B Cosmological Red Shift in a Perfectly Reflecting Box

[J O Linton]

TL;DR Summary

Does a photon in a box undergo cosmological red shift over time?

When a photon travels from a distant galaxy to us it undergoes an increase in wavelength due to the expansion of the universe during the time of flight. On the other hand, physical objects such as atoms and galaxies do not undergo a similar expansion because they are bound together by elctromagnetic and other forces. My question is this: suppose you put one or more photons into a box which has 100% perfectly reflecting walls. Will the photon(s) in the box experience a cosmological red shift over time or not? If so - why? and if not -why not?

 

[phinds]

The box would be a bound system, so the speed at which it is traveling (and likewise its recession velocity) relative to another object would be irrelevant.

 

[J O Linton]

As usual I have not expressed myself clearly enough. The box is not moving. It is stationary with respect to the universe. I put photons with a wavelength of 600nm in the box and then look at them 10 billion years later when the scale factor of the universe is twice its current value. What will I expect their wavelength to be?

 

[Ibix]

if not -why not?

You would see redshift if the distance between the box walls was growing with time. It isn't, so you'd expect the radiation to always have the same wavelength.

 

[J O Linton]

But this begs the question - what mechanism prevents the photons bouncing around inside the box from being cosmologically red shifted during those periods of time when they are traveling from one side of the box to another while their friends outside the box are expanding?

 

[Ibix]

But this begs the question - what mechanism prevents the photons bouncing around inside the box from being cosmologically red shifted during those periods of time when they are traveling from one side of the box to another while their friends outside the box are expanding?

The thing to realize is that the matter that fills an FLRW universe is a fluid of uniform and decreasing density. Each little elementary bit of it moves on a particular family of geodesics. The walls of the box do not follow that family and are not changing density, because they are tied together. This is also true of other bound systems such as galaxies, which is why cosmological redshift only appears at really large scales. Light (and it's better not to talk about photons because they are horribly complex bits of quantum field theory that just introduce unnecessary complexity) that interacts with the box walls is therefore not following the same kind of path as free traveling light.

 

[J O Linton]

That's very interesting. I think what you are saying is that, if the box is small (i.e. smaller than a galaxy!) quantum theory requires that the whole system, box plus light, is considered and that you cannot simply isolate the two components and treat them seperately. In such a box there would be no increase in wavelength.

This raises two interesing questions. Firstly, how big does the box have to be to show an effect - after all, if the box is as big as the universe, then presumably, we are back to the standard cosmological red shift.

Secondly, does general relativity on its own have an answer to this question - i.e. on the basis of physics as conceived by Einstein in about 1920? In other words, is GR complete or is it necessary to invoke quantum ideas to explain the behaviour of the universe on a cosmological scale. If this is the case, does this not suggest a profound link between the two theories which needs exploring?

 

[Ibix]

I think what you are saying is that, if the box is small (i.e. smaller than a galaxy!) quantum theory requires that the whole system, box plus light, is considered and that you cannot simply isolate the two components and treat them seperately.

No. Quantum theory does not come into this. The point I was making is that you should not try to include quantum mechanical objects (especially photons) in this because they are very complicated and classical physics is more than sufficient.

In an ideal FLRW universe, the whole of space is uniformly filled with matter and every piece is expanding away from every other piece. In this idealised case, two particles arbitrarily close together at some time will move further and further apart. You've added a box which does not expand because it not just a dust cloud - it's a bound solid object. Light interacting with it behaves differently to any other light in this spacetime because it's interacting with a solid object that is not expanding.

The real universe is like an FLRW universe at very large scales, hundreds of millions of light years and more. But smaller than that, the universe is very inhomogeneous and is full of bound systems like galaxies. So if we establish radio stations at opposite edges of the galaxy and they just bounce radio signals back and forth for the rest of eternity this is the same as your mirrored box.

Firstly, how big does the box have to be to show an effect

It isn't the size of the box. It's that the box walls are bound together and not just clouds of dust.

does general relativity on its own have an answer to this question

No quantum theory is needed for this. The only place quantum gravity is expected to enter cosmology is at the very beginning when the universe was super dense and high energy.

 

[J O Linton]

Well I am relieved to hear you say that quantum theory is not needed in order to resolve this puzzle because I am strongly biased in favour of GR. If GR is a Bach fugue, then QT is a piece by Schoenberg!

But the puzzle remains. Why do electromagnetic waves traveling through empty space change their wavelength while electromagnetic waves bouncing round inside a box stay the same? I think what you are trying to say is that the 'particles of matter' which you say fill entire space expand outside the box but not inside it. I am not entirely happy with the idea that 'space is uniformly filled with matter'. I would perhaps prefer to go back to the old (discredited?) idea of the 'aether'. Outside the box the aether (which is essentially a coordinate system, not a real substance) expands but inside a bound system the 'aether' does not expand.

 

[PeroK]

But the puzzle remains. Why do electromagnetic waves traveling through empty space change their wavelength ...

They don't. Redshift is always due to the relationship between the source and receiver.

 

[jartsa]

Summary:: Does a photon in a box undergo cosmological red shift over time?

When a photon travels from a distant galaxy to us it undergoes an increase in wavelength due to the expansion of the universe during the time of flight. On the other hand, physical objects such as atoms and galaxies do not undergo a similar expansion because they are bound together by elctromagnetic and other forces. My question is this: suppose you put one or more photons into a box which has 100% perfectly reflecting walls. Will the photon(s) in the box experience a cosmological red shift over time or not? If so - why? and if not -why not?

So the problem is this:

A pulse of light travels from a distant galaxy to us, undergoing an increase in wavelength due to the expansion of the universe while it's traveling.

There's another pulse of light too, but there's a mirror 3 light years ahead of it. So this pulse of light is in a box.

Now it seems absurd to say that the latter pulse of light is not expanding, while the former pulse of light is expanding.

Right?

 

[Ibix]

Why do electromagnetic waves traveling through empty space change their wavelength

They don't. Wavelength is not a property of a wave alone - it also depends who is doing the measuring. If you stand in a corridor and I run along it and we are both illuminated by a laser we will measure it to have different wavelengths because we see each other's rulers as length contracted. In cosmology this comes in because each succesive galaxy a wave passes is moving away from the light source faster and faster. The same is not happening to the light in the box - it's just bouncing backwards and forwards in the box.

I think what you are trying to say is that the 'particles of matter' which you say fill entire space expand outside the box but not inside it.

The point is that "the expansion of space" is just good old Newton's first law, complicated by curved spacetime. A cloud of dust completely filling spacetime expands because that's what an everywhere uniform cloud of dust does in GR. Your box is not a cloud of dust - you've bound its atoms together so they do not move apart any more than those of a cereal box.

I am not entirely happy with the idea that 'space is uniformly filled with matter'.

In reality it isn't. It is, to a very good approximation, uniformly filled with matter on scales so large that individual galaxies are indistinguishable dots. That means that a model universe everywhere uniformly filled with matter is a useful tool, but it does not mean that there is uniform matter everywhere at all scales. Manifestly there isn't. Kick the air and then a table if you don't believe me, but don't blame me if you break a toe on the latter.

 

[PeroK]

A pulse of light travels from a distant galaxy to us, undergoing an increase in wavelength due to the expansion of the universe while it's traveling.expandong

That's the same misconception as the OP has. The light is not changing. We measure a different wavelength from the source, but that's due to the relationship between the source and receiver, separated by expanding space.

 

[Ibix]

That's the same misconception as the OP has.

I was about to comment something similar, but I decided @jartsa was paraphrasing the OP. If so, the post could do with something like "I think you are asking..." at the beginning to make it clear.

 

[jartsa]

I was about to comment something similar, but I decided @jartsa was paraphrasing the OP. If so, the post could do with something like "I think you are asking..." at the beginning to make it clear.

It seems to me that you guys have a hard time understanding the question. I tied to make the question more obvious.

Dust cloud filling expanding space expands. Does photon gas cloud filling expanding space expand? I mean is there extra expansion because of the expansion of space?

 

[PeroK]

It seems to me that you guys have a hard time understanding the question. I tied to make the question more obvious.

Dust cloud filling expanding space expands. Does photon gas filling expanding space expand?

Photons don't experience anything, so this thread is more about trying to correct the OP's misunderstanding of redshift. The question itself is easily answered once you realize it's all about the receiver relative to the source.

 

[Ibix]

It seems to me that you guys have a hard time understanding the question.

I understood the question. I'm not sure that the OP has completely grasped what he's asking about yet, though.

A direct answer to your variant is that, as already stated, light does not expand and wavelength is a property of the interaction between the wave and the detector not just the wave. So neither the light that strikes the mirror nor the light that bypasses it can be said to have expanded, contracted, or anything else. However, the mirror at 3ly is not moving away from the light source so will see the same wavelength that was emitted. But if Hubble's law applied so close to the source, the mirror would be doing a few centimeters per second compared to a Hubble-expanding object (a co-moving observer) so would measure the light to have a different wavelength to a sensor attached to the mirror.

Does photon gas filling expanding space expand?

As usual, don't talk about photons. But a radiation dominated universe certainly expands, yes.

 

[phinds]

As usual I have not expressed myself clearly enough. The box is not moving. It is stationary with respect to the universe.

There is no such thing. All motion is relative and the universe is not a "thing" that you can take as "stationary". The closest you can come to what you seem to want is a co-moving observer but that's really irrelevant and my statement stands.

 

[PeterDonis]

physical objects such as atoms and galaxies do not undergo a similar expansion because they are bound together by elctromagnetic and other forces.

Correct.

suppose you put one or more photons into a box which has 100% perfectly reflecting walls. Will the photon(s) in the box experience a cosmological red shift over time or not? If so - why? and if not -why not?

No, for the same reason that you gave above for physical objects. The box itself is such an object, and does not expand; therefore the light you have contained inside the box will not experience any redshift, since that would require the box to have expanded.

 

[PeterDonis]

The box is not moving. It is stationary with respect to the universe.

This part does not matter. You could give the box a proper motion relative to comoving objects ("comoving" is the correct way to say what you are calling "stationary with respect to the universe") and it would still be a bound object and the light inside it would still undergo no cosmological redshift.

 

[Vanadium 50]

It seems to me that you guys have a hard time understanding the question. I tied to make the question more obvious.

It seems to me that you're introducing things that are not relevant and that is making a mess.

@Ibix answered this correctly in #4. If I am next to a mirrored box and watch it for a billion years, the wavelength is the same. If I am watching it from a distant galaxy, as the host galaxy recedes, the light in the box is redshifted.

If the objection is "yeah, but what does the One True Observer see?", there is no such observer.

 

[PeterDonis]

what mechanism prevents the photons bouncing around inside the box from being cosmologically red shifted during those periods of time when they are traveling from one side of the box to another while their friends outside the box are expanding?

You have it backwards. You should be asking, what mechanism makes light moving freely in the universe undergo cosmological redshift? And the answer is, the mechanism has nothing to do with the light: it has to do with the fact that comoving observers are moving apart--the distance between them grows with the scale factor of the universe. And cosmological redshift is the redshift observed by comoving observers, for light that travels between them.

In other words, the "wavelength" of the light is not an intrinsic property of the light to begin with; it's a property of the relationship in spacetime between the light, the emitter, and the receiver. "Cosmological redshift" describes the outcome a particular kind of relationship in spacetime between those things; your scenario of "light trapped inside a box" describes a different such relationship, and therefore has a different outcome (no redshift).

 

[PeterDonis]

Dust cloud filling expanding space expands. Does photon gas cloud filling expanding space expand?

You have it backwards. The dust cloud or the photon gas doesn't "fill expanding space" and expand because of that; the "expanding space" and the expanding dust cloud or photon gas are aspects of the same thing: the spacetime geometry and how it is determined by the distribution of matter and energy.

The photon gas cloud spacetime geometry is not the same as the dust cloud spacetime geometry, because the photon gas is relativistic (it has pressure equal to 1/3 its energy density) while the dust cloud is not (it has zero pressure), and pressure contributes to the stress-energy tensor. But in both cases the spacetime geometry and the distribution of stress-energy are linked by the Einstein Field Equation; neither one is determined in isolation.

 

[J O Linton]

I think Perok has put his finger on it when he said about wavelengths expanding

They don't. Redshift is always due to the relationship between the source and receiver.

That is exactly right. I think I see now. The wavelengths don't expand any more than objects traveling at near the speed of light physically contract. We are looking at the same object but from a different perspective. The truth is that, even though the galaxy which emits the light and the receiver on Earth are not moving with respect to each other (they are both 'comoving' observers) they do not share the same inertial frame. In the galaxies frame when the light was emitted it had a wavelength of 600 nm but in the Earth frame, 10 billion years later when the universe has expanded by a factor of two, the same wave appears to have a wavelength of 1200 nm.

On the other hand, the waves in the box stay in the same inertial frame the whole time because they are confined within a bound system. Their wavelengths therefore do not change.

Please correct me if this is incorrect.

 

[Vanadium 50]

The wavelengths don't expand any more than objects traveling at near the speed of light physically contract.

They do physically contract.

 

[phinds]

The truth is that, even though the galaxy which emits the light and the receiver on Earth are not moving with respect to each other (they are both 'comoving' observers)

The Earth is NOT a co-moving observer. The other galaxy might or might not be.

 

[PeterDonis]

The wavelengths don't expand any more than objects traveling at near the speed of light physically contract.

No. As @Vanadium 50 has pointed out (and as John Bell stressed in his paper "How to Teach Special Relativity", in which he introduced the Bell Spaceship Paradox), length contraction is real--it's not just a matter of "perspective". If you arrange your experiment right, a length contracted object can fit through an opening that it would not fit through if it were at rest relative to your apparatus. That means length contraction is a real, physical phenomenon.

Similarly, the difference between the "cosmological redshift" case and the "light trapped in a box" case is a real, physical difference, not just a matter of perspective. See further comments below.

The truth is that, even though the galaxy which emits the light and the receiver on Earth are not moving with respect to each other (they are both 'comoving' observers)

Wrong. The emitting galaxy and Earth are moving with respect to each other. The fact that they are called "comoving" observers does not change this; "comoving" is a technical term that means both observers see the universe as homogeneous and isotropic (actually, here on Earth don't exactly see that, since we observe a dipole anisotropy in the CMB that we have to remove before analyzing the data in order to draw conclusions about "comoving" observers in our vicinity, but we'll ignore that technical complication here).

they do not share the same inertial frame. In the galaxies frame when the light was emitted it had a wavelength of 600 nm but in the Earth frame, 10 billion years later when the universe has expanded by a factor of two, the same wave appears to have a wavelength of 1200 nm.

This difference has nothing to do with frames; you can do the computation of the emitted and received wavelengths in any frame you like and you will get the same answer. The difference in wavelengths is due, as I said before, to the relationship in spacetime between the emitter, the light, and the receiver. That relationship is invariant, independent of any choice of frame.

On the other hand, the waves in the box stay in the same inertial frame the whole time because they are confined within a bound system. Their wavelengths therefore do not change.

Again, this has nothing to do with frames; you can calculate the wavelength of the light using the global "comoving" frame for the universe and you will get the same answer. The wavelength being unchanging is, again, due to the relationship in spacetime between the light and the walls of the box (which play the role of "emitter" and "receiver" in this scenario), and that relationship is invariant, independent of any choice of frame.

 

[pervect]

They do physically contract.

I think that depends on what you mean by "physically contract". I would say that an object physically contracts if its proper length changes. Since the proper length is coordinate independent, velocity won't change it. So the only way to have a fast moving "physically contract" is to use a different definition of what that term means, i.e. it's a matter of semantics.

 

[PeterDonis]

I think that depends on what you mean by "physically contract". I would say that an object physically contracts if its proper length changes.

Yes, you can say this, but then you have to acknowledge that, in some experiments (for example, the scenario I described in post #27, where a fast moving object fits through an opening that it would not fit through if it were at rest relative to the apparatus), the proper length of the object is not the relevant length for predicting what will happen. Which in turn means that what meaning you want to give to a phrase like "physically contract" might depend on what experiments you plan to run.

 

[Sagittarius A-Star]

They don't. Redshift is always due to the relationship between the source and receiver.

Does this mean, that the following text and the animation above it are wrong?

Cosmological Redshift: the wavelength of the emitted radiation is lengthened due to the expansion of the Universe. In this animation, the galaxy on the left was formed a long time ago, while the galaxy on the right was formed more recently. Although each galaxy emits the same wavelength of the light, the light from the left hand galaxy has spent longer traveling through the expanding Universe, and has therefore experienced a greater ‘stretching’ (redshift). Astronomers are able to determine how far away distant objects are by measuring this wavelength expansion.

Source:
https://astronomy.swin.edu.au/cosmos/c/cosmological+redshift

 

[Ibix]

Does this mean, that the following text and the animation above it are wrong?

Source:
https://astronomy.swin.edu.au/cosmos/c/cosmological+redshift

The result is correct - nearer galaxies will be less redshifted. The explanation is less helpful. It's a very coordinate dependent explanation, and one can pick different coordinates and say that the emitted light was always redshifted.

The most generally correct statement is that the relationship between the light and each co-moving observer that it passes is slightly different than the relationship with the previous co-moving observer. This isn't really because the light is changing, but because co-moving observers' worldlines aren't parallel.

 

[PeroK]

Does this mean, that the following text and the animation above it are wrong?

Source:
https://astronomy.swin.edu.au/cosmos/c/cosmological+redshift

I learned from the master himself, @Orodruin, on this thread several years ago:

https://www.physicsforums.com/threads/cosmological-redshift.935943/

If we talk about light experiencing change as it travels, then explaining the misconception in the current thread is difficult.

 

[PAllen]

Yes, you can say this, but then you have to acknowledge that, in some experiments (for example, the scenario I described in post #27, where a fast moving object fits through an opening that it would not fit through if it were at rest relative to the apparatus), the proper length of the object is not the relevant length for predicting what will happen. Which in turn means that what meaning you want to give to a phrase like "physically contract" might depend on what experiments you plan to run.

I like the analogy that to fit a 20 foot long 2 by 4 through an ordinary door, all you need to do is rotate it. The fact that it will go through one way and not the other is obviously a physical phenomenon, which does not imply any change to the 2 by 4. Length contraction is exactly a rotation in spacetime relative to an opening, rather than in space, which achieves the same result. It is a real phenomenon, but just as with spatial rotation, does not imply any change to the object.

 

[PAllen]

I like the analogy that to fit a 20 foot long 2 by 4 through an ordinary door, all you need to do is rotate it. The fact that it will go through one way and not the other is obviously a physical phenomenon, which does not imply any change to the 2 by 4. Length contraction is exactly a rotation in spacetime relative to an opening, rather than in space, which achieves the same result. It is a real phenomenon, but just as with spatial rotation, does not imply any change to the object.

To expand on this, consider two different scenarios.

A rod is moving rapidly relative to a hole in a surface. The rod is parallel to the surface in the hole frame, but moving at a slight angle to the surface in this frame,such that it passes through the hole. No proper acceleration is involved. The proper length of the rod is bigger than the proper diameter of the hole. Then, in the rod frame, the hole and surface are not quite parallel to the rod, and are moving at an angle relative to the rod, and thus the hole passes around the rod.

Now consider the case of the rod moving parallel to the surface in the hole frame, as well as the rod itself being parallel in this frame. Then proper aceleration is required to get it through the hole. If the proper length of the rod is bigger than the proper diameter of the hole, then, in the initial rod frame (before proper acceleration), the rod is deformed to fit through the hole. Computation of the shear and expansion tensor show that this rod frame description captures the invariant physics. The rod fit through due to shear deformation rather than simply length contraction. In fact, for this case, for the rod to fit through with born rigidity, the proper diameter of the hole must be sufficiently large for its contracted width, in the initial rod frame, to be larger than the proper length of the rod.

 

[Orodruin]

I learned from the master himself, @Orodruin, on this thread several years ago:

https://www.physicsforums.com/threads/cosmological-redshift.935943/

If we talk about light experiencing change as it travels, then explaining the misconception in the current thread is difficult.

Assuming that the box is of fixed size and not expanding with the universe, the sides of the box will not be comoving. If you are then looking at everything relative to comoving coordinates, then the reflections on the sides (which are moving towards the light that is to be reflected) will upscatter the frequency of the light precisely by the amount necessary for the light to keep its frequency over time.

If the box does expand with the universe such that the sides are comoving, then the light will redshift over time.

 

[Ibix]

If the box does expand with the universe such that the sides are comoving, then the light will redshift over time.

It's probably worth drawing out the distinction between the construction of a box that does and does not expand with the universe. One that does not expand is an ordinary box like a shoebox. One that does expand is one whose ends must be disconnected from one another - a "practical" implementation would be a mirror here and another in a galaxy 100Mpc away. If the ends are connected then the material (more precisely, the electromagnetic forces between the atoms of the material) stops the box from expanding.

 

[Orodruin]

It is worth underlining that a light pulse does not have anything that can intrinsically be called its wavelength or frequency. Wavelength and frequency are only definable in relation to an observer. In the case of what is called cosmological redshift, the implicit understanding is that the family observers that is being considered is a family of comoving observers. In a local inertial frame, such observers are moving apart according to a Hubble law and the "cosmological redshift" in that coordinate system is explained through standard Doppler shift. Indeed, the observers close to the sides of the (non-expanding) box are moving away from the observer at the center of the box and eventually pass out of the box and according to those observers, the light redshifts (and then blueshifts when hitting the mirrors). However, for observers at rest in the local inertial frame, the light does not redshift and maintains its frequency during the reflections (because the mirrors are stationary). All in all, any physical prediction, such as the frequency of the light relative to an observer at the center of the box each time it passes, will be invariant and independent of the description we adopt.

I discussed this further in an advanced level Insight some years ago: https://www.physicsforums.com/insights/coordinate-dependent-statements-expanding-universe/

 

[jartsa]

Assuming that the box is of fixed size and not expanding with the universe, the sides of the box will not be comoving. If you are then looking at everything relative to comoving coordinates, then the reflections on the sides (which are moving towards the light that is to be reflected) will upscatter the frequency of the light precisely by the amount necessary for the light to keep its frequency over time.

So between the reflections the light pulse in the box expands. Right after reflection the pulse fits in a small normal box glued on the big box. Some time later it does not fit in another identical normal box. 'Normal' means "does not expand with the universe".

 

[jartsa]

If I am next to a mirrored box and watch it for a billion years, the wavelength is the same. If I am watching it from a distant galaxy, as the host galaxy recedes, the light in the box is redshifted.

Light in the box is redshifted?? According to a distant observer? How does the distant observer observe that? Some light leaks out of the box and travels to the observer? In that case the observer does receive redsifted light. But that does not prove that the light in the box is redshifted.

 

[Ibix]

So between the reflections the light pulse in the box expands

No. There is no invariant sense in which it expands, contracts, or does anything else. Observers attached to the box walls will say the pulse just bounces backwards and forwards, no red or blue shift. Co-moving observers floating inside the box would say that the pulse red shifts as it travels and is blue shifted by interacting with the mirror for zero total change. Box observers would say that each successive co-moving observer is moving a bit with respect to their predecessor, so the increasing red shift is increasing kinematic Doppler from the different states of motion of the co-moving observers.

Some time later it does not fit in another identical normal box.

It will always fit inside identically constructed small boxes.

Some light leaks out of the box and travels to the observer?

Yes - that's the only way it can become red shifted.

 

[Vanadium 50]

Light in the box is redshifted?? According to a distant observer? How does the distant observer observe that? Some light leaks out of the box and travels to the observer? In that case the observer does receive redsifted light. But that does not prove that the light in the box is redshifted.

That's your objection? THAT'S your objection? That once the light is observed by someone outside of the box the light is no longer in the box? That's philosophical navel-gazing.

Possibilities:

1. The light is really, really bright and the boxes are a little transparent.
2. There is a gizmo where an outside light source is made to match the wavelength of the light in the box, as verified by a local observer.
3. You have billions of boxes and once a year a box opens sending its light to earth.

 

[timmdeeg]

@jartsa My wording will be a bit different from that of the experts around here, being not one of those.

Let's compare two boxes, box A, the not expanding box, is glued so that the proper distance between the mirrors is constant over time. Box B, the expanding box is not glued, so that the proper distance between the mirrors increases according to to the expansion of the universe (more precisely proportional to the increasing scale factor).

Now imagine the two boxes side by side. After the light pulses has been emitted at one side (let's call this side of the box "pulse emitting side") we measure the frequency shift at the opposite mirrors.
The opposite mirror in box B moves away from the "pulse emitting side" while the light is traveling and we measure a redshift at this mirror corresponding to the expansion of the universe. This mirror "moves away" because it is comoving (from it's view the universe is homogeneous and isotropic).
The opposite mirror in box A can't move away from the "pulse emitting side" , so this mirror moves relative to the opposite mirror of box B towards the "pulse emitting side". Now imagine a comoving observer just before passing by. From the opposite mirror's (still box A) view this observer is blueshifted by the same amount of frequency shift as this observer sees the emitting side redshifted. Putting this together the emitting side of the box A is seen with zero frequency shift from the view of the opposite side of the box A as it should be because this box is not expanding.

Note, this presupposes that homogeneity and isotropy of the universe holds at all scales.
For simplicity it is assumed that the "emitting side" of both boxes is comoving.

It's my trial, dear experts please correct where necessary.

 

[Ibix]

Box B, the expanding box is not glued, so that the proper distance between the mirrors increases according to to the expansion of the universe

Note that it isn't necessary for the free-floating mirrors to co-move. It's perfectly possible to have them stay next to the ends of the solid box, and depending on how they were constructed and set up they might do one or the other. So you are imposing a condition on the initial state of motion of the mirrors here - fine, but should be stated explicitly.

 

[timmdeeg]

Note that it isn't necessary for the free-floating mirrors to co-move.

But I talk about a box which expands with the universe. Which other choice is there?

 

[Ibix]

But I talk about a box which expands with the universe. Which other choice is there?

Since there's a lot of confusion about what the expansion of the universe is and is not, I was just making explicit that free-floating things don't necessarily co-move. You would need to very carefully set up the situation, in general. It doesn't "just happen".

 

[timmdeeg]

Since there's a lot of confusion about what the expansion of the universe is and is not, I was just making explicit that free-floating things don't necessarily co-move.

Sorry, I don't get your point. I haven't been talking about "free-floating things" but instead about comoving mirrors. The proper distance between those increases proportional to the scale factor. I can't see any confusion here.

 

[Ibix]

Sorry, I don't get your point.

You wrote in #42 Box B, the expanding box is not glued, so that the proper distance between the mirrors increases according to to the expansion of the universe. That can be read as implying a causal relationship: the mirrors are not stuck together therefore they co-move. That is apparently not what you meant (which is good, because it's wrong) but it's one way to read what you wrote. I thought it important to clarify that free floating things can be co-moving but aren't necessarily co-moving. Extended objects that are glued together, on the other hand, cannot be co-moving except at at most one point along their length.

 

[PeterDonis]

It's perfectly possible to have them stay next to the ends of the solid box, and depending on how they were constructed and set up they might do one or the other.

If the mirrors are free-floating, the one thing they cannot do is stay next to the ends of the solid box, since by hypothesis the solid box is held together by internal forces and the two mirrors at its ends are not in free fall.

 

[Ibix]

If the mirrors are free-floating, the one thing they cannot do is stay next to the ends of the solid box,

Aargh! Yes, you're right. I've been thinking about Milne's cosmology recently where this is possible, but not in a general FLRW universe.

 

[timmdeeg]

You wrote in #42 Box B, the expanding box is not glued, so that the proper distance between the mirrors increases according to to the expansion of the universe. That can be read as implying a causal relationship: the mirrors are not stuck together therefore they co-move.

Ok, I see your point. Yes there is no causality. I was anticipating that saying "so that the proper distance between the mirrors increases according to to the expansion of the universe" clarifies that only comoving could be meant.