Photon in a mirrored box moving in direction of travel

[Herbascious J]

TL;DR Summary

How does a photon's behavior change for different observers watching a photon moving in the direction of travel in a mirrored box?

Imagine a special box made of inward facing mirrors. These mirrors have zero mass and are perfectly reflective. A single photon is inside the box bouncing from side to side between the two mirrors of the sides of the box. The photon is perfectly preserved in this state, and loses no energy. Putting aside the impossible nature of this scenario as much as possible, how does the behavior of the photon change for observers at rest with the box compared to an observer moving in the direction of travel of the photon? Meaning, the observer passes by the box in a side to side direction parallel with the motion of the perfectly trapped photon. For the purposes of this experiment the box can be considered a perfectly stable item which returns the full momentum to the photon as it is reflected if that is important.

I am having trouble with this problem because it seems like time is moving differently at either end of the box and I don't understand how the frequency is being observed by the moving observer, and what happens when the photon is reflected in the opposite direction. This photon is a precise color for the at-rest observer and I am specifically interested in how the color changes in both directions as seen by the moving observer, how the energy of the photon is different, as well as it's momentum and how time flows through out the experiment.

Currently, I'm under the impression that the photon moves slowly through time when traveling in the same direction as the box relative to the observer and that its wavelength is stretched out. However, when the photon reflects back in the opposite direction I don't have a clear understanding of how things appear to the moving observer.

 

[Dale]

Are you wanting to discuss an actual photon in quantum electrodynamics, or are you actually interested in a classical pulse of light?

TL;DR Summary: How does a photon's behavior change for different observers watching a photon moving in the direction of travel in a mirrored box?

This photon is a precise color

My QED is not strong, but wavefunction in a momentum eigenstate probably isn’t in a number eigenstate.

 

[PeterDonis]

Are you wanting to discuss an actual photon in quantum electrodynamics, or are you actually interested in a classical pulse of light?

This being the relativity forum, I would hope that the OP intended the latter.

My QED is not strong, but wavefunction in a momentum eigenstate probably isn’t in a number eigenstate.

It's worse than that: the term "photon" in QED, if it's going to mean anything more than handwaving, would have to refer to a specific Fock state, and Fock states are very hard to produce. A typical light source like a laser produces, at best, a coherent state, which is not an eigenstate of either momentum or photon number, and for which the term "photon" is not appropriate.

IMO the OP would be much better off with the classical light pulse, which is perfectly sufficient for discussing the scenario in question.

 

[Orodruin]

As for the scenario in question, it is a non-starter. The OP wants a massless box so the box will move at the speed of light. Thus there is no rest frame of the box. Perhaps OP would like to reformulate?

 

[Herbascious J]

As for the scenario in question, it is a non-starter. The OP wants a massless box so the box will move at the speed of light. Thus there is no rest frame of the box. Perhaps OP would like to reformulate?

Yes, this is difficult for me. It's extremely theoretical and not practical, so I'm prepared for it to be unusable. The basic thrust behind the question is to frame everything in terms described by Special Relativity. So, I'm really asking about light, traveling as an EM wave, with a precise wavelength and frequency, but of course c must be constant for all observers. I'm not thinking about QM intentionally, and the light doesn't need to be a quanta.

I do want to point out one thing about the box. This box is only a mental construction to reverse the direction of the photon. I guess there is no reason to give it a zero mass, and in fact, I made an error there because I am confusing the conditions with a different discussion. My apologies. The box may have normal mass, it need only sustain a continuing wave of light bouncing back and forth. The light may be simply a beam of a specific wavelength and color. Please let me know if I should re-post to clean things up. Thank you.

 

[Sagittarius A-Star]

As for the scenario in question, it is a non-starter. The OP wants a massless box so the box will move at the speed of light. Thus there is no rest frame of the box. Perhaps OP would like to reformulate?

I think it depends on what the OP understands under "photon" between mirrors. If he means a standing EM-wave, then this wave has energy in the box's (including wave) center-of-momentum frame, that means it has mass.

 

[PeterDonis]

it seems like time is moving differently at either end of the box

Why would it seem like that? In whatever frame you choose, both ends of the box have the same velocity.

 

[PeterDonis]

Currently, I'm under the impression that the photon moves slowly through time when traveling in the same direction as the box relative to the observer

What does "moves slowly through time" mean? Remember that a light pulse moves on a null worldline, not a timelike worldline.

and that its wavelength is stretched out.

The light pulse will be Doppler redshifted (longer wavelength, lower frequency) in this case, yes. At least, if we assume that the light pulse is ahead of the observer.

when the photon reflects back in the opposite direction I don't have a clear understanding of how things appear to the moving observer.

The light pulse is Doppler blueshifted (shorter wavelength, higher frequency) in this case--again, assuming that the light pulse is ahead of the observer.

 

[Herbascious J]

A place where I'm hung up is how the light seems to move at different velocities relative to the box as seen by a moving observer. If there is a small man inside the box, he just sees the light bounce back and fourth at the same speed, but if I zip past the box looking at it out of my side window so-to-speak, so that the light is traveling parallel next to me, and then in the opposite direction of me as I pass by, it seems like it takes longer to get to the head of the box from my perspective, before it turns around and then moves very quickly to get to the back of the box, because the back of the box is rushing up to the backward moving light beam at that point. I'm not sure how to reconcile this. I know the light takes the same amount of time for the man in the box regardless of direction. I've seen descriptions that say that the back of the box will be slightly ahead in time from the front of the box relative to the moving observer and I think that is the solution somehow.

 

[Orodruin]

So what you want is, in fact, not a massless box, but a box with infinite mass. If not, the box will change velocity when the light is reflected as this would be necessary to ensure energy and momentum conservation. Letting the box mass go to infinity makes the velocity change of the box go to zero.

In the rest frame of the box, the light bounces back and forth, keeping its wavelength the same in both directions.

In a frame moving relative to the box the light Doppler shifts. It reddens when moving in the same direction as the frame and blueshifts when moving in the opposite direction.

 

[Orodruin]

I think it depends on what the OP understands under "photon" between mirrors. If he means a standing EM-wave, then this wave has energy in the box's (including wave) center-of-momentum frame, that means it has mass.

It is mot about the light, it is about the container.

A standing wave is also typically not what people at this level mean when they say “photon”. They are typically imagining a little ball of light.

 

[Nugatory]

A place where I'm hung up is how the light seems to move at different velocities relative to the box as seen by a moving observer.

It does.
However, the light is always moving at speed $c$ in both frames (the one in which the box is at rest and the one in which the outside observer is at rest and the box is moving) and that is what’s required by relativity. To see this we can use the relativistic velocity addition rule $$w=\frac{u+v}{1+\frac{uv}{c^2}}$$Here $u$ is the velocity of something (in this case the flash of light, so $\pm c$ according to whether it’s going left or right) using the frame in which the box is at rest, $v$ is the speed of the box in some other frame, and $w$ is the speed of that something in that frame. Do the math, you’ll see that the speed of the flash is $c$ in both frames.

 

[Ibix]

The answer to understanding SR is always Minkowski diagrams.

Here is a Minkowski diagram of a pair of stationary mirrors, marked in blue, with a short light pulse (orange) bouncing between them. The mirrors are unaffected by the light pulse bouncing, implying that this is something like a mirrored box with ends that are joined by springs with a relatively long time constant (i.e., normal matter, with sound speeds $\ll c$).

There are three orange lines which are meant to represent the paths of three successive wave crests in our light pulse. At the bottom of the diagram the pulse is travelling to the right and bounces almost immediately to the left. The middle of the pulse passes through the origin for our drawing convenience. I've also added three fine grey lines, two vertical and one horizontal. The vertical ones mark the position of the front and rear wave crests at time zero, and the horizontal one marks the time when the center wave crest passes through $x=0$ again, and you can see that the front and rear wave crests pass through the grey line intersections - so the wave has the same wavelength going to the left as it does to the right.

Now let's say you're walking past this at a casual 0.2c. The diagram in this frame looks like this:

This time the blue lines are slanted showing that the mirrors are moving in this frame. I haven't transformed the grey lines - these are new ones, but again they mark the length of the right-going pulse and the return-to-origin time of the left-going pulse. You can see from this that the pulse is shortened when it's travelling to the left. Here's a zoomed in shot of the region:

You can clearly see that the grey lines (length of the right-going pulse) mark out a longer distance than the left-going pulse. So you can see that the pulse is blue shifted compared to the right-going one.

You can also look at the second diagram and see why the left-going pulse takes longer to travel than the right-going one. The left-going pulse is travelling in the same direction as the mirror it will bounce off, so when it gets to where the mirror was the mirror has moved away, so it takes more time than just length-of-the-box-over $c$. The right-going pulse is travelling in the opposite direction to the mirror it will bounce off, so the mirror covers some of the distance and it takes less time for the light to reach the end than just length-of-the-box-over $c$. The speed of the pulse is still $c$, however.

Hope that helps.

 

[Dale]

it seems like it takes longer to get to the head of the box from my perspective, before it turns around and then moves very quickly to get to the back of the box

Yes. That is correct. The second postulate says that light travels at $c$ in all inertial frames. It does not say anything about the time of bounces in a box.

I've seen descriptions that say that the back of the box will be slightly ahead in time from the front of the box relative to the moving observer and I think that is the solution somehow

Yes. This is the relativity of simultaneity. It is indeed the key to understanding most paradoxes in SR.

Here, suppose the observer in the box has two clocks, one on each end of the box. Then, to the outside observer, the clock that is positioned ahead will be offset from the clock on the back. So according to the outside observer the time is uneven, but the inside observer’s clocks are desynchronized so that they erroneously measure the time as being even

 

[Mister T]

Isn't this just the light clock in the longitudinal orientation? I think there are thorough analyses of that. Do a google search for "horizontal light clock".

 

[Orodruin]

Isn't this just the light clock in the longitudinal orientation? I think there are thorough analyses of that. Do a google search for "horizontal light clock".

Yes. Many times used to derive length contraction.

 

[PeterDonis]

the light seems to move at different velocities relative to the box as seen by a moving observer

The light moves at different velocity relative to the box in any frame when it is traveling in one direction vs. the other. Why is this a problem?

If there is a small man inside the box, he just sees the light bounce back and fourth at the same speed, but if I zip past the box looking at it out of my side window so-to-speak, so that the light is traveling parallel next to me, and then in the opposite direction of me as I pass by, it seems like it takes longer to get to the head of the box from my perspective, before it turns around and then moves very quickly to get to the back of the box, because the back of the box is rushing up to the backward moving light beam at that point. I'm not sure how to reconcile this.

You just did reconcile it in your description quoted above. Again, why is this a problem? It's exactly what you would expect in a frame in which the box is moving.

I've seen descriptions that say that the back of the box will be slightly ahead in time from the front of the box relative to the moving observer and I think that is the solution somehow.

Relativity of simulaneity, which is what you are referring to mere, is the "solution" to why the light can still have the same speed relative to you when the box is moving, while still having the rest of what you say being true. But it is the constancy of the speed of light relative to you that is a change in SR vs. Galilean relativity (the kind of relativity that Newtonian physics obeys). The rest of what you say would be just as true in a Newtonian universe.

 

[PeterDonis]

what you want is, in fact, not a massless box, but a box with infinite mass. If not, the box will change velocity when the light is reflected as this would be necessary to ensure energy and momentum conservation. Letting the box mass go to infinity makes the velocity change of the box go to zero.

But it doesn't make the velocity change of the light go to zero. The light still changes direction every time it bounces, and that changes its velocity relative to the box, and to anything else that is moving inertially, such as the observer in the OP's scenario. The velocity change of the light was what seemed to be confusing the OP, but it shouldn't be confusing at all, it's got to happen whenever the light bounces.

 

[Herbascious J]

But it doesn't make the velocity change of the light go to zero. The light still changes direction every time it bounces, and that changes its velocity relative to the box, and to anything else that is moving inertially, such as the observer in the OP's scenario. The velocity change of the light was what seemed to be confusing the OP, but it shouldn't be confusing at all, it's got to happen whenever the light bounces.

Ok, thank you, this is right where I was having trouble. Because I watch the man in the box see the light move the same speed in both directions, I believe the same amount of time passes for him as the light travels the length of the box in both directions. But when I watch the light bounce back and fourth, it takes longer in one direction, that it does in the other. This makes it seem like the man in the box is experiencing time at two different rates from the perspective of the moving observer and that time slows down in side the box and then speeds up (again this is from the perspective of the moving observer). And so my question was aiming at how can this be, and yes I can see now that having off set clocks at either end could explain this.

 

[Orodruin]

I believe the same amount of time passes for him as the light travels the length of the box in both directions

He does indeed.

But when I watch the light bounce back and fourth, it takes longer in one direction, that it does in the other.

Indeed.

This makes it seem like the man in the box is experiencing time at two different rates from the perspective of the moving observer and that time slows down in side the box and then speeds up (again this is from the perspective of the moving observer).

Not really. What you are seeing here are the effects of the relativity of simultaneity. The man in the box’s clocks do not seem to run at different rates to you, but the leading clock seems to be lagging behind. Same rates, but offset in time. This fully accounts for the effect you describe.

 

[Tomas Vencl]

Now let's say you're walking past this at a casual 0.2c. The diagram in this frame looks like this:
……..

Just a note, the blue lines representing the mirrors are closer together (for example, at the intersection with the x-axis) due to length contraction. Of course, at 0.2c, this will be negligible.

 

[Herbascious J]

Just a note, the blue lines representing the mirrors are closer together (for example, at the intersection with the x-axis) due to length contraction. Of course, at 0.2c, this will be negligible.

Something caught my attention here; the whole box does contract length-wise when moving. Does that mean that the light's wavelength also contracts over all? Does this mean that the light has more momentum as measured by the moving observer? I'm wondering if this is similar to the idea that mass increases as speed increases in SR.

 

[PeterDonis]

the whole box does contract length-wise when moving.

Yes.

Does that mean that the light's wavelength also contracts over all?

No. The light's wavelength behaves as I described in a previous post, according to the relativistic Doppler shift, which changes depending on which way the light is moving.

Does this mean that the light has more momentum as measured by the moving observer?

The light's momentum and energy change by the same Doppler factor as its wavelength and frequency.

I'm wondering if this is similar to the idea that mass increases as speed increases in SR.

No. "Relativistic mass" is an outdated concept. Relativistic Doppler is not.

 

[Ibix]

Does that mean that the light's wavelength also contracts over all?

What do you mean by "contracts over all"? Do you mean the time average of the wavelength or something? I'm sure you can figure it out if you want to.

Does this mean that the light has more momentum as measured by the moving observer?

Its average momentum will increase, yes, although the momentum of each leg of its motion will be alternately higher and lower than measured by the stationary observer. There are uses for the average momentum, but I don't really see the point here.

I'm wondering if this is similar to the idea that mass increases as speed increases in SR.

"Mass increases" relies on the concept of relativistic mass, which fell out of use due to the massive confusion it causes. "Mass" is taken to mean "rest mass" these days, which does not change with speed. You never need to use relativistic mass (it's just total energy divided by $c^2$), but if you do use it you should always include the word "relativistic".

"Total energy increases with speed" and "momentum increases with speed" are uncontroversial positions, yes.

 

[Herbascious J]

What do you mean by "contracts over all"? Do you mean the time average of the wavelength or something? I'm sure you can figure it out if you want to.

Its average momentum will increase, yes, although the momentum of each leg of its motion will be alternately higher and lower than measured by the stationary observer. There are uses for the average momentum, but I don't really see the point here.

"Mass increases" relies on the concept of relativistic mass, which fell out of use due to the massive confusion it causes. "Mass" is taken to mean "rest mass" these days, which does not change with speed. You never need to use relativistic mass (it's just total energy divided by $c^2$), but if you do use it you should always include the word "relativistic".

"Total energy increases with speed" and "momentum increases with speed" are uncontroversial positions, yes.

"Momentum increases with speed" So the moving observer sees a momentum increase in the light beam overall compared to the at rest observer? Of course the box's relativistic mass will increase, but I'm more interested in the light beam alone. That is an interesting effect I didn't know about if true.

EDIT: I should be clear, I am trying to see if the energy of the light alone, is acting like a stationary mass energy, if when accelerated and if the total energy goes up like in Special Relativity.

 

[Ibix]

So the moving observer sees a momentum increase in the light beam overall compared to the at rest observer?

On average. The beams in opposite directions have different momenta - one higher and one lower than in the box frame.

Of course the box's relativistic mass will increase

Don't use relativistic mass. You are just making trouble for yourself down the line.

I'm more interested in the light beam alone

On average, the energy increases. The beams in opposite directions have different energy - one higher and one lower than in the box frame.

I am trying to see if the energy of the light alone, is acting like a stationary mass energy,

No. That's one of the many reasons why we decided using relativistic mass was a bad idea.

 

[jbriggs444]

energy of the light alone, is acting like a stationary mass energy

An individual light pulse has zero mass but non-zero energy and non-zero momentum.

If you contemplate a continuous and uniform bouncing beam rather than a bouncing pulse then the beam as a whole has constant, non-zero invariant mass. Its has a center of momentum that is moving at a sub-light pace. This despite the forward beam and the return beam both being individually massless.

If you contemplate a bouncing pulse in conjunction with a box of finite mass then the two taken as a whole have a non-zero total mass and a center of momentum that is moving at a sub-light pace. The invariant mass of the box plus [massless] contents is greater than the invariant mass of the box alone.

Invariant mass is not an additive quantity in special relativity. The mass of the whole is not always equal to the sum of the masses of the parts.