B Is the concept of true distance compatible with relativity?

[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?

 

[PeterDonis]

it seems that the expansion scalar (being a single number) is defined by a "particular set of worldlines".

The expansion scalar is a property of a set of worldlines, yes. But every set of worldlines has one; it's not a property that only particular sets of worldlines (like the "comoving" worldlines in cosmology) have.

Lets consider a ball consisting of comoving particles. Are their worldlines such a particular set of worldlines?

Yes.

Assuming the cosmological principle the ball expands or shrinks spherically symmetric

If the ball is made up of "comoving" worldlines in our expanding universe, then it will be expanding (have a positive expansion scalar). If it is shrinking, it cannot be a ball of "comoving" worldlines in our actual universe; it must be a ball of "comoving" worldlines in some other spacetime.

I'm just guessing, yields energy density combined with pressure a scalar which decides if the ball and thus the universe expands or shrinks?

If we adopt standard cosmological coordinates, then in those coordinates the expansion scalar for the set of "comoving" worldlines appears as the quantity $\dot{a} / a$, where $a$ is the scale factor--i.e., the Hubble parameter. The quantity you wrote down, $- \left( \rho + 3 p \right)$, which is the RHS of the second Friedmann equation, is related to the rate of change of the expansion scalar. Ordinary matter and energy will always have $\rho + 3p$ positive, which means the rate of change of the expansion scalar will be negative--the expansion will get slower and slower, and might eventually reverse (depending on the initial conditions). Dark energy, however, has $\rho + 3p$ negative, which means it causes the expansion scalar to increase, not decrease. This is what is referred to as "accelerating expansion"; in this scenario, the expansion scalar will never become negative.

 

[PeterDonis]

this property doesn't seem to have an algebraic sign.

Right, it doesn't. A model of a contracting universe, in which the expansion scalar of the set of "comoving" worldlines is negative, not positive, is perfectly consistent. It just doesn't describe our actual universe. What tells us that the expansion scalar is positive in our actual universe is observation; you can't derive it just from abstract principles alone.

 

[timmdeeg]

If we adopt standard cosmological coordinates, then in those coordinates the expansion scalar for the set of "comoving" worldlines appears as the quantity $\dot{a} / a$, where $a$ is the scale factor--i.e., the Hubble parameter.

Ok, this makes sense, because the sign of $\dot{a} / a$ determines whether the universe is expanding or contracting and thereby determines the sign of the expansion scalar, whereas the sign of $\ddot{a} / a$ and $\ddot V/V$ respectively determines whether the universe expands accelerated or decelerated.
I am not sure what "appears" means. Is the expansion scalar identical with $\dot{a} / a$, or a function of it, or ...?

However so far this expansion scalar is not invariant, because we have chosen FRW-coordinates. Doing this I think we can use the time dependence of the proper distances as a criterion for expansion vs.contraction as well. Coming back to the origin of this discussion, is there a quantity named expansion scalar corresponding to $\dot{a} / a$ (and having a sign) but which in contrast to that is invariant (the time dependence of the proper distances fails, because proper distance is not invariant, as I understood during this Thread) however and if yes how is it expressed as a "property of a set of worldliness"? Or is this a mathematical expression which isn't explainable on a simple level?

 

[timmdeeg]

If we adopt standard cosmological coordinates, then in those coordinates the expansion scalar for the set of "comoving" worldlines appears as the quantity $\dot{a} / a$, where $a$ is the scale factor--i.e., the Hubble parameter.

After reconsidering this: If it is correct that in this case the expansion scalar is not invariant, then a scalar isn't invariant by definition, as I thought. Could you please clarify that?

 

[PAllen]

After reconsidering this: If it is correct that in this case the expansion scalar is not invariant, then a scalar isn't invariant by definition, as I thought. Could you please clarify that?

The expansion scalar for the co-moving congruence is invariant - any coordinates at all can be used to compute it. What Peter was noting was that in standard cosmological coordinates, it is the same as the coordinate dependent expression he gave. In any other coordinates, you wouldn't generally even be able to use that expression; but computed from the full defintion of expansion scalar, it would come out the same as that expression in special coordinates.

Relating this back to the example we were discussing of rockets connected by string with the rockets having a specific thrust profile, the inertial frame would compute that proper distance between the rockets was decreasing, but the string would still break. If the expansion scalar for the string congruence were computed in inertial coordinates it would show expansion of the string, despite the measured decrease of proper distance between the rockets over time.

 

[timmdeeg]

The expansion scalar for the co-moving congruence is invariant - any coordinates at all can be used to compute it. What Peter was noting was that in standard cosmological coordinates, it is the same as the coordinate dependent expression he gave.

Ah I see, thanks for clarifying.

Relating this back to the example we were discussing of rockets connected by string with the rockets having a specific thrust profile, the inertial frame would compute that proper distance between the rockets was decreasing, but the string would still break. If the expansion scalar for the string congruence were computed in inertial coordinates it would show expansion of the string, despite the measured decrease of proper distance between the rockets over time.

This is a good example. In contrast to proper distance the expansion scalar describes true physics (in the sense of being not observer dependent), the breaking of the string.

 

[PeterDonis]

I am not sure what "appears" means.

It means that the invariant quantity, the expansion scalar, is equal, in these particular coordinates, to the coordinate-dependent quantity $\dot{a} / a$. In other words, $\dot{a} / a$ is how that invariant is expressed in these particular coordinates.

 

[timmdeeg]

It means that the invariant quantity, the expansion scalar, is equal, in these particular coordinates, to the coordinate-dependent quantity $\dot{a} / a$. In other words, $\dot{a} / a$ is how that invariant is expressed in these particular coordinates.

Yes and thanks, I understand.

https://en.wikipedia.org/wiki/Congruence_(general_relativity)
"the expansion scalar represents the fractional rate at which the volume of a small initially spherical cloud of test particles changes with respect to proper time of the particle at the center of the cloud"

If this is correct that then while discussing the "ball" we have been quite close to an intuitive understanding was expansion scalar means.
Then I would expect that one obtains the same value of the fractional rate $\ddot V/V$ regardless if one uses FRW- or other coordinates, e.g. Fermi normal coordinates (because the ball is small).

EDIT

If we adopt standard cosmological coordinates, then in those coordinates the expansion scalar for the set of "comoving" worldlines appears as the quantity $\dot{a} / a$, where $a$ is the scale factor--i.e., the Hubble parameter.

So, it seems the expansion scalar is represented by $\ddot V/V$ or likewise by $\dot{a} / a$.

What troubles me is that a negative sign of $\ddot V/V$ doesn't necessarily mean that the universe contracts, it could expand decelerated as well (if I see it correctly). A negative expansion scalar however doesn't have the option for decelerated expansion, right?

I start being afraid to bother you.

 

[timmdeeg]

. If the expansion scalar for the string congruence were computed in inertial coordinates it would show expansion of the string, despite the measured decrease of proper distance between the rockets over time.

Is it correct and sufficient to say the expansion scalar for the string congruence is represented by the fractional rate at which the length of the string increases, measured in the proper time of the particle at its center?

 

[PeterDonis]

it seems the expansion scalar is represented by $\ddot V/V$

No; it's represented by $\dot{V} / V$. (Note that this is only true in local inertial coordinates; in other coordinates the volume $V$ does not have the same physical meaning. Any quantity involving "space", which volume does, is coordinate-dependent, because "space" itself is.)

 

[timmdeeg]

No; it's represented by $\dot{V} / V$.

Ok, thanks. I've erroneously linked 'rate of change of something' (see the wikipedia quote in #78) to the second derivative.

 

[rede96]

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).

I really appreciate your help, but unfortunately my mind keeps taking me back to the same place. I'll try and explain with a simple thought experiment. It is a bit long winded, I apologise for that, but see if it makes sense.

If I took two objects at rest wrt each other, separated by some arbitrary distance, then place a piece of rope between them and cut the rope so it exactly matches the distance they are apart, then any FOR that collects the rope and measures it's length will always measure the same length. This for me is the proper length of the rope and thus the proper distance between the two objects. (Albeit at the time I took the measurement) Other observers moving relative to the rope and two objects would of course measure a different distance between the two objects, but if they were to be able to place a a piece of rope between the two objects and measure it, they'd always get the same measurement as I did, as the two objects are always at rest wrt each other. I'll call this thought experiment the 'rope trick' for ease of reference.

So in the case where the two objects are receding, at any point in time I can do the rope trick and thus measure the proper distance between them at that given point in time relative to me. Any observers who then measured my piece of rope (EDIT: Just to be clear I mean measure the rope when it is at rest wrt to the measurer.) would agree the distance the two objects were apart at the time I took the measurement. I understand that if they did the rope trick, they might get a different length of rope, as it would depend on when they took the measurement as the two objects are receding. But in all cases who ever collected the rope and measured it, they would always get the same measurement as the frame that originally did the rope trick and thus always agree on the distance as measured in the original frame.

And there lies the problem for me, as distance is always absolute in the above thought experiment, by that I mean that all frames that collect rope and measure it, will always measure the rope to be the same length. No one will measure the same piece of rope to be a different length.

So if we take the case I mentioned previously where the two objects are moving together instead, if I do the rope trick at regular time intervals and measure each cut length of rope, I will measure each subsequent piece to be shorter than the proceeding one. This again is invariant, in that I can send my rope cuts to any FOR and they would measure the same thing. So even if they measure the two objects to be receding, it is impossible for them to conclude anything else but they are actually moving together. And moreover I would have thought if they did the same rope measurements, if would be impossible for one FOR to measure the rope cuts getting shorter and another FOR measure the rope cuts getting longer.

Of course it isn't feasible to do this actual measurement on cosmological scales, but there is nothing I can think of in the laws of physics that would suggest this isn't a valid way of measuring distances and a valid way for all frames to agree on if distances are receding or moving together.

 

[PeterDonis]

any FOR that collects the rope and measures it's length will always measure the same length

FORs don't make measurements. People make measurements, using measuring devices. A measuring device that is at rest relative to the rope will measure the rope to have what you are calling its proper length. A measuring device moving relative to the rope will not. A measuring device that is moving relative to the rope at close to the speed of light will measure its length to be close to zero.

Furthermore, you can find different measuring devices in different states of motion relative to the rope that will measure its "length" to be decreasing vs. increasing. The reason for this is that the "lengths" being measured correspond to different physical measuring processes and therefore different invariants. So even though the length measured by one particular method is indeed an invariant--all FORs will agree that a certain measuring device in a certain state of motion acting on the rope will measure its length to be such and such--there is no one single invariant that represents the "length" of the rope in all FORs. Different FORs--more precisely, different measuring processes--assign the term "length of the rope" to different invariants. There is simply no way around this.

I'm not sure what you mean by "collecting" the rope, but if you change the rope's state of motion, you are subjecting it to physical forces that might change its length, so any measurement you make of its length after that does not tell you what the rope's length was before you changed its state of motion.

 

[rede96]

I'm not sure what you mean by "collecting" the rope, but if you change the rope's state of motion, you are subjecting it to physical forces that might change its length, so any measurement you make of its length after that does not tell you what the rope's length was before you changed its state of motion.

Ok, so let's say that for the sake of argument that the rope doesn't change its state of motion, other observers who may have been moving relative to the rope will change their state of motion until they are at rest wrt to the rope then measure it. The point being that after the rope was used to measure the distance between two objects, once the rope was cut to represent that distance, this would be an invariant measurement. Irrespective of what observers moving wrt the rope may measure once the rope was in place. This shows that their moving measurements are in error. They don't represent the real distance, which is the basis for my argument.

Just because we can use different measuring devices in a different FOR doesn't mean they are all correct. And I was then assuming that this would be the same as saying just because we use different coordinate systems to measure distances doesn't make them all correct. My thought experiment seems to suggest that there is always just one proper distance between two objects. Which would be the one measured at rest wrt the rope.

A measuring device that is at rest relative to the rope will measure the rope to have what you are calling its proper length. A measuring device moving relative to the rope will not.

Of course, which seems to me to be equivalent as saying any measuring device that is at rest to the space between the two objects will measure the proper distance between the two objects. Because at some instant in time the rope fills the space between the two objects.

In any case, the point of my thought experiment was that no measurements of distance would be taken while observers were moving relative to the rope. The rope would be put in place by just one person in a FOR, then once the rope was cut to the correct distance, it can be measured.

Furthermore, you can find different measuring devices in different states of motion relative to the rope that will measure its "length" to be decreasing vs. increasing.

Yes in different states of motion. However if is impossible to have rope places between the two objects in the way I described above and for that rope to measured as both increasing in length and decreasing in length. It does one or the other, or stays the same.

 

[PAllen]

What if you are trying to measure the distance between two objects in relative motion? Or in GR, where there is no uniqe definition of distant objects being at rest at all, or what their relative velocity is? But even in SR, you have two object is relative motion: do you use a rope at rest with respect to object A, or object B, or something else? All may give different answers. You must abandon your idea of a true distance if you are ever to understand SR, let along GR.

[edit: I should add a further complication, that follows from a fundamental SR theorem (Herglotz-Noether): if the relative motion of the bodies is general (including accelerating through change of direction), there is no possible definition of an unstressed spanning rope at all, even in principle. In such a case, your attempt to define a real distance is doomed even if you invent some arbitrary answer to my questions above. ]

 

[PeterDonis]

after the rope was used to measure the distance between two objects, once the rope was cut to represent that distance, this would be an invariant measurement.

An invariant measurement of what? It would be an invariant measurement of the length of the rope, and therefore the distance between two objects, in a particular state of motion at a particular time. It would not be an invariant measurement of "the distance between two objects" under all possible cos"nditions. There is no such thing.

Just because we can use different measuring devices in a different FOR doesn't mean they are all ctrect.

A measurement is not "correct" or "incorrect". It just is. A particular measuring device outputs a particular result under particular conditions. The result is what it is. It doesn't make any sense to ask whether it's "correct" or not.

any measuring device that is at rest to the space between the two objects

There is no such thing. "The space between two objects" is not a thing that can be "at rest" or "in motion". The concept doesn't make sense.

no measurements of distance would be taken while observers were moving relative to the rope.

Then I don't understand the point. Obviously, if we take two objects that are at rest relative to each other and stay that way for all time, and put a rope between them, and cut the rope to the exact length it needs to be to just touch the two objects, then the length of the rope is the same as the distance between the two objects. But that reasoning only works because the objects are at rest relative to each other for all time.

The whole point of the "expanding universe" is that we are dealing with objects that are not at rest relative to each other for all time. There is no way to define a single invariant that represents "the distance between the objects" in that case. The best we can do is to define an invariant, the expansion scalar of the set of worldlines describing the objects, that tells us whether the set of objects, taken as a whole, is "expanding" or "contracting" (or neither, if the expansion scalar is exactly zero). But saying that "expanding" means "increasing distance between the objects" is an interpretation; it's not a statement of physics and it's certainly not an invariant. I realize your intuition is telling you that "expanding" ought to mean "increasing distance between the objects" in some invariant sense, but that intuition simply doesn't work in the context of a general curved spacetime.

 

[rede96]

But even in SR, you have two object is relative motion: do you use a rope at rest with respect to object A, or object B, or something else?

I think you may be missing my point or trying to over complicate it. I can, at least in principle, set off in a spaceship from Earth heading towards the moon with a long tape measure attached, then when I land on the moon I radio back to Earth where someone takes a measurement. Of course this is a ridicules way to measure the distance between the Earth and the moon, and it would only be relevant for the instant the measurement was taken. However, those two bodies are in relative motion and despite what distance any other observers moving wrt to either the Earth or moon may measure, I would maintain the the true distance is the distance taken from the tape measure. I guess my thinking is that at that snap shot in time, the earth, the moon and the tape measure are all at rest wrt each other, even if it is someone artificial.

So where I may be drawing wrong conclusions is that I thought if this was a valid way to measure over relatively short distances, then it must apply to very large distances, even if it is not feasible to do. And thus there is only one true distances between two objects. Even though I fully accept the in SR/GR there is no special FOR and different measurements will be made. I am not arguing against relativity in anyway.

The whole point of the "expanding universe" is that we are dealing with objects that are not at rest relative to each other for all time. There is no way to define a single invariant that represents "the distance between the objects" in that case.

As I've said I am not arguing against relativity at all. However very simply, if I can place an object between two bodies separated by some distance, the length of that object also represents the distance between them. As the length of that object can not physically change just by someone taking a measurement of it, neither can the distance.

So any other observer moving wrt to that object who measures the length will not measure its real length and hence not measure the real distance between the two bodies. Or in other words, if someone had to make an object to fit exactly between those two bodies (Assuming they are at rest for a moment) then there is only one length that will fit.

Now assume those two bodies are moving wrt each other, at any moment in time where someone would place an object between them, there is only one length of object that would fit, not many. That's how I was defining the 'real' length. or 'real' distance between them.

Of course we can't do this for bodies separated by large distances, but I don't see why the principle is any different? Just because we can't define or measure it, doesn't mean it doesn't exist.

As mentioned above, my thinking is that if we could freeze those two bodies in a moment of time, take a measurement of the distance between them from a FOR that was at rest wrt to those two bodies, then that would be the real distance between them.

 

[PeterDonis]

despite what distance any other observers moving wrt to either the Earth or moon may measure, I would maintain the the true distance is the distance taken from the tape measure.

And you would be wrong. Let me say it one more time: distance is coordinate-dependent. There is no such thing as "true" distance.

I am not arguing against relativity at all.

Yes, you are. You are arguing that there is such a thing as "true" distance. Relativity says there isn't.

I don't see the point of continuing to restate the same thing over and over. Thread closed.