Cosmological Red Shift in a Perfectly Reflecting Box

[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.

 

[timmdeeg]

If one goes into details one should mention that if the mirrors initially are kept in a fixed proper distance and then are freed they would be free-floating but to become comoving would take a long time. So to have them comoving initially requires to adjust them accordingly.

 

[PAllen]

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.

Specifically, initially free floating mirrors that start with exactly 0 mutual spectral shift for a back and forth signal, will gain redshift and move apart over time if the second derivative of the scale factor is positive; the reverse if it is negative. (This idealizes that somehow local matter is irrelevant, and the mirrors are responding only to the the global space time geometry). Note, this means that if the scale factor is decreasing, but the decrease is decelerating, the mirrors will move apart rather than converge, because the second derivative of the scale factor will be positive. This is distinct from cosmologically comoving mirrors, because the initial conditions (their state of motion) give them a starting spectral shift, such that they follow the scale factor over time.

 

[PAllen]

If one goes into details one should mention that if the mirrors initially are kept in a fixed proper distance and then are freed they would be free-floating but to become comoving would take a long time. So to have them comoving initially requires to adjust them accordingly.

Actually, they would never become comoving. At any point, there are an infinite number of inertial trajectories, exactly one of which is cosmologically comoving. Thus, a geodesic that is not-comoving at one event will never become comoving. Otherwise there must exist a point with two comoving geodesics, which is impossible.

 

[timmdeeg]

Actually, they would never become comoving.

So I have been misinterpreting the statement "As shown in Fig. 3, the untethered galaxy asymptotically joins the Hubble flow in each cosmological model that expands forever." in https://arxiv.org/pdf/astro-ph/0104349.pdf
because "asymptotically" means "never".
In Fig. 3 the authors write: "In the accelerating universe (ΩM, ΩΛ) = (0.3, 0.7), the perturbed galaxy joins the Hubble flow more quickly than in the decelerating universes ... ".

Thanks.

 

[PAllen]

So I have been misinterpreting the statement "As shown in Fig. 3, the untethered galaxy asymptotically joins the Hubble flow in each cosmological model that expands forever." in https://arxiv.org/pdf/astro-ph/0104349.pdf
because "asymptotically" means "never".
In Fig. 3 the authors write: "In the accelerating universe (ΩM, ΩΛ) = (0.3, 0.7), the perturbed galaxy joins the Hubble flow more quickly than in the decelerating universes ... ".

Thanks.

Not necessarily. What the paper is saying is that ratio of peculiar velocity to recession rate goes to zero over time, for an untethered galaxy with some initial peculiar velocity (all as measured by some reference comoving observer). This asymptotic behavior does not imply that untethered galaxy is ever exactly comoving, any more than the the function 1/x ever has a slope of zero.

 

[timmdeeg]

This asymptotic behavior does not imply that untethered galaxy is ever exactly comoving, any more than the the function 1/x ever has a slope of zero.

Yes, understand, thanks.

 

[votingmachine]

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?

I initially read this as a question about CMBR wavelength shifting. I has elements of that and elements of ordinary red-shift.

If it is strictly a question about CMBR, I think it has been answered that it won't shift. But the answers here confuse me whether they generalize to CMBR or not ... some mention velocity based red-shift. Can anyone specifically support that answer for CMBR?

 

[jbriggs444]

But the answers here confuse me whether they generalize to CMBR or not ... some mention velocity based red-shift. Can anyone specifically support that answer for CMBR?

There is no single way to split an observed wavelength into pieces so that "this piece comes from cosmological expansion" and "this piece comes from velocity". Such statements depend on the coordinates that one chooses. There is no one right choice of coordinates.

One can use so-called "co-moving" coordinates, decide that the CMBR was emitted at such and such a wavelength, red shifted (massively) by cosmological expansion, then red or blue-shifted a bit due to the peculiar velocity of the Earth. This will account for the observed wavelength when the CMBR is measured here. But that's just one choice of coordinates. There is no underlying physical truth to that particular accounting.

 

[jartsa]

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.

If you travel to the box to study the light, it's not redshifted or redshifting.

If the box travels to you, so that you can study the light, it's not redshifted or redshifting.

Seem that the box prevents any redshift happening to the light in the box.

 

[Demystifier]

In principle, the answer to the question is straightforward. One has to solve the D'Alembert wave equation in a time-dependent FLRW background and pick only those solutions which vanish on the boundary defined by the box. I haven't actually do that, but it seems pretty clear that before hitting the wall, a wave packet suffers a redshift expansion during the propagation. But when it hits the wall, it changes the packet in a way that I cannot guess easily without a calculation.

 

[PeterDonis]

it seems pretty clear that before hitting the wall, a wave packet suffers a redshift expansion during the propagation

It suffers a coordinate effect that many cosmologists misleadingly refer to as "redshift", but it suffers no actual effect at all unless it interacts with something.

To expand on "no actual effect at all": what seems "pretty clear" to me is that the only invariant thing you can say about the wave packet's propagation in the absence of interaction is that the wave vector is parallel transported along the null geodesic worldline of the light ray, and "parallel transport" along a geodesic translates to "unchanged".

 

[anuttarasammyak]

Expansion of the universe is derived from the equation where energy is distributed homogeneous. It is of super macro average scale. In our daily scale of planets, stars, and galaxies matters distribute inhomogeneous. I wonder it is inappropriate to apply the expansion law to such micro scale distance problems.

 

[Ibix]

I think it is inappropriate to apply the expansion to such micro scale worlds.

It isn't inappropriate. You can always find a state of motion in which you would see the CMB as isotropic, and someone with that state would be a co-moving observer. Two such people would move very slightly apart over time, absent other influences.

However, that last caveat is a huge one - on small scales, other gravitational and mechanical influences are more than enough to knock the observers out of their co-moving trajectories in extremely short order. That doesn't mean that using co-moving coordinates is wrong, but it's a bit like trying to use ground-fixed coordinates to calculate how to pick up your coffee on a train that's going over points. That is, it's complicated, and with an extra layer of disconnection between experiment and maths, but not wrong.

 

[anuttarasammyak]

It isn't inappropriate. You can always find a state of motion in which you would see the CMB as isotropic, and someone with that state would be a co-moving observer. Two such people would move very slightly apart over time, absent other influences.

Two such CMB-isotropic people near the Earth go down to the Earth or fall into the Sun by gravity, don't they? Such local geometry regarded as noise in universal average view is one of the concerns in my post with micro inhomogeneous distribution of matter.

 

[Ibix]

Two such CMB-isotropic people near the Earth go down to the Earth or falling into the Sun by gravity, don't they?

They're doing 600km/s with respect to Earth and escape velocity is only 11km/s, so they don't really fall. But their trajectories are affected, yes. That doesn't mean you can't analyse their trajectories with respect to observers who continue to see the CMB as isotropic.

 

[anuttarasammyak]

Two such people would move very slightly apart over time, absent other influences.

Thanks. I took absent here as "if" absent. I think the reality is that they may see on the Earth, fall into the sun or do something different from "very slightly apart" due to local inhomogeneous distribution of energy around them.

I watch the figure https://upload.wikimedia.org/wikipe...Hubble_constant.JPG/250px-Hubble_constant.JPG
and observe that around 1MPC is minimum distance threshold to show Hubble's law, so I assume one million light year is the minimum scale to show homogeneous distribution of energy.

We may need to prepare box of one million light year length with light mass enough to disregard its effect on spacetime geometry to show the result which OP expects.

 

[PAllen]

They're doing 600km/s with respect to Earth and escape velocity is only 11km/s, so they don't really fall. But their trajectories are affected, yes. That doesn't mean you can't analyse their trajectories with respect to observers who continue to see the CMB as isotropic.

I think asking about small scale behavior of FLRW spacetime is, in fact, just a theoretical exercise useful for gaining insight. In practice, local geometry would completely dominate. Consider that the the FLRW geometry implies a specific stress energy tensor. The interior of the milky way is described by some completely different stress energy tensor. The cosmological stress energy tensor only arises as a large scale average, and its geometry arises only on such large scales.

So while you can talk about whether or not a body in the solar system is comoving in the sense of seeing CMBR isotropy, I don’t think there is any valid way to discuss small scale dynamics of bodies in the solar system as being affected by cosmological geometry. We have had threads here go nowhere trying to even properly formulate such questions unambiguously.

 

[Ibix]

I don’t think there is any valid way to discuss small scale dynamics of bodies in the solar system as being affected by cosmological geometry.

I completely agree. The only point I was making is that you ought to be able to use a (notional) family of non-inertial worldlines that see the CMB as isotropic as the beginnings of a coordinate system. And that it probably isn't a good idea.

OP's experiment I would do deep in intergalactic space, where inertial observers could be co-moving on long timescales.

 

[PAllen]

I completely agree. The only point I was making is that you ought to be able to use a (notional) family of non-inertial worldlines that see the CMB as isotropic as the beginnings of a coordinate system. And that it probably isn't a good idea.

OP's experiment I would do deep in intergalactic space, where inertial observers could be co-moving on long timescales.

And your last suggestion gets at how tricky this question is (in the real world, rather than as a mathematical exercise in properties of an idealized geometry).

Consider that region of deep intergalactic space. To me that suggests average total energy density is much less than the overall average of the observable universe. Thus one might approach it by suggesting that it ought to locally have a vacuum metric, possibly with cosmological constant. Considering first the case without cosmological constant, then you have a solution with at most pure Weyl curvature. Further, arguments based on the shell theorem suggest that it should, in fact, have no curvature to a good approximation. However, the curvature producing geodesic divergence for initially 'parallel' geodesics in FLRW metric is pure Ricci curvature. Thus, the phenomenology of 'expansion' cannot occur at all in a region described as essentially vacuum

Allowing for a cosmological constant only modifies the argument a bit - the region in question should be described by a vacuum solution with cosmological constant, which will certainly have different dynamics over the region than pretending the global geometry was present (which, by the EFE requires the energy density to match the global average).

By virtue of some of these other threads where this was discussed, I have come to believe the common statement "bound systems don't see expansion" is true but almost irrelevant. The real issue is that global total average energy density (including dark energy/cosmological constant) determine the large scale geometry of the manifold. This large scale geometry allows the initial post BB state to evolve as expected (there is "room" for the matter to move apart; expansion of space really means just this). However, the large scale geometry is 'emergent' and only relevant over regions approximating the overall universal energy density. Any regions, however large, for which the regional energy density is substantially different from the universal average are not governed at all by the global geometry. They must, instead, be analyzed with the stress energy tensor over the region.

Sorry for the long post, but the inadequacy of the traditional "bound systems exception" has been bugging me for a while. The more correct statement is that any region too small to look like the universal average is not described by the geometry of the universal average - specifically, including any dynamical effects of the expansion (except those due to cosmological constant).

 

[vanhees71]

In principle, the answer to the question is straightforward. One has to solve the D'Alembert wave equation in a time-dependent FLRW background and pick only those solutions which vanish on the boundary defined by the box. I haven't actually do that, but it seems pretty clear that before hitting the wall, a wave packet suffers a redshift expansion during the propagation. But when it hits the wall, it changes the packet in a way that I cannot guess easily without a calculation.

But these are standing-wave solutions. The corresponding cavity photons are not propagating. Cavity QED is in this sense qualitatively different from "vacuum QED".

There is nothing hitting the wall being "before" somewhere else. Since the box is bound by electromagnetic interactions it doesn't participate in the Hubble redshift.