A couple of months ago, I was on Daniel Whiteson’s podcast, which is called “Daniel and Jorge Explain the Universe“. During the two-part episode in which I appeared, entitled “Is the Universe Made of Waves?” (Part 1 and Part 2), I explained some of the key points made in my book, “Waves in an Impossible Sea.”
One thing I emphasized is that while photons [“particles” of light] moving across empty space are always traveling waves moving at the speed of light c, electrons are different. When in motion they too are traveling waves, but unlike photons, they can slow down, and even be stationary. When stationary, they are standing waves, of a somewhat unfamiliar sort (as described here and here). All of this is discussed in detail in the book’s chapters 16 and 17.
Whiteson has sent me a couple of questions that listeners raised with him, and since I imagine some of you might have similar questions, I decided to answer them publicly here.
First Question: Frequency of Traveling and Standing Waves
- “How does the frequency of the traveling wave relate to that of the standing wave?
- It feels like they should be related because they are oscillations in the same field, but I think they don’t have a fixed relation because the mass is invariant and the traveling wave frequency is obviously velocity dependent.”
If a “particle” (which I prefer to call a “wavicle”, as quantum field theory views it is a wave with some particle-like properties) has a definite frequency, then by the Planck-Einstein quantum formula, that frequency f is related to its total energy E through the proportionality constant h, known as Planck’s constant:
A moving particle (a traveling wavicle) has more motion-energy [“kinetic energy”] than a stationary particle (a standing wavicle). Therefore it also has more total energy (its motion-energy plus its mc2 mass energy). That means it will have a higher frequency, too.
[I want to emphasize an important issue before I complete my answer. All motion is relative, and so are motion-energy and total energy. When I refer to a particle’s speed, I’m referring to its speed as you see it. Someone else, moving relative to you, may view the particle as having a different speed, a different motion-energy, a different total energy, and a different frequency. That’s relativity. So all formulas below are given from your point of view; someone else will have a different point of view. Einstein’s notion of relativity assures that all these points of view are logically consistent with one another.]
Suppose the particle has a rest mass mrest (a rare quantity on which all observers agree!) Its total energy when standing still is mrest c2. Let’s define Erest = mrest c2.
Now, if the particle is instead in motion with speed v, then its total energy is greater than its rest mass. The formula relating them is
We can now use the Planck-Einstein formula E = f h, and remembering E is the energy of a traveling particle and Erest is that of one that is standing still, we find
Since the square root is less than one, its reciprocal is greater than one, and thus a traveling wavicle always vibrates faster than when it is standing.
Note: I am leaving out several subtleties here, for lack of space-time; I will try to get back to them at another time. One subtle issue is the relationship between the speed v and the way in which the wave moves; it’s probably not quite what you imagine. (The key concept here is the difference between “group velocity” and “phase velocity.”) Another is that it’s not instantly obvious that the frequencies and appearances of the standing and traveling waves are all consistent with Einstein’s relativity. It takes some work to show that they are.
Second Question: Redshift and Mass in an Expanding Universe
- “Why does the traveling wave’s frequency get redshifted by the expansion of the universe but not the standing wave?
- If you think of the Universe as a single frame and expansion as a recession velocity, then you can use the Doppler effect picture to explain the lengthening of traveling waves. But if you think of galaxies as each having their own frame, where photons are redshifted due to expansion rather than recession velocity, then why don’t “standing waves” get stretched like traveling waves do? Why don’t particles lose mass as the Universe expands?”
What really gets redshifted by an expanding universe is wavelength, the distance between crests of the photon’s wave. The waves literally are stretched. But whether that affects a wave’s frequency depends on its details.
For a photon, stretched wavelength translates directly into a shift in the frequency. That’s because a photon’s wavelength λ is related to its frequency by
where c is the cosmic speed limit, also known as “the speed of light”. Consequently, if the universe doubles in size, it causes the photon’s wavelength to double and its frequency to drop in half.
But for an electron or any other particle with rest mass, this isn’t the correct formula. Instead the formula relating frequency and wavelength involves the particle’s rest mass (along with the cosmic speed limit and Planck’s constant.) It takes the form
where
(The origin and proper interpretation of this last expression is explained, in great detail and with great care, in chapters 5-17 of my book.)
This relation between frequency and wavelength has a very different behavior from that of photons! If the universe expands to infinite size, and the wavelength expands to infinity too, a photon’s frequency drops to zero. But for a particle with rest mass, only the first term in the square root goes to zero; the second term remains behind, leaving f = fstanding.
More completely
- If the wavelength is very small and the frequency very large, then fstanding is small compared to f, and the formula is almost the same as that for light waves, in which case a doubling of the universe’s size reduces the particle’s frequency by nearly half.
- But once the universe has expanded so much that the frequency of the particle is no longer much larger than fstanding, then doubling the size of the universe has a very limited effect on the particle’s frequency.
In short, the universe’s expansion has no impact on the particle’s fstanding or rest mass. If a particle has rest mass, then, when it has a relatively low frequency, redshift directly affects its wavelength, but it barely shifts its frequency at all. Its frequency has a minimum below which it cannot go: fstanding.
The Two Answers are Linked
Actually the two equations that appeared in the answers to these two questions
are secretly the same equation. You can see a hint of this in the two figures above; the two orange curves look almost like mirrors of one another. The relation between them isn’t instantly obvious; here’s where it comes from.
First, note that ftraveling in the first equation is literally the same thing as f in the second equation. The only difference is the notation; I didn’t emphasize, in answering the second question, that if the particle isn’t standing still, then, well, it’s moving — and so it is a traveling wavicle. And so, both equations give a relation between a traveling wave’s frequency and the corresponding standing wave’s frequency.
With that in mind, let’s drop the “traveling” subscript, take the square of both equations, and solve each one for (fstanding)2 . We’ll find
and these two equations are identical if
which happens to always be true.
[Notes: as a check of this last equation, notice that
- if v = c, as for a photon, we recover the photon’s relationship between frequency and wavelength;
- but as v goes to zero, λ can goes to infinity while f can remain finite.
Also, the above equation is the wave version of a better-known equation for objects in special relativity that relates their speed, their total energy, and their momentum p:
- v = p c2 / E
15 Responses
Matt, do you have plans to write another slightly more advanced book building upon the foundations you’ve laid down in your current book?
What strikes me about your articles, such as this one, is how they’re biased towards presenting the time-like picture only. I find it fascinating that energy and frequency above is one part of their respective vectors aka four-vectors; the other space-like components being momentum and wave-vector. Hence more generally: P = h_bar K or (momentum, energy/c) = h_bar (wave-vector, angular-frequency/c).
Admittedly, this is pushing your content towards physics undergraduate territory which might not benefit you and your publishers trying to make a living from selling your content to the more general public.
It is true that in the book I focused on energy rather than momentum, since that is of most concern in understanding rest mass and the Higgs field. But as this post shows, the momentum story is necessary to complete the picture for curious and mathematically comfortable reader of the book. Nevertheless, the audience for such explanations is smaller, and so most publishers would be less interested in such a book… and a scholastic publisher, who might be interested, would overcharge.
I am therefore inclined to put all the answers to questions such as yours on this website. That was part of the idea of having all the “Reader Resources” here to supplement the book. I also have the option of offering an on-line course for people who really want to go through it all systematically. Would that potentially interest you? (I’m still not clear of the technicalities of how I would do that…)
Have you thought about experimenting with Patreon like Sean Caroll?
I’m not interested in enrolling on online courses, private tutoring etc. since there’s already so much high quality ‘free’ content from MIT, David Tong, Lenny Susskind etc on YouTube. However, you answering questions in the comments at various levels is *very* valuable for me which may not be in your interest in the future for various reasons; as interest in your current book wanes. If that becomes the case, I’d be OK supporting you on Patreon if it wasn’t too expensive as with Sean Caroll.
Lenny Susskind’s Theoretical Minimum set of books was published by your current publisher in 2014; but are now published by Penguin. So I think a follow-on book to supplement your current one with ‘slightly’ more advanced stuff would likely interest your current publisher and fans like me of your current book.
That was 2014. As you say, there is now so much “free” content now that no one can afford to make anything better, because no one will buy it. If indeed no one will take a course in which they can ask questions of me directly, and would rather watch someone else’s class where they cannot ask questions, then I fear my book may be the last of its kind.
No one supports themselves decently on Patreon unless they go viral first. I may be too late for that.
Professor,
dear Matt,
I oedered your book as a Xmass present, and finally got hold of it (second printing) half may. It’s a nice read. Together with your blog (section on types 1 and 2 wavicles), it clarified my layman’s understanding of the concept restmass. Extra internal Menergy (to coin a phrase myself too) due to a degree of freedom for interacting with the H-field. Never thought of that. The Menergy was there all along, only now a part of it is frozen into non-motional restmass form.
Previously, I was a bit stuck on the interpretation given in
https://www.quantumdiaries.org/2011/06/19/helicity-chirality-mass-and-the-higgs/
where Flip Tanedo (in 2011 !) pointed towards the H-field mixing chiralities and so causing the speed limit to drop below c. (Are/were you aware of F. Tanedo’s blog? if so, how do you evaluate it?).
And now for something completely different:
Q: armed with the new insight into restmass, is there a bit of circular reasoning in booting physics concepts in class 101 from good old Newtonian concepts of inertia, force, momentum, acceleration and a universal clock? And from these minor fibs, elaborating and specialising to current knowledge?
Shouldn’t the definitions of Menergy and momentum be re-defined?
To me it’s like buying good money using ‘slightly’ counterfeit bills.
Thanks for your endeavour creating blog & book, more top scientists should do so.
Wouter.
Flip is a good scientist and blogger. Personally, I don’t love explaining things in terms of flipping (pun intended) chiralities and reducing the speed. It’s mathematically correct, and it has some intuitive value also, but it doesn’t really capture wat I see as the most important issue. The method I used is appropriate for any particle the Higgs gives mass to, no matter what its spin. Then there are details that apply to spin 1/2 particles (that’s what Flip is talking about) and other details that apply to spin-1 particles (described, as an introduction, in the early pages in my series on having an extra “triplet” Higgs field.)
As to your second question: as emphasized in my book, language is a mess, both outside of and inside of science. It’s always a problem. And language is circular; every word must be defined in terms of other words. So these issues cannot be evaded. Yes, if and when we know the final answer, it will probably be a good idea to start from scratch and redefine everything. But should we do it now, when we are sure that we are missing key insights? And even if we did it, would people use the new terms? (Consider the failure of Esperanto and the success of Google translate). Finally, how would we elect a committee to carry it out? Part of our terminology problem is that we lack a central international institution to make terminological decisions… which makes it hard to create a central international institution… etc.
Hi Matt Strassler, a question: Since “wavicle” doesn’t come up in any of the usual sources — it’s not in Wikipedia, for example — could you repeat your definition here? The best mental model I can come up with is an infinite superposition of equal-energy sinusoidal plane waves for all three axes, phase-synchronized at a small, nominally point-like region rest viewer’s xyz coordinate system, but with a UV cutoff to keep it from becoming infinite in energy. Is that in any way close to what you mean?
No, that’s way too complicated, and the energies in the superposition need not be equal, nor is phase synchronization necessary; and a UV cutoff isn’t typically needed either. For example, an electron in a superposition of the ground state and the first-excited state of hydrogen is still a single wavicle, but it satisfies none of your criteria.
A wavicle is a quantum of a wave. First, are we all clear on what a quantum of a vibration is — i.e., what a quantum of a pendulum’s vibration is, namely the vibration of the pendulum with the smallest possible amplitude? Let’s make sure we are all in agreement on that…
This always seemed intuitive to me, redshift saps energy from things and worked on motion. So a moving particle’s motion would be reduced towards zero, its minimum energy. A massless particle has no lower bound on its energy so its wavelength can stretch indefinitely. I’ve never come across the math behind it before and it’s nice to see it all laid out.
Right — although you said one thing that’s amiss. You wrote :”A massless particle has no lower bound on its energy so its wavelength can stretch indefinitely.” But the wavelength of a particle with mass can also stretch indefinitely. So that’s not the issue.
Instead, for a particle with nonzero mass, the wavelength going to infinity does not cause the energy to drop to zero. That’s because the formula for frequency and wavelength is different from what it is for a massless particle.
Dr.Strassler:
In your reply above… “Instead, for a particle with nonzero mass, the wavelength going to infinity does not cause the energy to drop to zero. That’s because the formula for frequency and wavelength is different from what it is for a massless particle”.
Isn’t this because a particle with mass has a minimum energy, rest energy, equal to MC^2?
Yes, that’s right. The minimum total energy for a particle is its rest energy (or intrinsic energy, as I call it in the book), which is Mc^2 where M its rest mass, also called invariant mass. Its frequency is its total energy divided by Planck’s constant, so its minimum frequency is its minimum energy divided by planck’s constant, namely Mc^2/h. As discussed in the book’s chapters 5-17.
Years of teaching undergraduate math have taught me to use “reciprocal” instead of “inverse”. When talking to other math-y people, either is fine (unless the context is overflowing with inverse functions).
Good point!
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