Is the concept of true distance compatible with relativity?

[PAllen]

Now I'm not a physicist and might get some slamming for this, so please correct me if this is wrong - but I was once told to think of the expansion of space as an increase in distances between objects without movement.

This has been written by some cosmologists, but it is not remotely justifiable in the math of GR. Given two 4-velocities of of separated bodies, there is nothing in the math of GR that distinguishes 'relative motion' from 'increase of distance without motion'. Nothing at all.

That helped me a lot because it delivers an easily understandable explanation for why special relativity does not apply. There is nothing moving apart faster than the speed of light, just distances increasing - and for particles, stars or galaxies separated by a very large distance the increase can be more than 300.000 km/s.

This also a common but strictly false statement. This is shown by the argument I gave earlier where maximal superluminal recession rate occurs in the special relativity limit of cosmological models. In particular, it shows that recession rate corresponds the special relativity quantity called celerity not relative velocity. In special relativity, celerity has no upper bound at all - it can be a thousand times c.

It kind of helped me to understand what goes on in inflation also. Nothing moves. No inertia. Just distances suddenly grow enormously.

Henrik

Inflation is a separate model than generic big bang cosmology (FLRW solutions in GR). In classical GR, inflation may be modeled by an initial, large, cosmological constant that then decreases to almost zero. However, viewed in classical GR terms, it remains true that there is nothing you can identify about two separated 4-velocities to say they do not represent motion.

 

[PeterDonis]

I am finding it difficult to understand how a spatial 'separation' between two objects can be coordinate dependent.

Because "space" itself is coordinate dependent.

Are there any simply examples you could point me to that show where this is the case?

Specific examples are going to be hard to find, because for practical purposes physicists like to choose coordinates that are convenient to use. If we have a congruence of worldlines with a positive expansion scalar (for example, the congruence of "comoving" worldlines in expanding FRW spacetime), it's much more convenient to choose coordinates in which the spatial distance between comoving objects increases. But "convenient" is not a statement about the physics; it's a statement about us humans and how we choose to model the physics. If you want to focus on the physics, the best thing is to forget about coordinates and look at invariants, such as the expansion scalar.

I sometimes generalize reheating as part of the inflation process.

That might not be a good idea, if only because it is likely to cause confusion since it is not standard terminology (at least, not as I understand it).

The problem I have conceptually with this explanation, taking the 2 sphere as an example, is that implies an extra dimension for the sphere to 'grow' in to.

No, it doesn't. The extra dimension helps us to visualize what is going on (at least in the 2-sphere case). But it is not logically required by the model.

I was wondering if it were possible to model the movement of matter in an expanding universe just in a 3 spatial dimensional context?

I'm still not understanding what you are trying to say here. The standard 4-dimensional spacetime model of the universe, including all movement of matter in it, does have 3 spatial dimensions. But you seem to be talking about something different from the standard 4-dimensional spacetime model. So I'm confused about what you mean.

 

[Hernik]

This has been written by some cosmologists, but it is not remotely justifiable in the math of GR. Given two 4-velocities of of separated bodies, there is nothing in the math of GR that distinguishes 'relative motion' from 'increase of distance without motion'. Nothing at all.

But in this link, provided by Chronos in post 12 as a paper that debunks misconceptions on the expansion of space: http://arxiv.org/abs/astro-ph/0310808 the following sentences page 5 seem excactly to claim that expansion is simply distances changing and not a process initially involving motion:

"The general relativistic interpretation of the expansion interprets cosmological redshifts as an indication of velocity since the proper distance between comoving objects increases. However, the velocity is due to the rate of expansion of space, not movement through space, and therefore cannot be calculated with the special relativistic Doppler shift formula."

Of course the result is relative motion, I understand that - but I understand from the text that objects in space are not moved away from each other by the expansion of space. Distances only grow. Where do I go wrong when I read that text?

Henrik

 

[PAllen]

But in this link, provided by Chronos in post 12 as a paper that debunks misconceptions on the expansion of space: http://arxiv.org/abs/astro-ph/0310808 the following sentences page 5 seem excactly to claim that expansion is simply distances changing and not a process initially involving motion:

This paper pushes a particular interpretation of the math that is not at all universally shared. As I noted in an earlier post, one of its co-authors distanced herself from this interpretation later (Tamara Davis). Steven Weinberg has also argued against this interpretation (even calling it the root of evil). I do not claim this interpretation is wrong (that is a category error unless an interpretation is internally inconsistent); what is wrong is not to recognize that it is just one interpretation, and is not mandated by the math of GR. In particular, this interpretation gives meaning to coordinate dependent quantities (technically, foliation dependent quantities). Since a key feature of GR is coordinate invariance, it is an error to overvalue such coordinate dependent quantities, let alone claim they are the 'truth'.

"The general relativistic interpretation of the expansion interprets cosmological redshifts as an indication of velocity since the proper distance between comoving objects increases. However, the velocity is due to the rate of expansion of space, not movement through space, and therefore cannot be calculated with the special relativistic Doppler shift formula."

The objectively true part of this statement is that the pure SR Doppler formula does not apply to distant objects in GR (this statement is true in general, not just for cosmology). However, GR generalizations of Doppler expressed in terms of invariant procedures do not distinguish two sources of Doppler. There are several formulations of general redshift in GR; none involve expansion of space in the expression. One example is to parallel transport one 4-velocity to the other along the null geodesic connecting the emission event and the reception event, and then apply the SR doppler formula locally. This procedure applies universally in GR. Note the complete absence of factoring into different causes. Note also, that the SR doppler formula is involved, but not just the SR doppler - the parallel transport on the null geodesic is a required feature of this generalization to GR, bringing in the effects of curvature. Another universal procedure is to follow two very nearby null geodesics from one world line to another, and find the ratio of the proper time difference. Again, no factoring into separate causes appears in this universal generalization of SR doppler.

Factoring redshift into different components requires a choice of coordinates, and is different for different choices.

Of course the result is relative motion, I understand that - but I understand from the text that objects in space are not moved away from each other by the expansion of space. Distances only grow. Where do I go wrong when I read that text?

Henrik

The error is by the authors in overstating their case.

 

[PAllen]

This has been written by some cosmologists, but it is not remotely justifiable in the math of GR. Given two 4-velocities of of separated bodies, there is nothing in the math of GR that distinguishes 'relative motion' from 'increase of distance without motion'. Nothing at all.

This quote of mine would more precisely be "nothing in the math of GR except choice of coordinates + interpretation". That is, there is no invariant in GR defining 'expansion of space'. Given a congruence of world lines (e.g. the comoving congruence, in the case of cosmology), there is an invariant definition of the expansion of that congruence. However, the steps from here to the notion of expanding space are:
- pick a coordinate system where spatial position is defined by this congruence (typically with other desirable properties, if possible, e.g. hypersurface orthogonality)
- Name the growth of distance between members of the congruence, in these coordinates, expansion of space.

The last step, in particular, is not part of the math or normal use of GR. It is an interpretation added to GR by many cosmologists. This is basis of my statement "nothing in the math of GR makes the claimed distinction". Some find this added interpretation useful, some do not.

[edit: to emphasize yet again, the conventionality of this, you can do this in flat, static, Minkowski space with the Milne congruence of inertial world lines. You then end up claiming extreme superluminal expansion of space for a static flat manifold.]

 

[rede96]

A 3-d analog of the balloon analogy is the raisin bread analogy. As cooking loaf rises, the distance between all the raisins increases isotropically and homogeneously (in the ideal) except near the edges. If you imagine there is no edge, you have a model of 3-d expanding congruence.

I'm still not understanding what you are trying to say here. The standard 4-dimensional spacetime model of the universe, including all movement of matter in it, does have 3 spatial dimensions. But you seem to be talking about something different from the standard 4-dimensional spacetime model. So I'm confused about what you mean.

Sorry for the confusion and not explaining this very well. In essence the balloon analogy or 2 sphere just never sat right with me personally. Partly because it implies a bound universe but mainly because in order for 'galaxies' to move apart on the surface of the balloon, the 'balloon' or sphere itself must grow into a dimension the wouldn't exist in just a two dimensional space. So mathematically, if I extend the 2 sphere to a 3 sphere, I get the same problem.

So instead of the surface of a balloon being used it it would make more sense to me to have a flat 2 dimensional area and just show everything moving away from everything else. Or for 3 dimensional space (ignoring the time dimension) the raisin bread analogy form PAllen above (thank you by the way) seems to make more sense to me. If I just imagine how the raisins move, ignoring the bread and edges etc, then this is how I imagined matter to be moving apart.

The only problem with thinking about it in this way is, unless the universe is infinite, then I can collapse the raisin bread example to a central point. Using the balloon analogy that doesn't happen.

 

[rede96]

Because "space" itself is coordinate dependent.

I'm probably looking at this in a very elementary way in that coordinates are only there for us to make measurements. So space is still space and does what it does independently of how we measure it. For example if I take two objects which are touching, then start them moving apart in such a way that they will always be moving apart, then I would have thought if one frame of reference can measure they are moving apart, all frames of reference will measure they are moving apart. They may measure different distances or different rates, but no one will say they measured them to be moving together or static wrt each other.

So relating that to expansion, I can imagine different coordinate systems will tell us different things, but I would thought measuring increasing distances is absolute. Is that not the case?

 

[PAllen]

I'm probably looking at this in a very elementary way in that coordinates are only there for us to make measurements. So space is still space and does what it does independently of how we measure it. For example if I take two objects which are touching, then start them moving apart in such a way that they will always be moving apart, then I would have thought if one frame of reference can measure they are moving apart, all frames of reference will measure they are moving apart. They may measure different distances or different rates, but no one will say they measured them to be moving together or static wrt each other.

So relating that to expansion, I can imagine different coordinate systems will tell us different things, but I would thought measuring increasing distances is absolute. Is that not the case?

Touching is invariant. Moving apart or together is coordinate dependent. This could even happen in SR. Consider two rockets accelerating uniformly, consistent with the Rindler congruence, such that each perceives their mutual distance constant (possibly surprisingly, this means the front rocket's proper acceleration must be slightly smaller). In an inertial frame, these rockets appear to be approaching each other rather than static. If you then have the front rocket accelerate a tiny bit more, but still less than the back, each rocket's standard accelerating frame would have them separating. However, an inertial frame would have them approaching. Each of these coordinate statements corresponds to the most natural measurements that may be made by the corresponding observer.

[edit: To add what you can say about this from an invariant perspective: if you treat the rockets that are receding in their frames, while approaching in the inertial frame, as part of a congruence, using the simplest way to fill in the congruence, you would find that the congruence has positive expansion scalar. While expansion of the congruence is invariant, the remarkable fact remains that in this case, as natural coordinates as Minkowski coordinates treat the world lines of the congruence as approaching each other. ]

 

[rede96]

Touching is invariant. Moving apart or together is coordinate dependent. This could even happen in SR. Consider two rockets accelerating uniformly, consistent with the Rindler congruence, such that each perceives their mutual distance constant (possibly surprisingly, this means the front rocket's proper acceleration must be slightly smaller). In an inertial frame, these rockets appear to be approaching each other rather than static. If you then have the front rocket accelerate a tiny bit more, but still less than the back, each rocket's standard accelerating frame would have them separating. However, an inertial frame would have them approaching. Each of these coordinate statements corresponds to the most natural measurements that may be made by the corresponding observer.

I'm still not sure I understand this. Imagine we tie a rope taut between the two rocket ships as they accelerated. The rope can only do one of three things. It can stay taut, it can become loose or it can stretch and snap. This is not coordinate dependent but still decribes how the distance between the two rocket ships changes.

 

[PAllen]

I'm still not sure I understand this. Imagine we tie a rope taut between the two rocket ships as they accelerated. The rope can only do one of three things. It can stay taut, it can become loose or it can stretch and snap. This is not coordinate dependent but still decribes how the distance between the two rocket ships changes.

Your question anticipates my addendum above. This is basically a way to measure expansion scalar in the case of a simple one parameter congruence.

 

[rede96]

Your question anticipates my addendum above. This is basically a way to measure expansion scalar in the case of a simple one parameter congruence.

I'll be honest I don't fully understand the terminology you are using in your Edit, so will need to do a bit of reading. But I take it you agree that the rockets can't both be receding and approaching simultaneously. In the same way that galaxies can't be both receding and approaching simultaneously. So how the actual distance changes between them is invariant (as in the rope example) but how we measure isn't.

 

[PAllen]

I'll be honest I don't fully understand the terminology you are using in your Edit, so will need to do a bit of reading. But I take it you agree that the rockets can't both be receding and approaching simultaneously. In the same way that galaxies can't be both receding and approaching simultaneously. So how the actual distance changes between them is invariant (as in the rope example) but how we measure isn't.

No, I don't agree unless you define receding as positive expansion scalar. Normally, receding is simply defined as increase in proper distance measured in a specified slicing of spacetime into space and time (called a foliation). This can call the same objects receding in one foliation and approaching in another. I gave you an example where both foliations that disagree are exceedingly natural, and this all happened in SR, not even GR.

 

[PeterDonis]

in order for 'galaxies' to move apart on the surface of the balloon, the 'balloon' or sphere itself must grow into a dimension the wouldn't exist in just a two dimensional space.

Once again: this does not logically follow. It is an artifact of the way you are visualizing the expanding balloon. It is not a logical implication of the actual model. If the balloon analogy bothers you because you feel forced to visualize it this way, the correct response is to discard the balloon analogy. It is not to try to make up alternate models that are incorrect. See below.

it would make more sense to me to have a flat 2 dimensional area and just show everything moving away from everything else.

It might seem like it makes sense to you, but this model does not match observations. Nor is it how FRW spacetime actually works. FRW spacetime is 4-dimensional, not 3 dimensional (and a "reduced" version of it with 2-spheres as spatial slices would be a 3-dimensional spacetime, not a 2-dimensional one).

 

[PeterDonis]

I would thought measuring increasing distances is absolute. Is that not the case?

No.

 

[rede96]

No, I don't agree unless you define receding as positive expansion scalar. Normally, receding is simply defined as increase in proper distance measured in a specified slicing of spacetime into space and time (called a foliation). This can call the same objects receding in one foliation and approaching in another. I gave you an example where both foliations that disagree are exceedingly natural, and this all happened in SR, not even GR.

If we take the SR example of the 2 rocket ships mentioned, if there is a taut, thin rope tied between them, if the rocket ships recede, the rope breaks. This is invariant in that every FoR will see the rope break. There is no FoR that will see the rope slacken. The rope, in the way I am understanding this, represents the proper distance between the two ships. In that if all frames of reference took turns in measuring the rope in their own FoR, they would all measure the same length.

So if there is a FoR that uses a specific coordinate system which measures the 2 rocket ships to be approaching each other, but observe the rope breaking, then it seems to me that the way the rockets are being measured is in error. If there are many ways to measure something but they all give a different result, they all can't be correct in terms of the proper distance.

Is that not correct?

 

[PAllen]

If we take the SR example of the 2 rocket ships mentioned, if there is a taut, thin rope tied between them, if the rocket ships recede, the rope breaks. This is invariant in that every FoR will see the rope break. There is no FoR that will see the rope slacken. The rope, in the way I am understanding this, represents the proper distance between the two ships. In that if all frames of reference took turns in measuring the rope in their own FoR, they would all measure the same length.So if there is a FoR that uses a specific coordinate system which measures the 2 rocket ships to be approaching each other, but observe the rope breaking, then it seems to me that the way the rockets are being measured is in error. If there are many ways to measure something but they all give a different result, they all can't be correct in terms of the proper distance.

Is that not correct?

No, it is almost as if you are choosing not to understand what I write. I will try again:

You are correct that the rope will break in all frames if it breaks in one. You are wrong that this requires the proper distance to increase in all frames. The proper distance cannot be defined without a space time slicing, and different slicings will have the rockets approaching rather than receding. Both cases are proper distance. The rope's tension is measuring expansion scalar (effectively) not proper distance. What you are missing is that in the inertial frame, the rope itself will be shrinking in length as measured in this frame, and the rockets will be getting closer together, but the exact rate of acceleration I specified for the rockets means the rope will be under increasing tension and will break. I have specified a situation where the length contraction of the [of the unstressed] rope [would] occur slightly faster than the distance between ships decrease, so the rope breaks in the inertial frame due to differential between these two shrinkages (the length the rope 'wants to be' versus the distance between the ships, both decreasing).

 

[rede96]

No, it is almost as if you are choosing not to understand what I write.

I will admit I am finding it difficult to get my head around this, but it isn't by choice! :-) It's a combination of age and no back ground in physics.

You are correct that the rope will break in all frames if it breaks in one. You are wrong that this requires the proper distance to increase in all frames. The proper distance cannot be defined without a space time slicing, and different slicings will have the rockets approaching rather than receding. Both cases are proper distance. The rope's tension is measuring expansion scalar (effectively) not proper distance. What you are missing is that in the inertial frame, the rope itself will be shrinking in length as measured in this frame, and the rockets will be getting closer together, but the exact rate of acceleration I specified for the rockets means the rope will be under increasing tension and will break. I have specified a situation where the length contraction of the [of the unstressed] rope [would] occur slightly faster than the distance between ships decrease, so the rope breaks in the inertial frame due to differential between these two shrinkages (the length the rope 'wants to be' versus the distance between the ships, both decreasing).

Ok, I think I understand your point about how the rope can break due to it's length contraction being slightly faster than the distance decreases, so if I take that on face value, can see how proper distance may not need to increase in all frames. Thanks for that.

But I'm still struggling with this. What about if ship A reflects a light signal off ship B and measures the round trip time? If the duration of the round trip grew, then couldn't the ships conclude that proper distance was increasing with time? And wouldn't this be absolute despite what other frames measured for proper distance?

 

[rede96]

It is not to try to make up alternate models that are incorrect.

I've been hanging around this forum long enough to know never to present personal theories or models! :-) I was simply using the 2d example and an analogy to help with my understanding.

FRW spacetime is 4-dimensional, not 3 dimensional

Yes, 3 spatial dimensions and 1 of time. But I was simply trying to understand how objects move within the 3 spatial dimensions of space, not the 4 dimensional spacetime. I made the assumption that as the universe is isotropic and homogeneous then it didn't matter which FoR I use for making measurements on expansion as they are all equally valid.

In any case, I think my last couple of post I made in response to PAllen cover my confusion. I am just finding it difficult to understand how 'increasing distances' between galaxies is coordinate dependent. So suspect I am going to have to do more reading.

 

[PAllen]

Ok, I think I understand your point about how the rope can break due to it's length contraction being slightly faster than the distance decreases, so if I take that on face value, can see how proper distance may not need to increase in all frames. Thanks for that.

But I'm still struggling with this. What about if ship A reflects a light signal off ship B and measures the round trip time? If the duration of the round trip grew, then couldn't the ships conclude that proper distance was increasing with time? And wouldn't this be absolute despite what other frames measured for proper distance?

Well, I already said (more than once) that if either ship measures their mutual distance, they will find it increasing. However, if an inertial observer measures their mutual distance it will be found to decreasing. Unless you want to claim that measurements by inertial observers are illegitimate, you have to admit that proper distance growth versus shrinkage is observer dependent, in general. In GR, this becomes even more arbitrary, because there are no global frames, or global preferred coordinates, just coordinate choices useful for some purposes. Proper distance is completely dependent on the particualar space-time slicing you use for your coordinates.

Instead of arguing over and over against the truth that proper distance between world lines is totally dependent on space-time slicing, for which no choice can be considered 'correct', why not try to accept that expansion scalar is the invariant you are looking for.

 

[rede96]

if either ship measures their mutual distance, they will find it increasing

Ok, great, and that was the point I was making. If the ships measure an increase in distance and someone else measures a reduction in distance, intuitively, one would think both cases can't be correct. I understand they can be measured by different FoR to be different, but I just can't help thinking there must only be one proper distance between two objects. Even if there is no way to know which FoR is measuring it.

Unless you want to claim that measurements by inertial observers are illegitimate, you have to admit that proper distance growth versus shrinkage is observer dependent

I don't know enough about this topic to make that claim, but I do seem to be thinking more along those lines with my current elementary level of understanding.

For example, to me an object has a certain absolute length, with a finite number of atoms that make that length. We know the size of atoms/particles etc, and we know they don't change. Just because 3 different observers may measure 3 different lengths doesn't mean the object has 3 different lengths. And that is where my head is stuck at the moment. Sorry.

Instead of arguing over and over against the truth that proper distance between world lines is totally dependent on space-time slicing, for which no choice can be considered 'correct', why not try to accept that expansion scalar is the invariant you are looking for.

I'm not arguing against anything, I'm just trying to improve my level of understanding, which I've said many times. However a lot of the terminology being used has no meaning to me, I don't understand what 'space time slices' are or what 'expanding comoving congruence' means.

So I accept I need to do more reading and it's probably pointless for me to continue with this until I do. However I do really appreciate the time people take to respond, but I now need to go put some time into this myself.

 

[timmdeeg]

No, I don't agree unless you define receding as positive expansion scalar. Normally, receding is simply defined as increase in proper distance measured in a specified slicing of spacetime into space and time (called a foliation). This can call the same objects receding in one foliation and approaching in another. I gave you an example where both foliations that disagree are exceedingly natural, and this all happened in SR, not even GR.

Talking about the expanding universe the usual notion is to imagine increasing ruler distances (proper distances) between comoving objects in a series of spatial slices. All observers will agree to that by measuring redshifts. Those with a high peculiar velocity whose distance to galaxies is decreasing in the direction of their motion can still calculate what they would measure if they were comoving and then will agree.
You explained already that in the context of SR accelerated and inertial observers don't agree regarding increasing and decreasing proper distances. If I understand this correctly then proper distance is not invariant. It is invariant only if one choses a distinct foliation of spacetime.

Coming back to the expanding universe It would be great if you could explain the foliation whereby in contrast to chose spacetime slicing proper distances are decreasing and how one should imagine observers who agree to that. A crude notion will be helpful, I doubt that a deeper understanding is possible without the knowing the math.
You mentioned the invariance of the expansion scalar. If possible, could you explain the meaning in simple words?
A final question, does the definition of proper distance necessarily include simultaneity along the distance?
Thanks.

 

[rede96]

You explained already that in the context of SR accelerated and inertial observers don't agree regarding increasing and decreasing proper distances. If I understand this correctly then proper distance is not invariant. It is invariant only if one choses a distinct foliation of spacetime.

This is the part I really struggle with too, as it would lead to different predictions from their movement where only one case would be correct. For example if the inertial observers measure distances increasing but the two ships (As mentioned above) are measuring their distances decreasing. At some point the ships will make contact, which must be observed in all cases, even though some inertial observers have measured the distances increasing. Those observers would have to conclude their measurements must have been wrong.

 

[Jonathan Scott]

I don't remember where it came from, but I've always found the model of an ordinary cone (for example made of paper) useful for visualisation, where time is the distance from the apex and space (one-dimensional) is around the circumference. It is clear that as time increases, the total amount of space increases, but locally nothing odd is happening. If you take paths which diverge uniformly from the apex, they represent comoving observers, like widely-spaced galaxies in the Hubble flow, and they move apart. If you take paths which are locally parallel, they represent paths which have an approximately fixed proper distance between them, like different stars in the same galaxy. If a path starts off parallel, then it remains parallel as long as the cone remains one which could be made of flat paper (for a constant rate of expansion).

I like J.A.Peacock's "Diatribe on Expanding Space" (a saved copy is available at http://arxiv.org/pdf/0809.4573.pdf ) which points out that poor terminology in this context creates a lot of confusion and is seriously misleading.

 

[PAllen]

Talking about the expanding universe the usual notion is to imagine increasing ruler distances (proper distances) between comoving objects in a series of spatial slices. All observers will agree to that by measuring redshifts. Those with a high peculiar velocity whose distance to galaxies is decreasing in the direction of their motion can still calculate what they would measure if they were comoving and then will agree.
You explained already that in the context of SR accelerated and inertial observers don't agree regarding increasing and decreasing proper distances. If I understand this correctly then proper distance is not invariant. It is invariant only if one choses a distinct foliation of spacetime.

Correct, it is invariant only given a particular foliation. In particular, if one picks a reference co-moving galaxy (no peculiar velocity), and builds a coordinate system 'as close as possible to SR Minkowski coordinates' [technical: called Fermi-Normal coordinates], then proper distance to some distant galaxy will be quite different from that using the standard foliation. Further, adopting the same definition of recession rate (change of proper distance by time - measured by the reference galaxy) will also be completely different, and I believe sub-luminal.

Coming back to the expanding universe It would be great if you could explain the foliation whereby in contrast to chose spacetime slicing proper distances are decreasing and how one should imagine observers who agree to that. A crude notion will be helpful, I doubt that a deeper understanding is possible without the knowing the math.

I don't think there is any useful foliation in cosmology where galaxy's proper distance is shrinking. That whole side discussion was just in support of the overall notion that expansion of proper distance is coordinate dependent. Peter first emphasized that in this thread, and I just wanted to describe a specific scenario where you could have expansion in one foliation and contraction in another.

You mentioned the invariance of the expansion scalar. If possible, could you explain the meaning in simple words?
A final question, does the definition of proper distance necessarily include simultaneity along the distance?
Thanks.

My best attempt at describing expansion scalar in words is that in the very local Minkowski-like frame (tetrad is the technical term) of a world line of a 'congruence' are the 'nearest' neighbor world lines getting further away versus closer.

Yes, one normally treats a spatial surface on which you compute proper distance is a simultaneity surface. However, since simultaneity is purely conventional, this adds no meaning. Any spacelike surface is a possible simultaneity surface.

 

[PAllen]

This is the part I really struggle with too, as it would lead to different predictions from their movement where only one case would be correct. For example if the inertial observers measure distances increasing but the two ships (As mentioned above) are measuring their distances decreasing. At some point the ships will make contact, which must be observed in all cases, even though some inertial observers have measured the distances increasing. Those observers would have to conclude their measurements must have been wrong.

Actually, in the example I gave, the inertial observer measured the ships as approaching, while each ship measures the other receding. However, as to your overall point, you just need to remember the notion of limits. They will never see a contradiction because the (if the ships never actually make contact) the rate of approach in a foliation where proper distance is decreasing, will get smaller and smaller. Thus, they can forever be approaching without meeting (and ratio of the proper distance measured in one coordinates versus another can grow without bound).

 

[timmdeeg]

Correct, it is invariant only given a particular foliation. In particular, if one picks a reference co-moving galaxy (no peculiar velocity), and builds a coordinate system 'as close as possible to SR Minkowski coordinates' [technical: called Fermi-Normal coordinates], then proper distance to some distant galaxy will be quite different from that using the standard foliation. Further, adopting the same definition of recession rate (change of proper distance by time - measured by the reference galaxy) will also be completely different, and I believe sub-luminal.

Very interesting and good to know. If I remember correctly, galaxies move away from each other picking Fermi-Normal coordinates , which is used to convince people who insist that they don't move but the space expands physically instead that this interpretation is coordinate dependent.

I don't think there is any useful foliation in cosmology where galaxy's proper distance is shrinking. That whole side discussion was just in support of the overall notion that expansion of proper distance is coordinate dependent.

This is very helpful. I was puzzled and couldn't believe that in GR too approaching vs. receding could depend on the foliation.

My best attempt at describing expansion scalar in words is that in the very local Minkowski-like frame (tetrad is the technical term) of a world line of a 'congruence' are the 'nearest' neighbor world lines getting further away versus closer.

Yes, one normally treats a spatial surface on which you compute proper distance is a simultaneity surface. However, since simultaneity is purely conventional, this adds no meaning. Any spacelike surface is a possible simultaneity surface.

I will read the Wikipedia article about congruence in GR, and eventually come back to this.

So it seems difficult to attribute the physical length of a ruler (which is not spacelike) to the proper distance between its end points. Is it perhaps the radar distance a possibility to do that?

 

[PAllen]

This is very helpful. I was puzzled and couldn't believe that in GR too approaching vs. receding could depend on the foliation.

Um, any SR example is also a GR example (SR is a subset of GR). My comment (about not knowing of any useful foliation in cosmological solutions that have galaxies approaching rather than receding) is specific to cosmological solutions (not GR in general). Also, one could easily construct useless foliations for cosmological solutions that have some co-moving galaxies approaching rather than receding. I am not sure there would be any way to construct a foliation where all galaxies are approaching each other.

So it seems difficult to attribute the physical length of a ruler (which is not spacelike) to the proper distance between its end points. Is it perhaps the radar distance a possibility to do that?

Generally, Fermi-Normal coordinates are taken to represent idealized ruler measurement. Radar would be yet a different distance. In general, distance per standard cosmological foliation, per Fermi-Normal, and per radar would all be different. Note, a ruler is typically taken to be a 1x1 congruence of world lines such that each curve 4-orthgonal to the congruence is a spacelike geodesic (and the expansion tensor of the congruence is zero). Fermi-Normal coordinates implement the closest possible to this for rulers measuring from a chosen origin world line.

[edit: Why doesn't anyone do cosmology with Fermi-Normal coordinates? Two big reasons: (1) you lose homgeneity and isotropy (except around the origin world line); (2) - all calculations would be intractable.]

 

[timmdeeg]

PAllen, thank you very much for your valuable answers.

 

[PeterDonis]

galaxies move away from each other picking Fermi-Normal coordinates

I don't understand what you mean by this. Picking coordinates is something humans do in order to model things. It's not something objects do when they move.

which is used to convince people who insist that they don't move but the space expands physically instead that this interpretation is coordinate dependent.

I don't understand this either. The fact that "space" is coordinate-dependent is a basic fact about coordinates. So all interpretations involving "space" (instead of "spacetime") are coordinate dependent. There's no need to prove it for any individual case.

 

[Chronos]

PeterDonis has hit upon a very important point - coordinates are invariably observer dependent. This same conclusion was reached by Einstein a century ago. A review of the Friedmann equation might be a useful point of reference.

 

[timmdeeg]

I don't understand what you mean by this. Picking coordinates is something humans do in order to model things. It's not something objects do when they move.

I don't understand this either. The fact that "space" is coordinate-dependent is a basic fact about coordinates. So all interpretations involving "space" (instead of "spacetime") are coordinate dependent. There's no need to prove it for any individual case.

Saying "galaxies move away from each other picking Fermi-Normal coordinates" I intended to say, if one uses Fermi-Normal coordinates then the galaxies move apart from each other. Hopefully this is correct now. I've often problems to express myself in English.
People agree that the galaxies are receding (of course), but some claim the reason for that is the generation of additional space (as I did a long time myself), others say no, they are just moving away. However that isn't true physics, because these interpretations depend on the choice of the coordinates (this I failed to express). I think from an invariant perspective the distances are increasing according to the time dependence of the scale factor, leaves some room for diverging interpretations. As I understand it, being a solution of the Einstein field equations the Friedmann equations too should be covariant.

Thanks for correcting.

 

[PeterDonis]

Saying "galaxies move away from each other picking Fermi-Normal coordinates" I intended to say, if one uses Fermi-Normal coordinates then the galaxies move apart from each other.

Ah, ok, that clarifies things. Yes, this will be correct, but note that Fermi Normal coordinates are different when centered on different galaxies. Also, their range is limited. A more precise way to say what you are saying here is that, if we choose a galaxy and construct Fermi Normal coordinates centered on its worldline, other galaxies that are within the region of spacetime that can be described by those coordinates will be moving away from the chosen galaxy.

from an invariant perspective the distances are increasing according to the time dependence of the scale factor

Not quite. The scale factor as it is usually defined is also coordinate dependent; you have to pick standard FRW coordinates for it to make sense. The invariant way of saying that "distances are increasing" is, as I said before, to look at the expansion scalar of the set of "comoving" worldlines, i.e., the worldlines of the set of observers who see the universe as always homogeneous and isotropic. The expansion scalar of that set of worldlines is positive, and this is the invariant measure of "increasing distances".

Note, btw, that the expansion scalar depends on the set of worldlines you choose; even in our expanding universe, it is easy to find sets of worldlines that do not have a positive expansion scalar. The reason the set of "comoving" worldlines is used is that the property of seeing the universe as homogeneous and isotropic is itself an invariant property, independent of coordinates--i.e., the set of "comoving" observers has an invariant definition; it can be defined without having to choose coordinates. So the expansion scalar of this particular set of worldlines has a meaning that is picked out by invariant properties of the spacetime; that's why it can be used as an invariant definition of "expansion of the universe".

 

[timmdeeg]

Note, btw, that the expansion scalar depends on the set of worldlines you choose; even in our expanding universe, it is easy to find sets of worldlines that do not have a positive expansion scalar. The reason the set of "comoving" worldlines is used is that the property of seeing the universe as homogeneous and isotropic is itself an invariant property, independent of coordinates--i.e., the set of "comoving" observers has an invariant definition; it can be defined without having to choose coordinates. So the expansion scalar of this particular set of worldlines has a meaning that is picked out by invariant properties of the spacetime; that's why it can be used as an invariant definition of "expansion of the universe".

So, it seems that the expansion scalar (being a single number) is defined by a "particular set of worldlines". Let's consider a ball consisting of comoving particles. Are their worldlines such a particular set of worldlines? Assuming the cosmological principle the ball expands or shrinks spherically symmetric, according to the sign of the rate of the volume change $\ddot V/V$, which is proportional to $-(\rho+3P)$. I'm just guessing, yields energy density combined with pressure a scalar which decides if the ball and thus the universe expands or shrinks?
I appreciate any help to understand the meaning of expansion scalar. There seems to be no non-technical literature available.

 

[PAllen]

So, it seems that the expansion scalar (being a single number) is defined by a "particular set of worldlines". Let's consider a ball consisting of comoving particles. Are their worldlines such a particular set of worldlines? Assuming the cosmological principle the ball expands or shrinks spherically symmetric, according to the sign of the rate of the volume change $\ddot V/V$, which is proportional to $-(\rho+3P)$. I'm just guessing, yields energy density combined with pressure a scalar which decides if the ball and thus the universe expands or shrinks?
I appreciate any help to understand the meaning of expansion scalar. There seems to be no non-technical literature available.

The world lines of a ball would not be an expanding congruence. It is the initial conditions of the 'big bang' - its isotropy and homogeneity over large scales - that ensures that galaxies that form share this attribute in their mutual relative motion. Any system formed independently of the big bang has no expectation of having such motion, and specifically, any bound system cannot have such motion. Thus, even galaxies in galactic clusters deviate from co-moving motion, because their mutual attraction modifies their motion from the co-moving initial condition. In the case of a ball, its formation inherits none of the big bang initial conditions, and its constituents are bound.

All of this gets at why, IMO, attributing the expansion to 'space' is misleading. Another example is that if, early in the history of the universe, you somehow got two well separated galaxies to move such that they observe no mutual redshift (by virtue of giving one the right peculiar velocity toward the other relative to co-moving motion), this feature would not change over time, nor would distance (as they each would mearsure it) between them increase. They need not be close enough to be a bound system for this to be true.

[edit: I see that you may have intended to arrange a sphere of test particles to have initial co-moving (expanding motion). Then, they would have the same expansion as the collection of all galaxies, as long as you rule out any mutual interactions (EM or self gravity).]

 

[timmdeeg]

[edit: I see that you may have intended to arrange a sphere of test particles to have initial co-moving (expanding motion). Then, they would have the same expansion as the collection of all galaxies, as long as you rule out any mutual interactions (EM or self gravity).]

Yes, the intention was to think of the universe as being filled with test particles that do not gravitate as an initial condition. It follows that each particle sees the universe homogenous and isotropic for ever. By this one avoids the formation of inhomogeneities like bound systems what might make it easier to focus on that abstract thing called expansion scalar. Would then the worldlines of the ball be an expanding congruence?

I understand that "seeing the universe as homogeneous and isotropic is itself an invariant property", as PeterDonis stated in his last post, but this property doesn't seem to have an algebraic sign. I'm still missing a notion how this property is related to the expansion scalar which has a sign. Is it possible at all to describe its meaning/definition with words? Hmm does the sign originate from the relative acceleration of neighboring particles? By the way, isn't there an invariant form of geodesic deviation?