Is the concept of true distance compatible with relativity?

[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.