Post by: Harri on 24/06/2021 20:21:49
Post by: Halc on 24/06/2021 20:49:26
Am I correct in thinking that there is space, and into this space has emerged our universe?'The universe' defines the space (and time) that we know, so it doesn't makes any sense to talk about space that is 'beyond' the universe, making 'the universe' a sort of object contained within something bigger rather than being the contrainer itself. It would involve a sort of political boundary where this space belonged to 'the universe' and this other space beyond the boundary did not. That's not what we mean by 'the universe'.
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And through this space the mass of our universe is expanding?The space itself is expanding. Nothing is expanding through it. The latter model makes predictions which are contradicted by what we see.
Post by: yor_on on 27/06/2021 14:11:12
Post by: Harri on 01/07/2021 20:13:20
Post by: yor_on on 01/07/2021 20:51:20
The expansion also can be thought of as going 'FTL' if you look far enough out in the universe. The further away the more 'new space' can be created, and every little patch adds up to expand the distance between you and that star. and so also the 'speed' of their recession, receding relative yourself.
and most everything is made of 'space', even mass.
Post by: Halc on 01/07/2021 21:05:49
The space itself is expanding. So is this expansion known as inflation?It's known as expansion. Inflation is something else, an exponential rate of expansion that doubles the size of a given space every (something like) 10-64 seconds. The inflation epoch of the universe was one of the earlier epochs, and completed in a tiny fraction of a second, after which the more or less linearly expanding space as we know it was set up.
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And does inflation occur at a faster rate in parts of the universe that has less mass, away from galaxies for instance?I think this is a misrepresentation of what is going on. Expansion isn't a force, so nothing is pushed apart that isn't already moving apart. Acceleration of expansion is a force, but only in places where gravity (or other forces) is insufficient to counter the effects of dark energy will objects be pulled away from each other. So for instance, Laniakea is a gravitationally bound mass (a huge one), but is sufficiently spread out that dark energy will eventually tear it apart rather than having it all eventually collapse into some common dense object.
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And within our own galaxy and solar system the expansion is in all directions but at a slower rate due to gravitational influences?Yes, the expansion occurs in these places, but it doesn't mean that Neptune is moving away since these things are bound to their respective structures, just like my car will always be the same length no matter what expansion is doing to it. But over time, the 'space' occupied by my car now will expand to a larger volume in billions of years, eventually leaving room for another car, but not in anyway stretching the car.
Post by: Eternal Student on 02/07/2021 14:15:47
So is this expansion known as inflation?I would support Halc's earlier answer. Inflation is a form of very rapid expansion and Inflation has been stopped in our region of the universe, while a slower rate of general expansion is still happening.
And does inflation occur at a faster rate in parts of the universe that has less mass, away from galaxies for instance?Halc said some reasonable things but I'm not sure you (Halc) really answered the question.
Low level answer:
It's actually really difficult to say. The easiest answer would be to say "yes", there is less expansion in matter dense regions of space such as space within galaxies and more in the space that is between galaxies.
Medium Level answer
Expansion is something that is best explained with General Relativity. There are models of the universe where a quantity called "the scale factor" is found. It is the growth of this scale factor with time that creates what would be described as the expansion of space. Anyway, the sad thing about these models is that they only apply on large, Astronomical scales. We can see that over large scales (more than the size of a galaxy), expansion should occur but the model does not show what happens in small local regions of real space. We can't be sure if the expansion is happening within a galaxy, in the space between it and another galaxy, or in both regions to some extent.
As regards real observations, we observe distant galaxies receding from us but not the nearby galaxies (those said to be within our local cluster of galaxies). We certainly don't notice any recession of the stars within our own galaxy. It's hard to say if there just isn't any expansion within our galaxy or if the expansion is just so minor that we wouldn't be able to measure it. The general consensus of opinion is that there shouldn't be an expansion of space within a gravitationally bound system like our local cluster.
High level answer
We can consider what happens on a universe-wide scale when more matter is put into our FRW universe models and then apply this result to small regions of space. It is seen that as the matter density increases, the acceleration of expansion becomes negative, i.e. expansion tends to be opposed. The expansion of space is slowed as time progresses and in some cases expansion is completely stopped and eventually reversed so that contraction happens. We can then naively assume that what applies on a universal scale should also apply on a smaller local scale: Therefore, if a local region of space is dense with matter then the local acceleration of expansion is negative etc. Or to phrase this another way - inside a galaxy expansion should be opposed. Meanwhile in regions of low matter density, such as between galaxies, expansion continues unabated. This is a simple or naive view - there really isn't any guarantee that just because a high density of matter tends to oppose expansion on a universe-wide scale it must do the same on a local scale. The more thorough approach is to recognise that on a local scale, a completely different local model of space must be used.
We can try to build local models using General Relativity - for example a local model of what is happening to space within a galaxy and another one for the space between galaxies. However we encounter two problems. The first problem is that the equations of General Relativity are too complicated to get exact solutions, we have to make some approximations. Within a galaxy it's impossible to consider all the sources of matter (e.g. all the planets and all the stars in that galaxy). So there is a tendency to simplify the model and obtain something like the Schwarzschild metric for that region of space. The details don't need to concern you too much, all that matters is that there isn't a quantity like the scale factor in these local models for space within a galaxy. This tends to make us think that there isn't any expansion of space here: The expansion of space doesn't seem to be happening in matter dense regions of space. However, as we said, the model is a hideous simplification of the reality.
(The second problem is that somehow we have to "join up" the model for space within a galaxy to the model for space outside a galaxy. If both models used the same co-ordinates then our job wouldn't be too difficult, however this is not what we have. I'm skipping all further discussion of the problems of joining different co-ordinate regimes to each other).
Next, Halc said something quite unusual:
QuoteAs outlined above, it is not at all obvious that expansion is happening within our own galaxy.
And within our own galaxy and solar system the expansion is in all directions but at a slower rate due to gravitational influences?
Yes, the expansion occurs in these places, but it doesn't mean that Neptune is moving away since these things are bound to their respective structures, just like my car will always be the same length no matter what expansion is doing to it. But over time, the 'space' occupied by my car now will expand to a larger volume in billions of years, eventually leaving room for another car, but not in anyway stretching the car.
However, even if there is some expansion of space then there are some other forces that we know of that will help to keep matter together. I think this is what Halc was talking about with his car. Large bodies of matter like a car are held together by electromagnetic forces between atoms and molecules. At a smaller scale, subatomic particles are held together by the strong nuclear force. These forces would pull the particles through space and keep them together if needed, even if the space around them was expanding.
We need to be a little carefull not to count "gravitational influences" twice but this is minor point and may cause confusion. The idea is that in General Relativity, gravity is not a force like the electromagnetic force described above and many of it's effects have already been accounted for in the expansion of space. If two objects appear to be attracted to each other due to gravity, then it is unlikely that the local bit of space between them would have been expanding anyway.
Best wishes to everyone.
Post by: Halc on 02/07/2021 20:40:12
Expansion is something that is best explained with General Relativity. There are models of the universe where a quantity called "the scale factor" is found. It is the growth of this scale factor with time that creates what would be described as the expansion of space. Anyway, the sad thing about these models is that they only apply on large, Astronomical scales.It’s not sad. GR handles the local digressions from absolute homogeneity just fine.
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As regards real observations, we observe distant galaxies receding from us but not the nearby galaxies (those said to be within our local cluster of galaxies). We certainly don't notice any recession of the stars within our own galaxy.As you shouldn’t. Given perfect linear expansion (a linear scalefactor), two objects (say a pair of pebbles a considerable distance apart) that are stationary relative to each other will remain at a constant proper separation forever in the absence of external forces like the gravity of other objects around it. Expansion of the space between the pebbles will have zero effect on their separation. So there’s no reason that things should pull apart unless they’re already moving apart.
Dark energy does change this since it constitutes a repulsion force which bends the scalefactor from being linear. That will eventually accelerate said pebbles apart in the absence of sufficient gravity between them countering it.
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The general consensus of opinion is that there shouldn't be an expansion of space within a gravitationally bound system like our local cluster.I’m not claiming to know otherwise, but do you have a reputable reference for this consensus? Given a local cluster of galaxies with such and such relative velocities between them, what empirical difference would there be between uniform expansion occurring everywhere and the same setup with expansion only happening in the far less dense voids between the clusters?
If there’s no empirical difference between the two situations, then it’s just an abstract coordinate difference, and it would seem more sensible to use the coordinate system most appropriate for the description at hand.
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It is seen that as the matter density increases, the acceleration of expansion becomes negative, i.e. expansion tends to be opposed. The expansion of space is slowed as time progresses and in some cases expansion is completely stopped and eventually reversed so that contraction happens.Again, reference please. Density results in gravity, which accelerates stuff through space but does not necessarily have any effect on expansion unless there is an empirical difference that can falsify one view or the other.
Yes, the general average density (and not just local density) of the universe has a negative effect on the rate of expansion. It slows it, as it did for the first say 7 billion years when the scalefactor had a negative curvature. It never went anywhere near negative, else the big crunch would have happened long ago. After those 7 billion years, the average density of the universe dropped low enough to where dark energy dominated, and the epoch of dark energy commenced, and accelerated expansion with it. Apparently the universe will eventually expand exponentially at a predicted rate of around 57 km/sec/mpc, but right now the scalefactor is still pretty close to linear.
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As outlined above, it is not at all obvious that expansion is happening within our own galaxy.Agree, but it there any evidence that expansion is not happening locally? If not, then it is simply a matter of choice of coordinate systems to describe things like galaxies, not a case of one being right or wrong about it.
Post by: Eternal Student on 03/07/2021 16:33:13
Thanks for your reply. You've given the monstrous task of finding references, which has made this post a bit longer than usual. You don't have to read it (and then maybe you won't notice that I've had to trim down one of the claims I made earlier).
It’s not sad. GR handles the local digressions from absolute homogeneity just fine.It (GR) may do but we (human beings) trying to use GR certainly don't...
Properties of Einstein's Field Equations:
As differential equations, these are extremely complicated; ....(Carroll describes the various appearances of the metric, it's 1st and 2nd derivatives and products of derivatives that appear) ..... The equations are also nonlinear, so that two known solutions cannot be superposed to find a third. It is therefore very difficult to solve Einstein’s equations in any sort of generality, and it is usually necessary to make some simplifying assumptions. Even in vacuum, where we set the energy-momentum tensor to zero, the resulting equations can be very difficult to solve. The most popular sort of simplifying assumption is that the metric has a significant degree of symmetry
[Reference: p. 165, Spacetime and Geometry, Sean Carroll. Similar reference available (free of charge) on-line - See page 16 of 32 close to equation 4.53 in this pdf: https://preposterousuniverse.com/wp-content/uploads/grnotes-four.pdf ]
I'll combine two of your (Halc) next sections, since they are both about recession:
Given perfect linear expansion (a linear scalefactor), two objects (say a pair of pebbles a considerable distance apart) that are stationary relative to each other will remain at a constant proper separation forever in the absence of external forcesThis actually looks correct. Obvioulsy it requires that the objects were stationary relative to each other and not that they were at fixed co-moving co-ordinates but that is exactly how you have stated it. There are some limitations on the statement, for example, if space is expanding very rapidly between them then the objects could not have been stationary w.r.t. each other to begin with, since one would require a local peculiar velocity through space > c. So the next sentence you made -
Expansion of the space between the pebbles will have zero effect on their separation.is not strictly true. However, you do mitigate the possibility that expansion was that rapid in the next sentence -
So there’s no reason that things should pull apart unless they’re already moving apart.I have to say that I was quite impressed with the analysis you presented there but I've got to link with the comment you made at the end of your reply....
QuoteYes, there should be some evidence and a discernible, measurable difference if space is expanding between two objects.
As outlined above, it is not at all obvious that expansion is happening within our own galaxy.
Agree, but it there any evidence that expansion is not happening locally? If not, then it is simply a matter of choice of coordinate systems to describe things like galaxies, not a case of one being right or wrong about it.
Photons should red-shift as they move through space that is expanding.
For a space with metric of this form:
ds2 = -dt2 + a2(t) δij dxi dxj
This is just the usual Minowski metric with a scale factor a(t) appearing in the spatial co-ordinates. It's a FLRW metric with zero curvature if you want to look at it another way (reference provided below).
A co-moving observer will identify a photon with Energy E1 at a scale factor a1 but they would identify the phton as having Energy E2 at a scale factor a2 where the following relationship holds:
Providing the usual cosmological red-shift formula:
[Reference: p 116, Spacetime and geometry, Sean carroll. Sorry, I don't think it's in his free on-line version]
Anyway, the main point is that if we were careful to set up a co-moving observer (I think we could also adjust the formulae for a non co-moving observer - but let's take the easy option), then we can see that light has travelled through expanding space.
If we take your (Halc) example of two objects that are stationary relative to each other but some fixed distance apart then let's make it easy and make one of those objects co-moving (the other one won't quite be but that doesn't matter). We'll call the co-moving object the radiation receiver, while the distant object will emit a photon toward that receiver. The two objects are stationary w.r.t each other so there is no conventional Doppler shift. The emitter creates a photon of known frequency in the usual way (they excite a Hydrogen atom and allow the electron to drop back to it's ground state etc.). At that time the scale factor was a1 and the receiver knows precisely what the energy of that phton had to be. It takes some time to travel to the receiver whereupon the scale factor is a2 so that the receiver identifies the photon as having a different frequency.
Now if the (local) space that the two objects occupy hadn't been expanding then the receiver would not have observed any red-shift.
Connect this with what I had said earlier: We do not observe a red-shift from anything in our galaxy other that that which can be explained by peculiar velocity through space. (Distance and hence peculiar velocity can be ascertained by other means, e.g. parallax observations etc.). We then have some reason to believe that space in our galaxy is not expanding (or at the very least is not well modelled by a metric of the form given above). However (as mentoned earlier) it could be that the distances (and hence photon travel times) weren't large enough and/or our measurements are of insufficient sensitivity.
Given a local cluster of galaxies with such and such relative velocities between them, what empirical difference would there be between uniform expansion occurring everywhere and the same setup with expansion only happening in the far less dense voids between the clusters?I can't answer that. I agree, there may be no difference to what we observe here on Earth. This is very much the same point made (by me) earlier, although for slightly different reasons:
...[Discussing the limitations of a FRW universe model since it only applies on Astronomical scales]....We can't be sure if the expansion is happening within a galaxy, in the space between it and another galaxy, or in both regions to some extent.
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I claimed a "consensus" opinion and you (Halc) asked for references:
Firstly, I didn't actually survey 100 physicists and ask for their opinions, "consensus" was used only as a turn of phrase.
The general consensus of opinion is that there shouldn't be an expansion of space within a gravitationally bound system like our local cluster.Here's a reasonable reference that applies on the scale of galaxies. There are others but I can't be bothered to find them all and reference them. I may have over-stepped the mark by going upto the sclae of the local cluster of galaxies. I might try and find references tomorrow but no one else will be interested anyway. Let's just re-phrase the statement I made and scale down "..........within bound systems like our galaxy." The general reasoning for why it applies all the way upto the local cluster is that we don't observe red-shift from these galaxies (see earlier - photons red-shifting if they travel through expanding space).
A clearer explanation is simply that on the scales of galaxies the cosmological principle does not hold, even approximately, and the FRW metric is not valid. The metric of space-time in the region of a galaxy (if it could be calculated) would look much more Schwarzchildian than FRW like, though the true metric would be some kind of chimera of both. There is no expansion for the galaxy to over-come, since the metric of the local universe has already been altered by the presence of the mass of the galaxy. ..... The expansion of space is global but not universal, since we know the FRW metric is only a large scale approximation.
[Section 2.6.3; page 7; Expanding Space: the Root of all Evil?; Francis, Barnes, James & Lewis, https://arxiv.org/pdf/0707.0380.pdf - Later accepted for publication in Publications of the Astronomical Society of Australia]
Also, a low value reference from Wikipedia:
Within the Local Group, the gravitational interactions have changed the inertial patterns of objects such that there is no cosmological expansion taking place
{Wikipedia. https://en.wikipedia.org/wiki/Expansion_of_the_universe
- - - - - -I also made this statement which you wanted references for:
[Concerning FRW universe models...] It is seen that as the matter density increases, the acceleration of expansion becomes negative, i.e. expansion tends to be opposed. The expansion of space is slowed as time progresses and in some cases expansion is completely stopped and eventually reversed so that contraction happens.For an FRW universe we have the Friedmann equations:
[Eqn 1]
[Eqn 2]
Where: a = a(t) = scale factor; κ = curvature; ρ = energy density; P = pressure; G = Grav. constant
[Reference: p. 334, Spacetime and Geometry, Sean carroll or other online sources, e.g. https://en.wikipedia.org/wiki/Friedmann_equations but note that Wikipedia treat the cosmological constant Λ separately, while in Carroll's version and the equations shown above this is included as an energy density component].
We have a > 0 for all times except at a Big bang or Big Crunch event. Then using Eqn 2, the right-hand side shows that
< 0 whenever we have (ρ+3P) >0. Increase the density of the matter component and ρ increases (while the pressure of the fluid is unaffected). To say this another way, the acceleration of expansion is negative when the matter density is high. Next we show the last part of the statement is true... in some cases expansion is completely stopped and eventually reversed so that contraction happens.
It is not enough that the acceleration of expansion is negative, it's often the case that this acceleration assymptotically approaches 0 before the the expansion has been stopped and then it can never be reversed. We will use [Eqn 1] to show that, for certain models, we can find a time when
. Since we have established that 
at all times, we know that 
will become negative shortly afterwards. To say that another way, we will have shown that expansion reverses and becomes a contraction.We can normailse the curvature, κ, so that it remains constant (with value +1, 0 or -1) by adjusting the scale factor (and co-ordinates, but we aren't even using the co-ordinates here) [See page 331-332, Spacetime and Geometry, Sean carroll]. We select a model with κ = +1, a universe with positive curvature. (You might ask, are we free to choose any curvature? The answer is yes - the curvature is determined by comparing the density against a critical density. So we are just choosing a high enough density. See page 337, Spacetime and Geometry, this post is already too long to discuss this further).
Recall that we are working with a matter dominated universe and that energy density from matter is proportional to
. O.K., we're finally ready to look at [Eqn 1] again: [Eqn 1]
The L.H.S. is squared, so it's always non-negative.
The R.H.S. has a positive contribution
which varies as 1/a3 and a negative contribution 
which obviously varies as 1/a2This means that as the scale factor, a, increases the positive contribution falls more rapidly. So there is a value of the scale factor where the R.H.S. becomes zero. The scale factor cannot increase beyond this value since we know the L.H.S. is always non-negative and more-over we have established earlier that the rate of acceleration is always negative in this model.
We're done. For a matter dominated FRW universe with positive curvture (sometimes called a matter dominated closed universe), the expansion must stop and reverse.
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OK. Bye for now and best wishes to everyone.
Post by: yor_on on 03/07/2021 21:03:20
" Anyway, the sad thing about these models is that they only apply on large, Astronomical scales. We can see that over large scales (more than the size of a galaxy), expansion should occur but the model does not show what happens in small local regions of real space. We can't be sure if the expansion is happening within a galaxy, in the space between it and another galaxy, or in both regions to some extent. "
And I guess this is why some people define it as being a result of just 'space'. As far as I'm concerned I will treat it the same way we do everything else, there being no 'special patches' in our universe. In that case this expansion is in every point.
Would you have a counterargument to it ES?
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btw I enjoyed the way you introduced mathematics in it. Even though I lack those skills :)
As long as you try to explain it as good as you can it will be cool with me.
Post by: yor_on on 03/07/2021 21:22:14
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I know, the correct word might be curvature but then again,. How would you get to a curved space without matter (proper mass)? One would have to define something we don't see (exist) to do it, wouldn't we?
Post by: Eternal Student on 04/07/2021 12:05:01
And I guess this is why some people define it as being a result of just 'space'. As far as I'm concerned I will treat it the same way we do everything else, there being no 'special patches' in our universe. In that case this expansion is in every point.
Would you have a counterargument to it ES?
A few ideas come to mind:
1. Mass warps spacetime. Let's consider the effect on time first. We have evidence for gravitational time dilation, e.g. clocks tick slower close to a massive object compared to clocks further away from it. Suppose that your idea was absolutely correct. Let's assume that locally expansion was occurring at the same rate at every point in space and consider two points, one very close to a massive object and the other much further away. Locally the rates of expansion are the same but due to time dilation, an observer at the distant point (away from the massive object) would see the space around the massive object expand more slowly.
Being in close proximity to a dense object is what caused that time dilation, so we might generalise the result to apply to regions of space with higher density (galaxies) compared to regions of space with low density (voids between galaxies).
2. Let's consider mass warping spacetime again and focus on spatial differences we can observe.
If we draw some geodesics (light ray paths that are our best interpretations of "straight lines") in space we see different things happening in empty space compared to space around a massive object.
Around a massive object, geodesics tend to converge (they get pulled together). This is not what we expect in expanding space. In space that is quite empty and static (not expanding) geodesics remain straight (they don't converge or diverge). Meanwhile in empty space that is expanding we would see geodesics diverging. If we wanted to see geodesics converging in empty (low density) space, then we need to make that space contract rather than expand.
3. "There being no special patches of space" is essentially the Cosmological principle. There is another discussion about the Cosmological Principle here: https://www.thenakedscientists.com/forum/index.php?topic=82561.0. I'm fairly sure you (Yor-on) are aware of it since you've put in a few posts yourself.
The Cosmological Principle only applies on large astronomical scales. I know that is a circular argument, it's just there's no point saying the same things here that have been said elsewhere.
4. This quote seems to be most relevant:
A clearer explanation is simply that on the scales of galaxies the cosmological principle does not hold, even approximately, and the FRW metric is not valid. ...... There is no expansion for the galaxy to over-come, since the metric of the local universe has already been altered by the presence of the mass of the galaxy. ..... The expansion of space is global but not universal, since we know the FRW metric is only a large scale approximation.
[Section 2.6.3; page 7; Expanding Space: the Root of all Evil?; Francis, Barnes, James & Lewis, https://arxiv.org/pdf/0707.0380.pdf - Later accepted for publication in Publications of the Astronomical Society of Australia]
I seem to recall you (Yor_on) said you were from Sweden? You probably speak better English than I do but I found the phrase I underlined above difficult to understand. They are using the words "global" and "universal" in ways that I would describe as being nuanced. For my own benefit, I would like to rephrase that section:
The expansion of space must apply over large scales but it does not apply everywhere in the same way.
That's not the way I, personally, would want to phrase it but I've got to try and retain what THEY said. I think it should be phrased "it doesn't have to apply everywhere in the same way and there is some reason to believe that it doesn't".
Best wishes to you, bye for now.
Post by: yor_on on 04/07/2021 21:04:26
My definition of it is my own. And it differs between what I call 'local definitions' and global. It's a headache as it puts it somewhere where I still can't define it. It's easy to define it locally. It's easy to build a picture from where we from 'repeatable experiments' reach this 'global definition' in where you need to include Lorentz transformations' to make it logical. It places it as a cerebral exercise. But practically I wouldn't define it as being cerebral.
And as you say, I use the principles in where we define no 'patch' of this universe as special. Looking at it that way a expansion should hold everywhere. You can't make something of nothing and then lock it to a 'empty space'. You have to give me a better reason. As for the rest of it, the headache I get is in defining this global definition of a universe.
Post by: yor_on on 04/07/2021 21:13:00
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I'm somewhat inebriated so I might want to look this over tomorrow, but the principle I use makes time dilation's a exception, as no one is set in stone. You can't define it as long as speeds are 'relative'. It's a cerebral exercise at best. But you can define being in a same frame of reference, even if ideally. So I took the other turn and define it as no patch in this universe is special, including time dilation's and LorentzFitzGerald contractions. This makes being in a same frame of reference as something holding globally, not only locally. But it makes the global definition somewhat of a headache. It makes 'c' equivalent to your clock. And to me it makes perfect sense, locally defined.
to accept it you just need one thing. Accepting that all 'repeatable experiments', constants etc. is defined purely locally.
Post by: yor_on on 05/07/2021 06:23:06
Yesterday it (what I wrote) was perfectly logical, and most of it is today too, it might just be me being easily satisfied though. I think you put the spotlight on it in this sentence " Let's assume that locally expansion was occurring at the same rate at every point in space and consider two points, one very close to a massive object and the other much further away. "
I can't make that work, but I want a way to make it work. I've been thinking about that one too.
Post by: Halc on 09/07/2021 14:29:29
Then you shouldn’t have much trouble with the concept of bound objects like solar systems and galaxies not being affected by expansion. That was my point there.Quote from: HalcGiven perfect linear expansion (a linear scalefactor), two objects (say a pair of pebbles a considerable distance apart) that are stationary relative to each other will remain at a constant proper separation forever in the absence of external forcesThis actually looks correct.
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Obviously it requires that the objects were stationary relative to each other and not that they were at fixed co-moving co-ordinates but that is exactly how you have stated it. There are some limitations on the statement, for example, if space is expanding very rapidly between them then the objects could not have been stationary w.r.t. each other to begin with, since one would require a local peculiar velocity through space > c.I deny that limitation for the case specified. The zero-energy solution to the FLRW model is Minkowskian spacetime, meaning the proper separation between a pair of objects can be constant regardless of the magnitude of that separation. It’s only a coordinate difference, but the hyperbolic coordinates of expanding space measures velocity differently, and recession rates are not an inertial velocity, but rather a proper velocity, which can exceed c.
So for example, while in comoving (hyperbolic) coordinates, a distant galaxy (redshift z=11) might be increasing its proper separation from us at ~2.4c (along a line of equal cosmological time), that same worldline relative to inertial coordinates corresponds to a recession velocity of only about 0.983c. Of course real spacetime is not free of energy, so spacetime in reality is not Minkowskian and so talk of inertial coordinate systems at that scale is inapplicable. My comment was only applicable to the zero energy solution with linear scalefactor.
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So the next sentence you made -It is strictly true. Only a physical force could alter the proper separation of those pebbles, not a mere abstract coordinate choice. Gravity could do it, as could dark energy, but this was the linear solution which lacks those things.Quote from: HalcExpansion of the space between the pebbles will have zero effect on their separation.is not strictly true.
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Yes, there should be some evidence and a discernible, measurable difference if space is expanding between two objects.Disagree. Observed redshift is due to increasing separation distance, not to a choice of coordinate system. In my example with arbitrarily distant separated pebbles, light from either pebble as observed at the other will not appear redshifted. Of course this is my linear scalefactor.
Photons should red-shift as they move through space that is expanding.
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For a space with metric of this form:I must presume that for a linear a(t), no redshift. I would expect otherwise for a different scalefactor, but for something small like a galaxy, the scalefactor has no significant time to diverge from a linear approximation over the short time it takes light to traverse its width. Hence there will be no measurable expansion in any observation within the scale of a galaxy, and no way to tell if expansion is present or absent. That’s been my argument.
ds2 = -dt2 + a2(t) δij dxi dxj
This is just the usual Minowski metric with a scale factor a(t) appearing in the spatial co-ordinates. It's a FLRW metric with zero curvature if you want to look at it another way (reference provided below).
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Providing the usual cosmological red-shift formula:Sorry, don’t know what all the symbols are. It seems to be a formula for receding objects with negligible peculiar velocity, which doesn’t really apply to the edges of a bound object like a galaxy. If I had a non-rotating ‘stationary’ pizza 100k LY across, the edges would all have inbound peculiar velocity and thus not be redshifted as viewed anywhere on the disk. All the anchovies would be coming to me but never getting here.
[Reference: p 116, Spacetime and geometry, Sean carroll.]
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Anyway, the main point is that if we were careful to set up a co-moving observer (I think we could also adjust the formulae for a non co-moving observer - but let's take the easy option), then we can see that light has travelled through expanding space.How do you know if they’re comoving if you don’t know if space expands within a galaxy or not? Such a setup seems to require begging your answer before your measurement.
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If we take your (Halc) example of two objects that are stationary relative to each other but some fixed distance apartOpposite ends of a long rigid rod say, which also acts as a tape measure. Sure, one end with zero peculiar velocity, and the other end significantly far away. We need to consider the real universe and not my linear one since the goal is to test for expansion within a galaxy or the lack of it, and the linear case doesn’t have high density matter concentration, that density being everywhere zero. This is problematic since we need more separation than the size of a galaxy gets us, and we also need less separation than that to test the claim.
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We'll call the co-moving object the radiation receiver, while the distant object will emit a photon toward that receiver. The two objects are stationary w.r.t each other so there is no conventional Doppler shift.You’re switching reference frames without being explicit about it. That’s always going to result in confusion. Relative to some inertial frame, the two objects have identical velocity.
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At that time the scale factor was a1 and the receiver knows precisely what the energy of that phton had to be. It takes some time to travel to the receiver whereupon the scale factor is a2 so that the receiver identifies the photon as having a different frequency.Relative to the cosmological frame, they don’t have identical velocity. The far one has a peculiar velocity inward which cancels the expansion redshift and results in no Doppler shift at all, conventional or otherwise. You seem not to be taking into account the peculiar velocity of the emitter.
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Agree, but to demonstrate the situation one way or another (said consensus), some empirical difference must be observed, else it is just a matter of choice of coordinate systems.Quote from: HalcGiven a local cluster of galaxies with such and such relative velocities between them, what empirical difference would there be between uniform expansion occurring everywhere and the same setup with expansion only happening in the far less dense voids between the clusters?I can't answer that. I agree, there may be no difference to what we observe here on Earth.
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A clearer explanation is simply that on the scales of galaxies the cosmological principle does not hold, even approximately, and the FRW metric is not valid. The metric of space-time in the region of a galaxy (if it could be calculated) would look much more Schwarzchildian than FRW like, though the true metric would be some kind of chimera of both. There is no expansion for the galaxy to over-come, since the metric of the local universe has already been altered by the presence of the mass of the galaxy. ..... The expansion of space is global but not universal, since we know the FRW metric is only a large scale approximation.That just says that spacetime is neither flat nor homogeneous within a galaxy. It doesn’t say that space isn’t expanding there, only that the metric (which applies to flat homogeneous space) doesn’t describe what is a ‘local digression’ any more than the metric describes the curvature of spacetime here on Earth.
[Section 2.6.3; page 7; Expanding Space: the Root of all Evil?; Francis, Barnes, James & Lewis, https://arxiv.org/pdf/0707.0380.pdf - Later accepted for publication in Publications of the Astronomical Society of Australia]
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Also, a low value reference from Wikipedia:That does indeed assert it, but it’s wiki. I don’t agree with the statement since there is no empirical difference between local ‘inertial patterns of objects’ with and without expansion, as per my argument above.
Within the Local Group, the gravitational interactions have changed the inertial patterns of objects such that there is no cosmological expansion taking place
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For an FRW universe we have the Friedmann equations:This is a better reference, but if I read it correctly, it seems to reference the curvature, energy density etc. of the entire universe. None of the figures are local. It is used to show that the universe had decelerating expansion while it was matter dominated, but it is no longer matter dominated. But locally it is, so if that metric was applied locally instead of universally, it could be read as space expanding only in the thin areas. This sort of corresponds to the absolutists that talk about the ‘aether wind’ which has space continuously collapsing into matter-dense areas like Earth. So the radius of Earth is constantly growing as the surface has positive peculiar velocity outward countering the negative space expansion going on locally.[Eqn 1][Eqn 2]
Where: a = a(t) = scale factor; κ = curvature; ρ = energy density; P = pressure; G = Grav. constant
[Reference: p. 334, Spacetime and Geometry, Sean carroll or other online sources, e.g. https://en.wikipedia.org/wiki/Friedmann_equations but note that Wikipedia treat the cosmological constant Λ separately, while in Carroll's version and the equations shown above this is included as an energy density component].
Let me know if you think I’m just talking nonsense. I’m wondering myself. But as per my argument above, this produces no empirical difference than an Earth with locally not expanding or contracting space.
Post by: Eternal Student on 09/07/2021 15:43:39
It'll take me some time to reply (and I'll try to keep it short).
Post by: Eternal Student on 10/07/2021 13:07:19
I think we'll take this point first:
How do you know if they’re comoving if you don’t know if space expands within a galaxy or not? Such a setup seems to require begging your answer before your measurement.We can establish if an observer is co-moving and we can do this independently of whether we assume space is expanding, contracting or static. In the real universe (and also in standard cosmological models), the Cosmic Microwave Background Radiation (CMBR) will be (almost) isotropic if an observer is co-moving. While if they had a peculiar velocity with respect to the CMB frame then they will observe a dipole anisotropy in the CMBR.
The expansion of the universe will affect the wavelengths received but it does this uniformly in all directions. By changing the expansion we can make the CMBR more blue-shifted or red-shifted but we can't make it happen more in one direction than another. The existence of a dipole anisotropy is purely the result of the observer's motion relative to the CMB frame.
We can establish the following equivalent set of statements:
The CMBR is observed to be isotropic iff (if and only if)
A dipole anisotropy in the CMBR is NOT observed iff
The observer is at rest in the CMB frame iff
The observer is co-moving iff
The observer has constant spatial co-ordinates (X,Y,Z) for all co-ordinate times. We're going to use Cartesian co-ordinates to describe our 4-dimensional spacetime. Our global metric is defined in those co-ordinates and another way to describe what it means to be co-moving is that our observer moves on a worldline described by (X,Y,Z, t) with constant X,Y,Z and t varying.
[references available if required].
Now the main point of disagreement between you (Halc) and me seems to be what causes a red-shift:
1. Halc's view:
Observed redshift is due to increasing separation distance, not to a choice of coordinate system. In my example with arbitrarily distant separated pebbles, light from either pebble as observed at the other will not appear redshifted.2. While I maintain that red-shift is caused by the properties of the space through which a photon has travelled. In particular, if space is expanding then photons lose energy.
We both recognise that a Doppler effect applies: A photon's energy is influenced by relative motion of receiver and emitter. So the statement No.2 that I made above needs to be clarified: When I say that a photon loses energy I must specify the reference frame. I will choose to measure all photons in the CMB frame (more precisely, a photon has energy E iff an observer at rest in the CMB frame would measure energy E).
With this in mind, we have a method to identify if space is expanding in some region. If it is not expanding, then the metric is of the usual form for flat Minkowski space and Euclidean geometry rules the day. If our emitter and receiver (or Halc's two pebbles) maintain constant proper distance then they are both at rest in the CMB frame as soon as any one of them is at rest in the CMB frame*.
* The proof of this is trivial but the vocabulary can make it seem complicated. Let's try and go through it here:
Without loss of generality, assume the emitter and receiver lie along a line parallel to the x-axis (we can rotate our axis to ensure this happens since the space is Minkowski so it will retain the form of a Minkowski metric in the new rotated co-ordinates). Assume the receiver is at rest in the CMB frame. So by the equivalent statements listed at the top of this reply we have that the receiver is co-moving. By definition then, it has spatial co-ordinates (X,Y, Z) where X,Y,Z are constant for all co-ordinate times. Now consider the emitter, it lies along a line of constant Y and Z so it has co-ordinates (x , Y, Z) where Y,Z are constant and only x(t) may vary with co-ordinate time. We know that the emitter and receiver remain at constant proper distance, d, and the metric is Minkowski, hence we have (X - x)2 = d2. Therefore, x = X +/- d, we're not going to allow x = X+d for some values of co-ordinate time and x=X-d for other values of co-ordinate time, we insist that co-ordinate changes must be continuous with respect to time. Hence x = X+d or X-d at all co-ordinate times. So the emitter has fixed spatial co-ordinates, so it is co-moving, so it is at rest in the CMB frame. We probably didn't need to take this long, it's just simple - in Minkowski space, Euclidean Geometry rules.
OK, so if space is not expanding then the emitter can create their photon of known energy as usual (excite a Hydrogen atom etc.) and send it to the receiver. I choose to measure the photons energy in the CMB frame always but that's fine. Both emitter and receiver were at rest in the CMB frame, everyone reports the same energy for the photon. We're all done with the case of non-expanding space between emitter and receiver.
Now, Halc's argument is that if the space between the emitter and receiver had been expanding then everyone will still report the same frequency for the photon. This is because the emitter must now have some velocity toward the receiver (measured as a co-ordinate velocity in the receivers local frame) because constant proper distance must be maintained. So there is a Doppler shift to be applied which will precisely cancel the red-shift of the photon as it moves through expanding space. The implication is that we have no way to measure the photons energy other than using the local rest frames of the emitter (when it was there) and then the local frame of the receiver (when it gets there). Halc is almost right here.
My argument is that we can choose a frame of reference. I choose to measure a photons energy in the CMB frame always. Wherever we are in the universe we can always identify the CMB frame (just observe the CMBR and adjust your motion so that it is isotropic and then you are at rest in the CMB frame). We can repeat the reasoning (or sketch proof) given above where the space between emitter and receiver is expanding --> use the earlier proof as a blueprint with the new metric ds2 = -dt2 + a(t) δijdxidxj ---> We see that if the receiver is at rest in the CMB frame then the emitter has x-axis co-ordinates X +
d where a(t) is the scale factor. So the emitter cannot be co-moving unless a(t) ≡ constant. To say this another way, the emitter is not at rest in the CMB frame unless space is static (not expanding or contracting). The observers at the emitter must report the photons energy as it is measured in the CMB frame (because that's what I told them to do. I always choose to measure photons energy in the CMB frame). They will report that the photon had a higher energy (well, higher if space is expanding) when it was released then the receivers identify when they measure the photon some time later in the CMB frame.The argument pivots around the idea that a CMB frame can always be identified at every point in space. The original construction of the argument using red-shift is perhaps needlessly clumsy. All that's important is that there are intrinsic differences that can be measured when space is expanding compared to when it isn't. That photons lose energy (when measured in the CMB frame) while they're in expanding space is one such intrinsic difference.
On a practical level, we have to somehow get observers to the distant emitter with CMBR observation equipment and so it'll take few years to actually do this experiment. However, we are theorists not practical scientists. The intrinsic difference is there regardless of whether we can measure it today.
Another way you might prefer to see this is that Euclidean geometry fails to describe the situation if space is expanding. With Euclidean geometry, when one object is at rest in the CMB frame and another object is at a distance but has no velocity relative to the first then we expect the second object to also be at rest in the CMB frame. In expanding space we have non-Euclidean geometry and the expectation does not hold - one object can be at rest in the CMB frame but the other object is not despite it having constant proper distance from the first object.
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That's enough for now. I may address some other points later. Best Wishes to everyone.
Post by: Halc on 11/07/2021 17:53:05
I think we'll take this point first:Comoving only has meaning relative to an expanding metric, so no, it cannot be done relative to a static metric. A contracting metric is an expanding one with negative expansion rate.Quote from: HalcHow do you know if they’re comoving if you don’t know if space expands within a galaxy or not? Such a setup seems to require begging your answer before your measurement.We can establish if an observer is co-moving and we can do this independently of whether we assume space is expanding, contracting or static. In the real universe, the CMBR will be (almost) isotropic if an observer is co-moving.
The comment is interesting though. Let’s say we’re on zero-peculiar-velocity-Earth, except all the mass of Earth is a compressed into about a meter at the center and we’re on a thin glass shell at Earth radius, which is a smaller radius than the precision of our isotropy measuring device. Point is, same gravity, but we can see in all directions, even down. Does the observer on Earth on one side see the same CMBR as the other? No. I think the red/blueshift is the same for an observer anywhere on Earth, but the intensity depends on which direction you look. It is always brighter up than down since most of the light comes from that way. So the CMBR cannot be isotropic as viewed from space within a local gravity well. So it isn’t quite clear how you’re going to establish that you’re comoving.
Maybe you only consider the redshift and not the intensity. But if you do that, I think it will show that space within a galaxy (or a glass planet) is expanding since a pair of comoving (and properly accelerating) observers on opposite sides of the galaxy will be increasing their proper separation per the Hubble constant. I think you’ve been trying to say the opposite.
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Now the main point of disagreement between you (Halc) and me seems to be what causes a red-shift:I’d like to qualify that a bit better: It depends on the separation rate at time of emission. So imagine a universe where the expansion just happens to be stopped today but was expanding in the past and will big-crunch in another 13.8 BY. Distant galaxies will be redshifted because the light was emitted in the past when they were receding, despite the fact that those galaxies are currently not receding.
1. Halc's view:Quote from: HalcObserved redshift is due to increasing separation distance, not to a choice of coordinate system.
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That is a coordinate effect though, as you acknowledge below. But are you still asserting that my pebbles will appear redshifted to each other? They’re maintaining constant inertial separation distance (they’re adjacent to a rigid rod with markings on it every cm and not moving relative to the local rod). This is different than maintaining constant separation in an expanding metric which requires constant proper acceleration of at least one of the pebbles.QuoteIn my example with arbitrarily distant separated pebbles, light from either pebble as observed at the other will not appear redshifted.2. While I maintain that red-shift is caused by the properties of the space through which a photon has travelled. In particular, if space is expanding then photons lose energy.
I’ve recently worked out the numbers of a real example in another thread. Point is, I don’t care about a coordinate effect like “photons are losing energy”. I care about what is measured, and observers at each pebble will not measure redshift when viewing the other. Do you assert otherwise? The non-comoving pebble has a peculiar velocity inward, the Doppler blueshift of which exactly cancels the redshift from the photon losing energy.
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So the statement No.2 that I made above needs to be clarified: When I say that a photon loses energy I must specify the reference frame. I will choose to measure all photons in the CMB frame (more precisely, a photon has energy E iff an observer at rest in the CMB frame would measure energy E).Then I agree. An inertial rock similarly loses kinetic energy. All massive objects tend to approach zero peculiar velocity in the absence of forces accelerating them. The energy of an object (rock, photon) at a given time becomes a property of the object and not a relation to an another arbitrary object/frame.
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With this in mind, we have a method to identify if space is expanding in some region. If it is not expanding, then the metric is of the usual form for flat Minkowski space and Euclidean geometry rules the day.I never suggested that space wasn’t expanding in the pebble scenario. The example assumed everywhere linearly expanding space, and such space is equivalent to Minkowski space. The “in some region” part only applies to a real universe, not to the simplified pebble case. The real universe has mass and energy where we’re discussing whether or not local concentrations of that energy might empirically affect the local expansion rate. Your analysis of the no-expansion case assumed Minkowskian spacetime, which doesn’t apply to a region locally distorted by the mass of a galaxy.
The point of the pebble scenario was to demonstrate that there is no empirical difference between linearly expanding space and a static space with objects simply moving apart within it. The only hope of an empirical difference therefore lies in the fact that expansion in the real universe isn’t linear.
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Now, Halc's argument is that if the space between the emitter and receiver had been expanding then everyone will still report the same frequency for the photon. This is because the emitter must now have some velocity toward the receiver (measured as a co-ordinate velocity in the receivers local frame)Correction: measured as a peculiar velocity, which isn’t relative to anybody’s local frame. The emitter has zero velocity as measured by the receiver’s inertial frame.
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So there is a Doppler shift to be applied which will precisely cancel the red-shift of the photon as it moves through expanding space.Said Doppler shift only occurs due to nonzero peculiar velocity, but none due to zero inertial velocity. So the Doppler shift is entirely a coordinate effect in this special case. You’re changing coordinate systems furiously without being explicit about it, which is giving rise to confusion.
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The implication is that we have no way to measure the photons energy other than using the local rest frames of the emitter (when it was there)I implied no such thing. You say how to do it below.
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To say this another way, the emitter is not at rest in the CMB frame unless space is static (not expanding or contracting).Again, there is no one ‘the CMB frame’ since it is a local inertial frame relative to a specific location in space, making it meaningless without specification of said location. Just say the emitter has a peculiar velocity towards the receiver. The term ‘the CMB frame’ only has meaning to a species that has never been anywhere else but one location. It loses meaning in any scenario with observers in more than one place, which is what we’re doing here.
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They will report that the photon had a higher energy (well, higher if space is expanding) when it was released then the receivers identify when they measure the photon some time later in the CMB frame.Fine. We agree on that. So that energy degrades on the way to the receiver who measures the same thing regardless of coordinate system choice.
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The argument pivots around the idea that a CMB frame can always be identified at every point in space.In spacetime free of distortion (as with the pebble scenario), it can be. In spacetime distorted by a real galaxy, it is less clear if this can be locally determined, as discussed near the top of this post. The CMBR is never completely isotropic due to the local distortion, but at least the redshift can be isotropic, so we can go with that.
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The original construction of the argument using red-shift is perhaps needlessly clumsy. All that's important is that there are intrinsic differences that can be measured when space is expanding compared to when it isn't. That photons lose energy (when measured in the CMB frame) while they're in expanding space is one such intrinsic difference.OK, but that doesn’t make our observer at the comoving pebble reciever measure anything different given one coordinate system or the other. Looking at each other (a local test) tells them nothing. Only a non-local measurement (that of the CMB) would indicate that one of the pebbles has a peculiar velocity, but it also does not have an inertial velocity relative to the rigid rod that goes between them. I can make a photon lose half its energy in 5 minutes if I choose an appropriate reference event on which to base my hyperbolic coordinates, and the comoving coordinate, at least in the linear case, is nothing but hyperbolic coordinates in Minkowskian spacetime.
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On a practical level, we have to somehow get observers to the distant emitter with CMBR observation equipment and so it'll take few years to actually do this experiment.Well, for one, it’s a thought experiment on a forum, so practical considerations are moot. In reality, any assertion of (general consensus of) space empirically expanding or not within a galaxy has to be based on observations from only one point that have already been because we’ve never been anywhere else. Any non-local test is just conjecture.
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However, we are theorists not practical scientists. The intrinsic difference is there regardless of whether we can measure it today.I suggest otherwise, at least if the expansion is linear. Any intrinsic difference has to be do the non-linearity of the scalefactor, which is effectively immeasurable in the time it takes light to cross something small like a galaxy.
Another way you might prefer to see this is that Euclidean geometry fails to describe the situation if space is expanding.
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With Euclidean geometry, when one object is at rest in the CMB frame and another object is at a distance but has no velocity relative to the first then we expect the second object to also be at rest in the CMB frame.Nope. We expect the 2nd object to have a nonzero peculiar velocity since its unaccelerated worldline does not intersect the selected reference event. This is essential to our disagreement I think. I would hesitate to say ‘Euclidean geometry’. We’re talking Minkowskian geometry in which space is Euclidean, but spacetime is not. The frame rotations are different.
Post by: Eternal Student on 12/07/2021 17:14:39
Late editing: Crumbs, this is a long post. Sorry. Some stuff has been trimmed out and you could also skip to the summary at the end.
Comoving only has meaning relative to an expanding metric, so no, it cannot be done relative to a static metric.That's a disposable comment. A rectangle doesn't stop being a rectangle when it's a square. Yes, we could describe a square as something else like a Rhombus if we want to confuse everyone but we won't. We'll choose to describe a square with characteristics and parameters of rectangles.
Yes, technically I am setting everything in the context of a large scale model that is well approximated by spacetime with the FLRW metric. Where space is not expanding, we continue to look upon the metric in this form, just with a constant scale factor. So, an observer is co-moving <=> Observer has fixed spatial co-ordinates (in the co-ordinates of the metric) <=> They observe the CMBR isotropically.
You (Halc) made a comment later about a universe that expands, stops and then contracts again. A co-moving observer is well defined under the expansion or contraction and they don't stop being a co-moving observer when the expansion is zero.
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Comments about anisotropy in intensity:
Yes, this is fine and agreed. We are using wavelength. We're also idealising space as required so that nothing obstructs our view of the CMBR.
We've had previous discussion about (the lack of) isotropy in intensity before: (Link to another thread may be added later, when I find it. We discussed geodesically incomplete spacetime such as when a black hole stops an observer from seeing the CMBR beyond it).
You (Halc) also mentioned gravitational potential wells. This will influence redshift but we're going to keep your pebbles or my emitter and receiver identical in mass and radius, i.e. we're just going to idealise the situation and ignore gravitational redshift of this kind. However, if light was sent from a high density pebble (in open space) to a receiver on a low density pebble (in open space) then some additional redshift occurs. [Actually, the practical procedure of trying define a gravitational potential that applies globally (or atleast covers both emitter and receiver across expanding space) is not trivial and may not even be possible. We may prefer to treat the pebbles as "test particles", they have no mass of their own, they experience the metric in their region but do no influence it].
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My view of red-shift was stated as:
2. While I maintain that red-shift is caused by the properties of the space through which a photon has travelled. In particular, if space is expanding then photons lose energy.Halc's response and question was:
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That is a coordinate effect though, as you acknowledge below. But are you still asserting that my pebbles will appear redshifted to each other?Well, yes it is a co-ordinate effect, if you want to look upon it that way. However, the co-ordinates are not completely arbitrary. They can be identified or singled out by observations in the real universe (upto translations and spatial rotations). My observers can try to use a different frame but they will know if it is co-moving because they can observe the CMBR and check for isotropy. Having fixed the motion of the frame, defining lengths and times is just about standard physics and S.I. units. They can try to define their time in a strange way - but I'll tell them that in their CMB frame they have to define a second the same way the receiver does and give them an atomic clock. Similarly for lengths - I'll give them some light and tell them to use it sensibly with their atomic clock (release the light and let it travel for a fraction of a second etc.) Now, we're assuming that atomic clocks and light do behave identically everywhere and in every inertial frame - but that's standard physics and a topic for another thread if we want to discuss that. (Note: I mentioned earlier that your pebbles will be test particles and for simplicity I am placing them in resonably open space, far enough away from a significant gravitational source. In this way "inertial frames" just means the usual thing as in Special Relativity).
This is important, so I'm going to say it another way: Co-moving co-ordinates are not arbitrary or completely abstract. We can construct a local frame that has many of the properties of the co-ordinate system that is used in the FLRW metric and large-scale models of FRW universes. At any point in open space (away from gravitational sources) we can identify a local inertial frame such that an observer remaining at the origin of our frame will observe the CMBR isotropically. This is the local CMB frame. For an observer remaining at the origin of this CMB frame all of the following hold:
(i) The observer has fixed spatial co-ordinates in both co-ordinate systems. [It's just (0,0,0) in the local system and (A,B,C) with fixed A,B,C in the co-moving co-ordinates because the observer is co-moving ].
(ii) Local co-ordinate time, t and universe-wide co-moving time, T show no dilation (they pass at the same rate)
Late editing: Maths trimmed from points (iii) and (iv) below. Not enough time to double-check. Errors are possible - ES
(iii) Local spatial co-ordinates (x,y,z) and the corresponding co-moving spatial co-ordinates (X,Y,Z) are closely related to each other. The local co-ordinate system (x,y,z) is arbitrarily orientated - there's nothing special about the x-direction compared to any other direction. There is one orientation of the spatial co-ordinates such that the following holds: x is a function only of the corresponding co-moving co-ordinate X; similarly y = y(Y) and z = z(Z). Furthermore, they are directly proportional to each other if we stay local to the origin of our CMB frame. For small x, y, and z we have x ≈ aX ; y ≈ aY and z ≈aZ where a = a(T) is the scale factor that appears in the FLRW metric.
(iv) We can improve the correspondance between the local co-ordinate system and the co-moving system. The scale factor is quite arbitrary in most models, we're usually only concerned with the ratio of two scale factors at different times. We can set a(time=now) to be 1, so that we have x ≈ X etc. for times close to "now". Then, locally (in both space and time) our CMB frame is a good representation of the co-moving co-ordinate system with a(now) = 1. (We can do a similar trick by insisting distances in both the co-moving co-ordinates and local co-ordinates are measured in metres - defined as distance along a null path over a fraction of a second. This is slightly more complicated to explain and everyone has stopped reading already and more Maths isn't going to bring the audience back).
OK, what doesn't match up bewteen our local co-ordinate system and the co-moving co-ordinate system is essentially translations in space and time and also rotations of the spatial co-ordinates. Looking back to (i) above, the origin was (0,0,0) in the local system but this is just some arbitrary point (A,B,C) in the co-moving co-ordinates. Also (ii) tells us that the time co-ordinates progress at the same rate but there is nothing to stop a fixed translation t = T + c. Indeed typically we set t=0 at the instant we started to use that local frame while we have set T=0 at the instant of the big bang. Meanwhile (iii) makes it clear that spatial rotations are arbitrary.
Fortunately, we don't need to worry about this. A photon's energy is independant of translations and spatial rotations applied to the frame of reference in which you measure it.
OK, I think that just about covers things. In answer to your (Halc) question: Yes, photons from your emitter pebble will lose energy while they're in expanding space and co-ordinate time passes. The receiver pebble will identify a redshift (provided the photon's energy was measured in the local CMB frames at both emitter and the receiver). You (Halc) made this comment:
I don’t care about a coordinate effect like “photons are losing energy”. I care about what is measured,The main point I would assert is that it isn't just an artifiact from "arbitrarily choosen" co-ordinates. The CMBR allows us to measure something objectively and fixes enough properties of our local frames (the CMB frame centred at the emitter and also the CMB frame centred at the receiver) to be objective about how the photon's energy will be measured.
If we coulld not identify a CMB frame, then we would have to use the local rest frames of the emitter and then the local rest frame of the receiver. Exactly as you (Halc) stated, measuring a photon's energy in the local rest frames, the emitter and receiver would report the same frequency. They have no way of knowing if space was Minkowski or if it was expanding. Since, if it was expanding the emitter had some peculiar velocity and then a blue-shift Doppler effect would have applied which would precisely cancel the red-shift from the expansion of space.
That's how it is (I think). Using local rest frames at both emitter and receiver, there is no way to know if space is expanding. But since we can use a CMB frame at emitter and receiver there will be a difference we can measure.
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I'm not sure that this comment:
This is different than maintaining constant separation in an expanding metric which requires constant proper acceleration of at least one of the pebbles.is consistent with this comment made by you (Halc) in a earlier post:
Given perfect linear expansion (a linear scalefactor), two objects (say a pair of pebbles a considerable distance apart) that are stationary relative to each other will remain at a constant proper separation forever in the absence of external forces
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Let's go back and consider something from a much earlier post. Where I suggested that expansion doesn't have to be happening in dense regions of space like galaxies:
Quote from ES was:
A clearer explanation is simply that on the scales of galaxies the cosmological principle does not hold, even approximately, and the FRW metric is not valid. The metric of space-time in the region of a galaxy (if it could be calculated) would look much more Schwarzchildian than FRW like, though the true metric would be some kind of chimera of both. There is no expansion for the galaxy to over-come, since the metric of the local universe has already been altered by the presence of the mass of the galaxy. ..... The expansion of space is global but not universal, since we know the FRW metric is only a large scale approximation.
[Section 2.6.3; page 7; Expanding Space: the Root of all Evil?; Francis, Barnes, James & Lewis, https://arxiv.org/pdf/0707.0380.pdf - Later accepted for publication in Publications of the Astronomical Society of Australia]
Response from Halc was:
That just says that spacetime is neither flat nor homogeneous within a galaxy. It doesn’t say that space isn’t expanding there, only that the metric (which applies to flat homogeneous space) doesn’t describe what is a ‘local digression’ any more than the metric describes the curvature of spacetime here on Earth.
I just can't see how you (Halc) can interpret that quote (from the paper of Francis et.al) in this way.
They say: There is no expansion for the galaxy to over-come, since the metric of the local universe has already been altered by the presence of the mass of the galaxy.
You (Halc) say: It doesn’t say that space isn’t expanding there... only stuff about the metric
I can only say: The metric is all about the expansion.
Expansion just means that the metric has the right general form so that a term called the scale factor a(t) appears in the metric as a multiplier for the spatial co-ordinates. If the metric doesn't have the right form and we can't identify this term, a(t) then there is no "expansion" that can be sensibly identified or associated with the behaviour of spacetime in that region. For example, if the metric inside a galaxy was more like the Schwarzschild metric then "expansion" isn't easily identified, there are other strange effects (e.g. the role of a radial co-ordinate and the time co-ordinate completely switch over inside the Schwarzschild radius and we can imagine that space is being dragged into the central gravitating source) but nothing like conventional "expansion". For example, time slices through the spacetime don't look like 3-D Euclidean space with an expansion as time progresses.
In later posts you (Halc) made this comment:
Your analysis of the no-expansion case assumed Minkowskian spacetime, which doesn’t apply to a region locally distorted by the mass of a galaxy.Which is almost exactly the point that Francis et.al were making. In dense regions of space such as inside a galaxy, the metric is not well approximated by the FLRW metric, so "expansion" doesn't have to be happening there and it cannot be identified by looking at the metric that applies in that local region.
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About the CMB frame
Again, there is no one ‘the CMB frame’ since it is a local inertial frame relative to a specific location in space, making it meaningless without specification of said location. Just say the emitter has a peculiar velocity towards the receiver. The term ‘the CMB frame’ only has meaning to a species that has never been anywhere else but one location. It loses meaning in any scenario with observers in more than one place, which is what we’re doing here.Yes, I agree. I was there in the other thread where this was recently discussed (Might add the link later, if I find the original thread). At that time I was the one wanting to make it clear that a CMB frame is specific to a location and there isn't one universal CMB frame. I have tried to be careful to say "the CMB frame" and tacitly brush over the issue because it confused enough people already. Where I have said "the CMB frame" I am referring to a whole class of different inertial frames, one such frame existing at each point in space.
To be clear then, where I have said "I choose to measure the photon in the CMB frame always" I did mean that we choose to use the CMB frame centred at the point where the photon is currently located. Thus, the emitter pebble measures in their CMB frame and the receiver pebble measures in their CMB frame. This doesn't affect the general idea, the CMB frames did not have to be the same, they just had to be CMB frames.
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Then there's this section:
Quote from ES was:Which is not right. However, it comes down to the use of the phrase "Euclidean geometry" which you have picked up on. Some people do say "Euclidean" to mean "Minkowskian" or that the spacetime has a standard "Lorentzian" metric and I am one of those people.
With Euclidean geometry, when one object is at rest in the CMB frame and another object is at a distance but has no velocity relative to the first then we expect the second object to also be at rest in the CMB frame.
Response from Halc was:
Nope. We expect the 2nd object to have a nonzero peculiar velocity since its unaccelerated worldline does not intersect the selected reference event. This is essential to our disagreement I think. I would hesitate to say ‘Euclidean geometry’. We’re talking Minkowskian geometry in which space is Euclidean, but spacetime is not. The frame rotations are different.
In standard Minkowski space (which is a spacetime with the FLRW metric but scale factor set to 1 at all times), we are in a very fortunate position. The class of CMB frames as discussed above (each CMB frame being specific to one point in space) become unusually degenerate (in a good way). Specifically, the CMB frame at a point P really is just a translation (and possibly rotation) of the CMB frame centred at some other point Q. The geometry of the spacetime is straightforward. There is no difficulty extending a local frame across the entire spacetime to become a global (or universal) reference frame. Motion (velocity) at a point P can be determined in the CMB frame centred at P, in the co-moving co-ordinates of the entire universe, or in the (suitably extended) CMB frame located at Q. They will all be the same (upto rotations of the axis). The metric is so straightforward that constant separation between the two objects is precisely enough to know that both will be at rest in their own CMB frame if any one of them is.
You mentioned rotations and you're right, the CMB frames at different points could be rotations of each other. However, for objects that are "at rest" in the CMB frame, the velocity is 0 which remains 0 no matter how you rotate the axis. You mentioned that the worldines don't intersect which is also correct but that is achieved because the CMB frames are fixed translations of each other.
[Late editing: You may have meant 4-D rotations are possible as opposed to just 3-D rotations. For Minkowski spacetime, all the time axis in every CMB frame and the co-moving co-ordinate frame are aligned. The greatest difference between them could only be a translation. There will only be rotations of the spatial axis ]
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Summary
I've enjoyed reading what everyone has said here and writing some stuff of my own. Thank you to everyone.
I maintain these views:
1. Space doesn't have to be expanding inside a galaxy. There are at least two good reasons to think that it isn't:
(i) The FLRW metric is not a good approximation to the metric of space in a dense region.
(ii) Using the FRW universe models and the Friedmann equations we can see that increasing matter density causes decelleration of expansion. We can naively assume that what happens on a universe-wide scale should also happen locally.
2. There are ways that we can "know" if space is expanding inside a galaxy but they'll take a few years to actually do.
I adjust my position on this issue:
3. Having worked through the issues and based on comments from others. I acknowledge that using red-shift from emitters and receivers or two pebbles is not the easiest way to determine if space in a galaxy is expanding.
Post by: Halc on 13/07/2021 15:03:46
Crumbs, this is a long post. Sorry. Some stuff has been trimmed outFor long posts, I try to make my replies shorter than the post to which I’m replying, but I failed on my last one. Guess that means the conversation is interesting.
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OK, I’ll grant that. ‘Comoving’ is defined as a measurement and thus is a relationship with the material all around you and isn’t really a coordinate system (CS) dependent thing as my comment implies. My house is always comoving with the ground under it, and thus is always stationary relative to that ground regardless of any alternate choice of CS is made, and I can be doing physics in the alternate CS but still measure if I’m comoving with the nearby ground.Quote from: HalcComoving only has meaning relative to an expanding metric, so no, it cannot be done relative to a static metric.That's a disposable comment. A rectangle doesn't stop being a rectangle when it's a square.
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We discussed geodesically incomplete spacetime such as when a black hole stops an observer from seeing the CMBR beyond itBut it doesn’t. You see the entire CMBR (and more!) from an accelerated position near a black hole. There’s no obstruction despite the fact that all the light comes from the direction in which you’re accelerating. Light from the other side bends around and finds you. It can only be blocked by physical obstruction like the Earth under you.
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You (Halc) also mentioned gravitational potential wells. This will influence redshift but we're going to keep your pebbles or my emitter and receiver identical in mass and radius, i.e. we're just going to idealise the situation and ignore gravitational redshift of this kind.I used pebbles in an effort to minimize mass, because they’re supposedly in a zero-energy solution scenario, so they must be arbitrarily small.
Our two obervation points on opposite sides of a galaxy do need to be at identical potentials. That scenario was the real universe, not the Minkowskian one.
You don’t need to say “You (Halc)”. You’re responding to my quotes. Everybody knows who “You” is in this context. Second-person sentence structure is appropriate, but you switch to 3rd person sometimes, I suppose to address the perceived audience.
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My view of red-shift was stated as:It is critical to my point, so yes, I want to look at it both ways.
I maintain that red-shift is caused by the properties of the space through which a photon has travelled. In particular, if space is expanding then photons lose energy.
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Yes [that] is a co-ordinate effect, if you want to look upon it that way.
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However, the co-ordinates are not completely arbitrary. They can be identified or singled out by observations in the real universe (upto translations and spatial rotations). My observers can try to use a different frame but they will know if it is co-moving because they can observe the CMBR and check for isotropy.No argument. I’m just saying light doesn’t slowly redshift relative to an inertial frame like the one in which both pebbles are stationary. The redshift only occurs relative to the expanding CS, that being a property of a hyperbolic CS like that.
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This is important, so I'm going to say it another way: Co-moving co-ordinates are not arbitrary or completely abstract.Agree, but the rotating frame in which my house is stationary is also not completely abstract. I don’t mean to trivialize it with that comment. Comoving coordiantes are universe whereas the frame of my house is not. I can think of no other CS that foliates most of the (real) universe like that, but any arbitrary inertial frame does foliate the entire zero-energy universe in which we put the pebbles, even events not foliated by the comoving CS.
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We can construct a local frame that has many of the properties of the co-ordinate system that is used in the FLRW metric and large-scale models of FRW universes.I thought the coordinate system used by the FLRW metric was not local at all. I was unaware that the FLRW metric referenced a local CS at all except in the usual GR way that says spacetime is locally Minkowskian except at a physical singularity.
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At any point in open space (away from gravitational sources) we can identify a local inertial frame such that an observer remaining at the origin of our frame will observe the CMBR isotropically. This is the local CMB frame.Some choose the term “CMB frame” to mean the universal comoving coordinate system, which is why I balk at the term. Of course “local CMB frame” implies a local inertial frame, but the adjective is usually left off, leaving the reader to wonder if we’re talking about the comoving frame or a local inertial one. You cannot speak of a local frame when discussing expansion across a galaxy since it is exactly the divergence from locality that we’re trying to measure.
On the other hand, in the zero energy scenario, all inertial frames are indefinite sized, and there is no ‘local’ about it. Your discussion that follows seems to reference the zero-energy scenario since you’re specifying “away from gravitational sources” which is impossible except in the absence of gravitational sources.
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For an observer remaining at the origin of this CMB frame all of the following hold:Hence the importance of the absence of gravitational sources since even a perfectly uniform distribution of matter will sink your local clock deep into a gravity well. The only way universe-wide co-moving time is then meaningful is if it is defined as the time on a clock at average gravitational depth, but that average depth keeps changing as the density drops with expansion. Fun stuff.
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(ii) Local co-ordinate time, t and universe-wide co-moving time, T show no dilation (they pass at the same rate)
Anyway, I’ll grant this point, lest our goal become out of reach.
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(iv) We can improve the correspondance between the local co-ordinate system and the co-moving system. The scale factor is quite arbitrary in most models, we're usually only concerned with the ratio of two scale factors at different times. We can set a(time=now) to be 1, so that we have x ≈ X etc. for times close to "now". Then, locally (in both space and time) our CMB frame is a good representation of the co-moving co-ordinate system with a(now) = 1. (We can do a similar trick by insisting distances in both the co-moving co-ordinates and local co-ordinates are measured in metres - defined as distance along a null path over a fraction of a second. This is slightly more complicated to explain and everyone has stopped reading already and more Maths isn't going to bring the audience back).Can only do that locally. Distances, even without time progressing, are very different in the two coordinate systems, but they do match locally. This is why the radius of the visible universe can be ~48 BLY today whereas in a universe where an inertial frame applies, it could only have a radius of 13.8 BLY if that’s what T is. Of course, in the zero-energy solution, the size of the visible universe isn’t limited to 48 BLY. There are no event horizons.
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Exactly as you stated, measuring a photon's energy in the local rest frames, the emitter and receiver would report the same frequency.Good. Now I’ll add that it is not necessarily the case in the real universe, only our special (pebble) one with no gravity involved.
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They have no way of knowing if space was Minkowski or if it was expanding.Just a coordinate difference, so there’s no physical difference between the two. And they can always look at the CMB which gets you a peculiar-velocity meter for the one CS and gets you a location of the reference event for the Minkowskian CS. But space isn’t one or the other in the special case since the difference is purely abstract. In the real case, there is gravity and such so of course Minkowski spacetime does not describe the universe.
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I'm not sure that [these comments are consistent]:Good catch. 2nd comment is a crude approximation and thus wrong. The latter comment is true only in an inertial CS. Those same points are increasing their proper separation in the expanding CS, so in order to keep that distance constant in that CS, proper acceleration is needed outward to slow the inbound inertial velocity required. That is what the first comment references.Quote from: HalcThis is different than maintaining constant separation in an expanding metric which requires constant proper acceleration of at least one of the pebbles.Quote from: HalcGiven perfect linear expansion (a linear scalefactor), two objects (say a pair of pebbles a considerable distance apart) that are stationary relative to each other will remain at a constant proper separation forever in the absence of external forces
So to word it another way, using the expanding CS: You have one pebble stationary and a 2nd one with a peculiar velocity towards the first, maintaining a fixed distance from it at time T. In the absence of acceleration outward or any other forces acting on it, the 2nd pebble will reach the first eventually. Hence the necessity of outward proper acceleration to keep it at that fixed distance.
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- - - - -I’d rather consider what can be measured, and not what coordinate system is best chosen to describe a local collection of mass.
I suggested that expansion doesn't have to be happening in dense regions of space like galaxies:
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I just can't see how you (Halc) can interpret that quote (from the paper of Francis et.al) in this way.
Put a comoving observer on each side of the galaxy, and a tape measure between them. Are they moving apart, together, or staying essentially at constant distance? If the latter, then I’d accept that expansion is absent in a galaxy, but I think I can demonstrate the first case.
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They say: There is no expansion for the galaxy to over-come, since the metric of the local universe has already been altered by the presence of the mass of the galaxy.Keep in mind that I consider ‘expanding’ to be a coordinate effect, so whether space is expanding is nothing but an abstract choice. The paper says there’s no reason to choose the expanding CS.
You (Halc) say: It doesn’t say that space isn’t expanding there... only stuff about the metric
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I can only say: The metric is all about the expansion.The metric is an abstract choice. Or so I suggest at least. Hence I think an empirical test would server better than a choice of one’s favorite CS. My whole point with the pebbles is to demonstrate that expansion or not is just a coordinate difference, not a physical one, so long as scalefactor is linear. Hence the CS choice is arbitrary.
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Where I have said "the CMB frame" I am referring to a whole class of different inertial frames, one such frame existing at each point in space.That would make it more or less the way I use the phrase. Such a frame is the cosmological frame, or comoving frame, and it isn’t inertial at all, so doesn’t have the usual inertial properties.
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Thus, the emitter pebble measures in their CMB frameI had the emitter measure it in the emitter frame. If I have a laser that puts out frequency F, then it will be measured at F in the emitter’s inertial frame regardless of its peculiar motion. Anything else is a calculation, not a direct measurement. Let’s at least be explicit with the frame references so we’re at least clear.
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Then there's this section:Fine. I’m not. Euclidean spacetime does not have invariant intervals for instance. I know people use the word differently, so I’d rather we just kept away from depending on the term and one’s local interpretation of it.Quote from: HalcWhich is not right. However, it comes down to the use of the phrase "Euclidean geometry" which you have picked up on. Some people do say "Euclidean" to mean "Minkowskian" or that the spacetime has a standard "Lorentzian" metric and I am one of those people.QuoteWith Euclidean geometry, when one object is at rest in the CMB frame and another object is at a distance but has no velocity relative to the first then we expect the second object to also be at rest in the CMB frame.Nope. We expect the 2nd object to have a nonzero peculiar velocity since its unaccelerated worldline does not intersect the selected reference event. This is essential to our disagreement I think. I would hesitate to say ‘Euclidean geometry’. We’re talking Minkowskian geometry in which space is Euclidean, but spacetime is not. The frame rotations are different.
Perhaps my disagreement stems from the ambiguous reference to “the CMB frame” used in your comment. Is it the CMB frame of the 1st or 2nd object? See why I don’t like the term?
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SummaryIncreasing matter density of the whole universe causes deceleration of the whole universe. The equations don’t imply that local collections of matter, emptying out nearby regions to do so, has any effect on the local expansion. So I have one better reason to think that it is expanding in a galaxy:
I maintain these views:
1. Space doesn't have to be expanding inside a galaxy. There are at least two good reasons to think that it isn't:
(i) The FLRW metric is not a good approximation to the metric of space in a dense region.
(ii) Using the FRW universe models and the Friedmann equations we can see that increasing matter density causes decelleration of expansion. We can naively assume that what happens on a universe-wide scale should also happen locally.
(iii) Two comoving observers on opposite sides of the galaxy will find themselves receding from each other. If they do, then it demonstrates local-density-independent expansion. Each views the CMB frequency as isotropic, so there is no arbitrary selection of equations going on. I haven’t demonstrated this, but I can. If a measuring rod is already in place, they can quickly view their motion relative to it.
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2. There are ways that we can "know" if space is expanding inside a galaxy but they'll take a few years to actually do.Just a few years? I was thinking 6+ digits of years if any signals need to be passed. Want to do it quicker? Get a smaller lab and more precise instruments. Hence my planet that was a 1 meter dense ball surrounded by a radius 5000 km glass shell.
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I adjust my position on this issue:There was no galaxy in the pebble example. The scenario was meant to illustrate something else.
3. Having worked through the issues and based on comments from others. I acknowledge that using red-shift from emitters and receivers or two pebbles is not the easiest way to determine if space in a galaxy is expanding.
Post by: Eternal Student on 14/07/2021 01:05:28
There's nothing wrong with short replies. There are several other threads that need more urgent attention.
Bye for now and best wishes to you.
LATE EDITING: I've had even more time to read through what you've said Halc. There are a few points I could pick up on but I'm thinking that the thread has drifted off topic from the Original Post quite a lot already. So I was going to leave the thread alone now.
The argument about whether expansion is happening inside a galaxy or mainly in the voids of space between galaxies (along with exactly what difference would appear in a co-ordinate independent measurement) still seems to be a frequent topic of discussion. I've seen a couple of related threads appearing in some other forums (I won't name those other forums here) in the last week. There are pundits (experts and/or self-proclaimed experts) making several different statements.
Post by: Europa on 19/07/2021 01:57:27
Am I correct in thinking that there is space, and into this space has emerged our universe? And through this space the mass of our universe is expanding? Warping and rippling this space as it does so?There is insufficient data to define what the universe is
Post by: yor_on on 21/07/2021 14:25:09
Post by: yor_on on 21/07/2021 14:28:03
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If you want it even more complicated you can add a inflation and a accelerating expansion to it.
Post by: TommyJ on 28/07/2021 15:54:53
Speaking of the Universe size and expansion/infinity/beyond concepts.
The observation starting point is defined as unknown (although all researches are built on the Big Bang). That’s why the main focus is understanding what is going on and what is crucial to humanity’s nearest proximity.
Now observed expansion is increasing speed, but there are 3 options: it will stop at a certain balanced point, slow down but never stop (infinitely), start to collapse back.
Existing observations have several approaches.
Including it’s expansion as a sphere or a flat matter expansion. In theory gravity should slow down this expansion (this is where dark matter and its mass matters).
Spherical theory tells us geography, geometry and the evident conclusion, that no matter how long you travel on the sphere surface, you will never find an ending obstacle.