Why is photon gas pressure = photon energy density (per volume) divided by 3?
Basically it's down to a photon's energy being equal to its momentum, and the division by three comes from the three dimensions of space (energy is a scalar, independent of direction, but momentum has direction, and pressure is the average momentum of particles traveling in a given direction).
Not to hijack the thread, but is there a physical limit to how high electromagnetic energy density can go, like a certain point where you cant pack anymore photons in a given volume at all? I imagine now there would come a point where a black hole would form, but in the beginning, it seems like you could have just about any amount of photons in a cerain space.
If you just take classical electromagnetism, yes, you can pack as many as you like into whatever volume you like.
In practice, however, the number and frequency of photons you typically get within a region of space-time depends upon the temperature. At very high temperatures, virtual particles start to appear regularly alongside the photons. For example, once a significant number of photons have enough energy to collide and produce electron/positron pairs, you end up with a gas consisting of photons as well as electron/positron pairs. As the temperature increases further, more types of particle/anti-particle pairs are produced. This tends to "spread out" the energy, so that it requires much greater energy to go to higher temperatures. It's sort of like boiling water: the water gets to 100C, and then you have to keep dumping energy into it until it all boils before it will go any higher in temperature.
At some point, you get to high enough temperatures that the physics we know doesn't work any longer. Then, well, we don't know what happens.
Thank you Chalnoth.
I know this is an old thread but, since I was stuck on the same question and have worked out a derivation, I may as well post it. The derivation for a photon gas (for which the relationship is exact) is in the first part of the post. The second part is trying to derive it as an approximate relationship for a relativistic gas of massive particles such as electrons. I'm not happy with that bit yet, so it's best ignored.
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