Energy of Light in Accelerating Universe: Comparing Cases

[Pony]

Assume two observers very far from each other, so far, that the accelerating expansion of the universe matters. (edit: But not outside of each others event horizons.) They will send light beams each other, and measure the energy of it. Also tie them together with a very long rope to fix their position in a sense.

I am interested in the cases where the cosmological constant is 0, and when greater than 0; the 30 years old, and current physics. (I believe that where the expansion is caused by the cosmological constant, they can't just float in a fix distance, their rope will tense up, and some force will appear in it; while in the former models, where the expansion is more like an "initial velocity" of the matter in the universe, they can just float in a fixed distance with a loose rope. But I don't think this matters.)

As far as I know, when the cosmological constant is zero, they can send energy each other, one of them sends 1 Joule of light, and the other receives that much of amount while if the cosmological constant is greater than zero, the light they send will suffer redshift, and some energy of it will disappear.

Is this correct?

 

[PeroK]

Assume two observers very far from each other, so far, that the accelerating expansion of the universe matters. They will send light beams each other, and measure the energy of it. Also tie them together with a very long rope to fix their position in a sense.

I'm not sure why they need to be tethered. You can search for the "tethered galaxy problem" if you want.

As far as I know, when the cosmological constant is zero, they can send energy each other, one of them sends 1 Joule of light, and the other receives that much of amount while if the cosmological constant is greater than zero, the light they send will suffer redshift, and some energy of it will disappear.

Expansion does not depend on a cosmological constant. There would be redshift in any expanding universe. And, the universe must be expanding (or contracting) according to General Relativity. The cosmological constant is responsible for accelerated expansion; but not for expansion itself.

The energy does not disappear, as such. There is a relationship between the source and the receiver that means that the light has less energy in the local reference frame of the receiver than it did at the source. But, that also applies where you have simple relative motion between source and receiver.

What you do not have on a cosmological scale is a well-defined notion of total energy. What you can say is that energy is conserved locally. See, for example:

https://www.preposterousuniverse.com/blog/2010/02/22/energy-is-not-conserved/

 

[Pony]

There would be redshift in any expanding universe.

If they are floating far away, but with a loose rope between them, fixing their distance? Are you sure about this?

Also I am particularly interested in the effect of the cosmological constant on the traveling light, and not galaxies flying left and right and causing redshift. You don't seem to address that, or maybe a "doesn't matter"?

 

[PeroK]

If they are floating far away, but with a lose rope between them, fixing their distance? Are you sure about this?

Assuming the rope does not stretch, then its irrelevant whether the universe is expanding or not. There would be no redshift. I can't see the purpose of having an expanding universe but preventing things moving along with the expansion?

 

[Halc]

Assume two observers very far from each other, so far, that the accelerating expansion of the universe matters. They will send light beams each other, and measure the energy of it.

If they're far enough apart that the acceleration matters, the light from either might never reach the other. In non-accelerating expansion, light will always get from A to B given enough time.

Also tie them together with a very long rope to fix their position in a sense.

As already mentioned, this is pointless. The rope will break if the observers are far enough apart even if the rope isn't tied to anything. It would be like dangling your friend into the event horizon of a black hole and then trying to haul him back out.

I am interested in the cases where the cosmological constant is 0, and when greater than 0; the 30 years old, and current physics. (I believe that where the expansion is caused by the cosmological constant, they can't just float in a fix distance, their rope will tense up, and some force will appear in it; while in the former models, where the expansion is more like an "initial velocity" of the matter in the universe, they can just float in a fixed distance with a loose rope. But I don't think this matters.)

Right. With a constant of zero, the rope need not have any tension except where local gravitational disturbances accelerate portions of it one way or another. With a positive constant, there's a limit to the length of your rope.

As far as I know, when the cosmological constant is zero, they can send energy each other, one of them sends 1 Joule of light, and the other receives that much of amount while if the cosmological constant is greater than zero, the light they send will suffer redshift, and some energy of it will disappear.

If the energy is sent from opposite ends of the rope (or say a rigid rod), it will be received at the other end at the same frequency as it was sent. But the ends of the rope will have nonzero peculiar velocity, which means that relative to the expanding frame, the emitters are moving towards each other. The blueshift from that motion cancels the redshift of the expanding frame over the time it takes to make the trip.

Light energy (and other energy like kinetic energy) disappear relative to an expanding reference frame, but the rope/rod defines an inertial frame where such energy is conserved.

 

[Pony]

As already mentioned, this is pointless. The rope will break if the observers are far enough apart even if the rope isn't tied to anything. It would be like dangling your friend into the event horizon of a black hole and then trying to haul him back out.

Not that much away. Say some thousands of light years. I assume the cosmological constant will cause some force in the rope, but not a black hole like infinite force.

 

[PeroK]

Not that much away. Say some thousands of light years. I assume the cosmological constant will cause some force in the rope, but not a black hole like infinite force.

There was a thread about this recently:

https://www.physicsforums.com/threads/cosmic-tidal-force.1045127/

 

[vanhees71]

Assuming the rope does not stretch, then its irrelevant whether the universe is expanding or not. There would be no redshift. I can't see the purpose of having an expanding universe but preventing things moving along with the expansion?

Even if there's "a rope" keeping the sender and receiver at a fixed distance, the emitted light wouldn't be restricted by this but undergo the usual redshift due to Hubble expansion. The only difference to the usually considered "co-moving observers" is that the "tethered observers" are accelerated, i.e., there's the force holding them at constant distance from each other.

 

[Vanadium 50]

There is no red-shift.

Acceleration is irrelevant. Velocity matters, but the tether ensures no velocity difference.

If you work in a coordinate system where the light is red-shifted by the expansion, the target galexy is not in a comoving frame, and is being pulled towards the source galaxy by the tether. The ensuing blue-shift cancels the red-shift.

 

[Pony]

Pony: if Λ=0 there will be no redshift, if Λ>0 there will be a redshift
PeroK, Vanadium: no redshift both cases
Vanshee: Hubble redshift both cases

We may need a poll about this

 

[Vanadium 50]

Science is not done by poll.

I don't think you are stating @vanhees71 position correctly.

But most importantly, it sounds like you are not asking a question, but instead advocating for an answer.

 

[vanhees71]

It's of course true that in addition to the cosmological red-shift for the receiver teithered to the sender you have an additional Doppler effect due to the velocity of the receiver relative to the co-moving frame, which compensates the cosmological red shift. To make this quantitative, we should discuss a concrete physical example with calculations. For sure in the hard science you don't figure out "the truth" by a democratic vote or "by authority" but by clear mathematical reasoning within theory and finally by observation!

 

[PeroK]

Pony: if Λ=0 there will be no redshift, if Λ>0 there will be a redshift
PeroK, Vanadium: no redshift both cases

Imagine you have a source at rest relative to the receiver. It emits a light signal. No redshift.

Now, imagine that space starts an accelerated expansion while the light signal is in flight. At various points on its journey the light passes intermediate comoving sources (identical to the original and that emit light of the same source frequency). These intermediate sources are, however, moving relative to the original rest frame. Their light is, therefore, redshifted in this frame. There is no physical reason, however, that the light from the original source is redshifted in the original frame.

Likewise the receiver is held at rest in the original frame and measures no redshift in the original signal. But, does measure some redshift in the signals from the comoving sources. But, not as much as a comoving receiver would.

Comoving means moving according to the accelerated expansion.

 

[PeroK]

PS redshift depends on the relationship between the source and receiver and not on the expansion of space, per se.

 

[Pony]

PS redshift depends on the relationship between the source and receiver and not on the expansion of space, per se.

Yes but the space between them can super mess it up, e.g. when they are at different gravitational potential.

E.g. just like in the case of two gravitating object (say Sun and Earth), you can integrate the effect of the "tidal force" you mentioned earlier on the light, and you get how much the energy of the light have changed.. or do you?

.. anyway, I was thinking about it, and I am joining the no redshift camp too. As Vanadium noticed this was in my mind as an end goal, but now I think it doesn't work this way. Now I wonder why everyone says that the light loses energy due to the expansion of space, when it is literally just two galaxies having speed.

Thank you for the answers.

 

[PeterDonis]

redshift depends on the relationship between the source and receiver and not on the expansion of space, per se

While this is true, it does not mean spacetime geometry is irrelevant. So far, everyone's reasoning in support of the "no redshift" conclusion has been that if the source and receiver are at rest relative to each other (which in this case is enforced by the tether between them), there can be no redshift. But that is obviously false, since, for example, as the OP notes in post #11, observers at rest relative to each other at two different altitudes in a gravitational field will observe nonzero redshift when they exchange light signals.

So since that heuristic is not valid, we either need to find another heuristic that works, or actually do the math. Fortunately, for this case doing the math is not at all difficult, since we can use as our model de Sitter spacetime in static coordinates. This means our two observers are at constant spatial coordinates and the only part of the metric that matters is $g_{00}$:

$$
g_{00} = 1 - \frac{\Lambda}{3} r^2
$$

where $\Lambda$ is the cosmological constant. (We could rewrite this in terms of dark energy density, but it doesn't really matter since we don't need the actual numerical value of the constant $\Lambda / 3$, we just need to know that it's a constant.) The square root of $g_{00}$ is the redshift factor, so for two observers stationary at $r_1$ and $r_2$, with $r_2 > r_1$, the redshift for light going from $r_1$ to $r_2$ is simply the ratio of their redshift factors:

$$
\frac{\Delta \nu}{\nu} = \sqrt{\frac{1 - \frac{\Lambda}{3} r_2^2}{1 - \frac{\Lambda}{3} r_1^2}}
$$

This is nonzero, so there will be redshift. But note that, if $\Lambda = 0$, the redshift vanishes, so for the case of non-accelerating expansion, two observers that remain a fixed proper distance apart will not observe redshift if they exchange light signals.

 

[PeterDonis]

Now I wonder why everyone says that the light loses energy due to the expansion of space, when it is literally just two galaxies having speed.

You are correct that this (unfortunately common) phrasing is sloppy, but it is not as simple as "two galaxies having speed" either. One has to make at least two key distinctions:

(1) The usual "expanding space" terminology refers to comoving observers. The observers in your scenario are not comoving (or at least they cannot both be comoving), so the usual "expanding space" terminology cannot be used.

(2) As I noted in post #16, the actual redshift, or lack thereof, depends on the spacetime geometry as well as the worldlines of the observers. So you have to look at what the actual spacetime geometry is; you can't just automatically apply heuristics like "redshift depends on relative speed, if the objects are at rest relative to each other, there is no redshift".

 

[PeterDonis]

the rope/rod defines an inertial frame

Not in the presence of a nonzero cosmological constant.

 

[PeterDonis]

If you work in a coordinate system where the light is red-shifted by the expansion, the target galexy is not in a comoving frame, and is being pulled towards the source galaxy by the tether. The ensuing blue-shift cancels the red-shift.

As my analysis in post #16 shows, the redshift is not zero, so the cancellation you refer to in the coordinates you refer to (which are not the same as the ones I used in post #16) cannot be exact.

 

[PeterDonis]

As Vanadium noticed this was in my mind as an end goal, but now I think it doesn't work this way

I should note that you are correct that it does not work the way you assumed in that other thread, and nothing I have said in this thread conflicts with that statement.

 

[Pony]

Is your model stationary in time, that is, if they wait a billion years or so, will everything be the same? I am fairly convinced (convinced myself) that if the two of them are symmetrical, and the configuration/universe is stationary, then they must 'see' each other without any redshift, if one emits a laser beam for a period t and with a frequency f, the other must see it that way. At least I can't draw any counter examples on paper.

edit: note that in the two simple cases of time dilation, moving at a constant speed, and sitting in different gravitational potential, either they are not symmetrical, or their configuration is not stationary

 

[PeterDonis]

Is your model stationary in time, that is, if they wait a billion years or so, will everything be the same?

Yes. The density of dark energy does not change with time, and that is the only parameter in the model. For your scenario, that is perfectly fine, since you have specified that the two observers maintain a constant proper distance from each other.

A model which took into account that in our actual universe, there is matter present as well as dark energy, would have an effective parameter $\Lambda$ that changed slowly with time. This would not change the qualitative conclusion, but it would slightly change the actual numerical value of the redshift.

I am fairly convinced (convinced myself) that if the two of them are symmetrical, and the configuration/universe is stationary, then they must 'see' each other without any redshift

In the model I described in post #16, the two observers are not symmetrical. Heuristically, they are at different heights in a gravitational potential.

Note that it is possible to have an alternate model to the one I gave in which the two observers are symmetrical: put them both at the same value of $r$, but on opposite sides of the origin. For example, put them both in the "equatorial plane" $\theta = \pi / 2$, but with one at $\phi = 0$ and the other at $\phi = \pi$. Then, since both are at the same value of $r$, both will have the same redshift factor and there will be no net redshift between them.

So the question now becomes one of specification of the problem: which of those two scenarios did you intend to specify in your OP? In somewhat more intuitive terms, the two scenarios differ as follows: the origin $r = 0$ can be thought of as being at the top of a potential "hill", and we have:

(1) In the scenario I described in post #16, the two observers are both on the same side of the hill, with one lower than the other, so the light passing between them goes either uphill (redshift) or downhill (blueshift).

(2) In the alternate scenario I described above, the two observers are on opposite sides of the hill, and both at the same height, so the light passing between them goes first uphill, then downhill, with the two effects canceling each other.

This is a good illustration of how actually doing the math can uncover aspects of the problem that one's intuition did not expect.

Another way of describing the difference, btw, is that, if we consider all of the observers "hovering" at constant $r$ in between the two observers who are exchanging light signals, in scenario (1) above, none of those observers are comoving, while in scenario (2) above, there is one that is comoving, the one at $r = 0$.

 

[Pony]

I still don't think they can see each other in redshift, say they can see each other to speed up 2 times. Say there is a clock at the middle:

A_______clock_______B

A watches that clock, it ticks at a given rate. But then it ticks at the same rate for B too, so A can watch what is happening on the clock 2 times faster if she just watches B's journal about the clock. It starts with a delay, but then they can accumulate arbitrary many time this way, A just sits there for a billion year, and she sees that B saw the clocks state ahead of 2 billion years. This can't happen, if A can travel to the clock in a constant amount of clocktime, regardless when she starts, because she will see in B's journey that she's already arrived before she even departed.

 

[PeterDonis]

I still don't think they can see each other in redshift

As I said in post #22, it depends on how you specify the scenario.

say they can see each other to speed up 2 times

As I have already pointed out, the scenario in which there is a redshift, scenario (1) in post #22, is not symmetrical. If one sees the other's clock speeded up by a factor of 2, the other will see the first's clock slowed down by a factor of 2. So your reasoning is simply wrong for this scenario.

For the other scenario, scenario (2) in post #22, there is no redshift, so your reasoning does not even apply to this scenario in the first place.

In other words, your reasoning is based on the invalid premise that there is a scenario which is (a) symmetrical, and (b) has a nonzero redshift. But there is no such scenario.

Say there is a clock at the middle

Is that clock the comoving clock at $r = 0$ in scenario (2) in post #22? Or is it a clock at some nonzero value of $r$ in between $r_1$ and $r_2$ in scenario (1) in post #22?

If you didn't realize this was even a question before, well, that is why I went to the trouble of posting post #22. There is no point in continuing to just assert your intuition if you are not going to take the time to read post #22 and think about what it says.

 

[vanhees71]

For a discussion of the cosmological redshift and arbitrarily moving observers (derived from the eikonal approximation of electrodynamics in a FLRW spacetime), see

https://itp.uni-frankfurt.de/~hees/pf-faq/gr-edyn.pdf