and bust cycle. The claims, blurry photos, faked evidence, and breathless eyewitness accounts all seem superficially fascinating. I got sucked into this when I was around 10. I remembering thinking that something this huge, aliens visiting the Earth, would come out eventually. All the suggestive evidence was interesting, but I knew deep down none of it was conclusive. At some point we would need the big reveal – unequivocal evidence of alien visitation.

As the years rolled by, the suggestive blurry evidence and wild speculation became less and less interesting. You can only maintain such anticipation for so long. Eventually all it took was for me to hear Carl Sagan say that all the UFO evidence was crap, and the entire house of cards collapsed. Now, 40 years later, nothing has changed. We have mostly the same cast of dubious characters making the same tired claims, citing mostly the same incidents with the same conspiracy theories. The only difference is that their audience is a new generation that hasn’t been through it all before.

Perhaps the boom bust cycle is faster now because of social media and the relative short attention spans of the public. I suspect the Pentagon report will have the effect of forcing those with a more casual interest off the fence – either you have to admit there is simply no evidence for alien visitation, or you have to go the other way an embrace the grand UFO conspiracy theory. Or perhaps the current generation simply does not care about evidence, logic, and internal consistency and will just believe whatever narrative generates the most clicks on Tik Tok.

The post Pentagon Report – No UFOs first appeared on NeuroLogica Blog.

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A conversation with Leah Goldstein on becoming a kickboxing world champion, ultra-endurance cyclist, and an elite commando combating terrorism. For this she was to be honored at the International Women’s Day event… until she was disinvited and canceled.

This is her story.

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