# Do photons have a maximum density per unit area?

I've been sitting on this thought for a while, and I figure it's time to pass it off to someone more educated than myself.

Consider the following hypothetical scenario:

In a vacuum, we have six panels, each of which has one face which is a perfect white body (known to the layman as "a mirror"). The other face is translucent - it allows some light through, though the exact amount is irrelevant. These panels are similar to the two-way mirrors seen in police interrogation rooms on TV when viewed from the observer's room - essentially, light can enter the observer's room, but it can't leave. Any light that hits the panel that would otherwise enter the interrogation room is reflected back "completely and uniformly in all directions" (per wikipedia definition of a white body).

Using these six panels, we create a box where all white body faces are directed inwards, such that if we shine a light at the box from the outside, light can enter the box, but it cannot leave the box.

If we shine the light at the white body box for a short amount of time, and then turn it off, the photons (should) bounce around inside the box indefinitely.

If we shine the light at the white body box and do not turn it off...?

Question: Does light have a maximum density per unit area? (Can we "fill" the box with light? If so, what happens? Why?)

Bonus Question: Does the effect differ if we were to shine a different wavelength of light (e.g. radio wave, xray, etc.)?

PS: I'm aware that Density may not be the proper term to describe the question I'm asking. The idea of whether or not light has mass seems ambiguous to me - while it seems to be the general consensus that light does not have mass, it does have energy, so... let me know in the comments.

PSS: I'm aware that my box must be constructed of unobtanium (read: doesn't exist), but just bear with me.

• One-way mirrors are not exactly one-way. There is no way to have something that transmits more in one direction than in the other, this would be essentially equivalent to a Maxwell's demon setup violating the 2nd law of thermodynamics. Your proposed box just doesn't exist. Dec 17, 2015 at 21:46
• Well, of course it doesn't. I'll clarify the question to reflect that. Dec 17, 2015 at 21:54
• I believe the essence of this question is "Do photons have volume, given that they're both a particle and a wave?" Am I right? Dec 17, 2015 at 23:26
• We can look at it that way. Do photons have volume, and if so, how many can I cram into a unit area? But seriously, what happens to the box, man! ;) Dec 17, 2015 at 23:30

Every photon has the energy $e=h\cdot f$ where $h$ is Planck's constant and $f$ the frequency. Since mass and energy are related through $m=e/c^2$ the mass of your box would become larger the more photons you put in. Since the Schwarzschildradius is $r_s=2\cdot G\cdot m/c^2$ the Schwarzschildradius of the box would after a while exceed it's radius and become a black hole, assuming of course that no radiation leaks out and your box can handle the temperature and pressure. If for example you heat up a can of water with a laser it also gains mass, which it radiates away when it cools down. If you shine in more than your box can radiate away it will become heavier. Since photons are bosons which can in contrast to fermions co-exist in the same state in the same place and time there is in priciple no upper limit of how many photons you can put into your box.

• I had a feeling I was basically asking what happens at the center of a black hole, but I wasn't sure. And I guess I never thought of that, but logically, a particle with more energy must also weigh more. To clarify, does that mean the mass of a photon increases linearly with an increase in wavelength (and likewise, my box would be heavier if I shined blue light in it as opposed to red light)? I'm going to sit on it for a day to watch for other answers, but I'll gladly upvote when I get the required rep. Dec 17, 2015 at 21:53
• >*does that mean the mass of a photon increases linearly with an increase in wavelength*< - No, it increases linearly with an increase of frequency and decreases linearly with an increase of wavelenght [: Dec 17, 2015 at 23:28
• Whoops... That's definitely what I meant. That's interesting. Science fiction leap here - does that mean we could create artificial mass (and therefore gravity) by packing a very very very large number of photons into a single area? Dec 17, 2015 at 23:46
• According to Susskind and Damour , it seems that there is some upper limit ( a q-bit per Planck volume ??!!). Probably, it is why black holes grow in space ...
– user46925
Dec 18, 2015 at 2:57
• @igael Do you have a source link? Dec 18, 2015 at 7:10

It is impossible to construct the panels as you say. The main reason is that it would violate certain laws of thermodynamics. Technically there is no such thing as a 'one way mirror'. The proverbial 'one way mirror' is simply a partially mirrored sheet of glass (plexiglass, etc.). The reason it is called 'one way' is because the environmental light on one side is substantially less than on the other side, so people on the bright side see very little light coming from the dim side, but those on the dim side see substantial light coming from the bright side compared to the ambient light in their room. If you turn up the light on the dim side to that of the otherwise bright side, then people on each side will see the same amount of light through the 'mirror'. All of the above is the case for passive devices. In theory an active device (one that takes power from a separate source) could be created that truly is a one-way mirror, but it still could not violate any laws of physics and would have various limitations (maximum light output, etc.). The simplest of these would be a camera and monitor.

The simplest way to look at it is this: If you have a mirrored box, the only way to get light in there is to open it, which would cause part of it to not be mirrored. That is, you can't create a device that reflects all light AND emits more light when you add power to it.

• I guess I forgot to mention that my panels are made of an Unobtainium/Handwavium composite :) The point of this question isn't really "can we build the box", so much as it is, "can we fill up an area with light such that there is no room for anything else?" Dec 17, 2015 at 18:35
• If that's the question, then why not just imagine a super-powerful light source? If the violation of the laws of thermodynamics is besides the point, let's not complicate the issue unnecessarily by violating the laws of thermodynamics. Dec 17, 2015 at 18:48
• OK, let's say the question is reworded like this: You have completely reflective mirrors and you only very briefly open the box to shine a massive light pulse in there, so that the box is once again closed before any reflected light can make it back to the opening. In that case I would say that there is still a limitation in that when light reflects off a surface, the atoms are briefly in an excited state, and when they are excited the reflectivity will go down. In other words, if the light is strong enough then the mirror surface must store so much energy that it will alter the material. Dec 17, 2015 at 18:57
• But doesn't that go against the definition of a white body? All light must be reflected, so that would mean that it can't convert any of the photons to energy, correct? The question I'm trying to get at is this: If we keep stuffing more and more energy into one space, is there any limit? Does it eventually become like the singularity before the Big Bang? Dec 17, 2015 at 20:52
• Yes, it goes against the idea of a white body. I'm saying that there is a limit to the behavior of a white body (there is no theoretical white body). Specifically, it will change its behavior when the excited states on its surface become excessive. Dec 17, 2015 at 20:59