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

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

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 ** mc^{2}** mass energy). That means it will have a

*[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 *m*_{rest} (a rare quantity on which all observers agree!) Its total energy when standing still is *m*_{rest} ** c^{2}**. Let’s define

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

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** =

More completely

- If the wavelength is very small and the frequency very large, then
*f*_{standing}is small compared to, 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.*f* - But once the universe has expanded so much that the frequency of the particle is no longer much larger than
*f*_{standing}, 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 *f*_{standing} 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: *f*_{standing}.

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 *f*_{traveling} in the **first** equation is literally the same thing as ** f** in the

With that in mind, let’s drop the “traveling” subscript, take the square of both equations, and solve each one for **( f**

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 c**^{2}/ E