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Ok. A photon has no rest mass for the good reason that it has no rest energy. But what about a standing wave? There is fair belief that particles are standing waves, so perhaps nothing rests.

So. Take a photon and bottle it. This presumes a bottle with a perfect mirror finish but not impossible in principle. The photon is now a standing wave. So does it have mass?

If it does then put the bottle in a gravitational field. The photon's path will be bent by gravity acting on a mass. Will that path be different for an unbottled photon?

Edit: Um. Not editing this, because I already have the answer I needed. Also because frankly I have no idea how to edit it to make it what you want. By all means close it or edit it to your preference. Me, I find it exciting that a basic assumption used in astronomy has so clearly been refuted (clearly, admittedly not necessarily correctly), but if you feel they have no need to read an answer that points to a very exciting paper then by all means kill it.

Me, I'm excited because as far as I can figure it if light is bent by simple gravity acting on mass rather than curved space then there are no black holes. Things very much like them but not singularities. Just as Einstein always insisted.

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If you could bottle a photon would it have mass?

I'd say the photon itself wouldn't, but the "system" would have extra mass, yes.

Ok. A photon has no rest mass for the good reason that it has no rest energy.

It has no rest mass because it isn't at rest. But it does have its energy E=hf and momentum p=hf/c. Since you can't reduce the energy without reducing the momentum it's better to think in terms of energy-momentum. This is a measure of resistance-to-change-in-motion for an electromagnetic wave propagating linearly at c.

But what about a standing wave? There is fair belief that particles are standing waves, so perhaps nothing rests.

The standing wave system does have rest mass because it's at rest, even though the wave inside isn't really at rest. Hence when you open your bottle the photon comes flying out at c. Not from a standing start - it was always moving at c. See https://arxiv.org/abs/1508.06478. That's a paper by Martin van der Mark and Gert (not the Nobel) ‘t Hooft called Light is Heavy. It talks about a photon in a mirror-box, and says the box is heavier and more massive when the photon is in there. Think of mass as resistance to change-in-motion for a wave going round and round at c. When you open the box, it's a radiating body losing mass as per Einstein's E=mc² paper.

So. Take a photon and bottle it. This presumes a bottle with a perfect mirror finish but not impossible in principle. The photon is now a standing wave. So does it have mass?

I'd say the photon itself wouldn't, but the "system" would have extra mass, yes.

If it does then put the bottle in a gravitational field. The photon's path will be bent by gravity acting on a mass. Will that path be different for an unbottled photon?

The photon's path is bent by gravity, not by gravity acting on a mass. Because of this, the mirror-box is heavier when the photon is in it. Here's the picture from Light is Heavy:

enter image description here

Note this in the discussion section at the end: "What is intriguing is that matter's most basic building blocks, the elementary particles, all have non-zero spin, intrinsic angular momentum, which seems to imply that they all must have some sort of intrinsic dynamics." When your system is called an electron, it does indeed have mass. When you annihilate it with a positron, that's like opening one box with another and vice versa, whereafter photons fly out and there's no boxes left. See Hans Ohanian’s 1984 paper what is spin? He says “the means for filling the gap have been at hand since 1939, when Belinfante established that the spin could be regarded as due to a circulating flow of energy”.

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