Are Geodesics in Outer Space Curved or Straight Paths?

[Mentz114]

This might be so on grounds of convenience but doesn't address my point about GR's requirement of general covariance. Or do you imply that expansion is indeed a coordinate feature not preserved under transformation and thus, if we are to believe what Einstein held as the backbone of GR besides the Equivalence principle, not physically real, but that we maintain it due to its convenience?

That's far to complicated for me to understand.

It's not a matter of convenience. If the universe is homogenous, isotropic and expanding, then it should be seen as such.

Earlier you said

... [only] properties that are invariant under changes of coordinates are physically real, ...

Expansion isn't a property in that sense, it's a frame dependent phenomenon.

 

[TrickyDicky]

That's far to complicated for me to understand.

It's not a matter of convenience. If the universe is homogenous, isotropic and expanding, then it should be seen as such.

Let's see if we can get this right, what is a matter of convenience in GR is the choice of coordinates and metric which shouldn't affect the underlying physics. I believe this to be GR 101 so I hope is not so complicated to understand.

Expansion isn't a property in that sense, it's a frame dependent phenomenon.

I am not talking about frame dependence like for instance the gravitational field or momentum are frame dependent, What Ich talked about and bcrowell agreed is that the phenomenon of expansion is coordinate-dependendent, which according to general covariance would qualify it as not phisically real.
Since this would be an utter ATM claim, please might they or somebody knowledgeable clarify where the mistake here is (either mine or theirs)?

 

[bcrowell]

Let's see if we can get this right, what is a matter of convenience in GR is the choice of coordinates and metric which shouldn't affect the underlying physics. I believe this to be GR 101 so I hope is not so complicated to understand.

I am not talking about frame dependence like for instance the gravitational field or momentum are frame dependent, What Ich talked about and bcrowell agreed is that the phenomenon of expansion is coordinate-dependendent, which according to general covariance would qualify it as not phisically real.
Since this would be an utter ATM claim, please might they or somebody knowledgeable clarify where the mistake here is (either mine or theirs)?

I'm not sure there's really any disagreement here on the facts. I think it's more of a matter of taste. We all seem to agree that the FRW spacetime has a certain set of coordinates in which it looks particularly simple. I think we're just differing in how much we emphasize that.

 

[TrickyDicky]

bcrowell, this is what I call gaslight, are you saying that considering expansion as physical fact(as I and many others do) or as a coordinate artifact is just a matter of taste?

 

[JesseM]

Two earlier threads on how there's no necessity to understand the FRW metric in terms of "expanding space":

https://www.physicsforums.com/showthread.php?t=176380
https://www.physicsforums.com/showthread.php?t=320958

One way of seeing that "expanding space" is dependent on a choice of coordinate system is to consider the case of a FRW metric where the matter density is set to zero (a Milne universe)--in this case, if you use the type of cosmological coordinate system usually used for the FRW metric you get a universe expanding at a constant rate and with space at any instant having negative curvature, but since the matter density is zero this is really the same metric as flat Minkowski spacetime, just described using an unusual choice of non-inertial coordinate system. See the two diagrams here comparing worldlines of different observers in the Milne universe when graphed both in the cosmological coordinate system and in an ordinary SR inertial frame.

 

[bcrowell]

bcrowell, this is what I call gaslight, are you saying that considering expansion as physical fact(as I and many others do) or as a coordinate artifact is just a matter of taste?

The amount of space between galaxy A and cosmologically distant galaxy B increases over time. It's a matter of taste whether to ascribe that increase to an expansion of space itself or to motion of the two galaxies.

 

[Mentz114]

Let's see if we can get this right, what is a matter of convenience in GR is the choice of coordinates and metric which shouldn't affect the underlying physics. I believe this to be GR 101 so I hope is not so complicated to understand.I am not talking about frame dependence like for instance the gravitational field or momentum are frame dependent, What Ich talked about and bcrowell agreed is that the phenomenon of expansion is coordinate-dependendent, which according to general covariance would qualify it as not phisically real.
Since this would be an utter ATM claim, please might they or somebody knowledgeable clarify where the mistake here is (either mine or theirs)?

Expansion manifests itself as relative motion. Relative motion can be transformed away by a suitable choice of coordinates. Does that mean that the motion is not 'real' !

I don't think it does.

If an observer has suitably expanding rulers they won't see any expansion. The laws of physics are not broken - they just say that 'man with expanding ruler sees no expansion'.

 

[TrickyDicky]

Two earlier threads on how there's no necessity to understand the FRW metric in terms of "expanding space":

This is not what I'm talking about, besides according to GR there can't be such thing as "expanding space" since space doesn't exist by itself, only to describe matter relations:

"All our spacetime verifications invariably amount to a determination of spacetime coincidences. If, for example, events consisted merely in the motion of material points, then ultimately nothing would be observable but the meeting of two or more of these points." (Einstein, 1916, p.117)
"People before me believed that if all the matter in the universe were removed, only space and time would exist. My theory proves that space and time would disappear along with matter."

What I am trying to understand here is the concept of general covariance, do you agree that according to this principle a real motion affecting the whole universe should be preserved under any coordinate change?

The amount of space between galaxy A and cosmologically distant galaxy B increases over time. It's a matter of taste whether to ascribe that increase to an expansion of space itself or to motion of the two galaxies.

By applying general covariance you cannot strictly ascribe that increase to space itself as It's pointed out above.

Expansion manifests itself as relative motion. Relative motion can be transformed away by a suitable choice of coordinates. Does that mean that the motion is not 'real' !

Well if affects all gallaxies in the universe is not a relative motion but a rather absolute motion. If it really can be made to vanish by a suitable choice of coordinates then according to general covariance it can't affect all the galaxies(i.e. all reference frames in the universe if there is isotropy as we all assume) must be as you say a "relative motion", but that is not what standard cosmology says, standard cosmology says that expansion must be perceived from any point in the universe (no special or privileged point of view must exist). That is a real and absolute motion that cannot be tranformed away by coordinate choice as if it were the relative motion of a train from a station.

Perhaps if someone explains his/her interpretation of general covariance, I could see if I got it wrong somewhere.

I don't think it does.

If an observer has suitably expanding rulers they won't see any expansion. The laws of physics are not broken - they just say that 'man with expanding ruler sees no expansion'.

Then you have to justify the expanding ruler.

 

[Mentz114]

Then you have to justify the expanding ruler.

Exactly. The coordinates in which we don't see expansion don't correspond to reality, so they can't be justified on physical grounds.

Amongst the coordinate choices , it seems there's only one that can be thought of as the 'real' observer. George Jones says more or less this in post #16.

I'll also refer you you Mike Fontenot's posts in another thread, where he claims that from amongst the available choices, there's only one set of coords that describe an accelerating frame.
(https://www.physicsforums.com/showthread.php?t=425142&page=9.)

That is a real and absolute motion that cannot be tranformed away by coordinate choice as if it were the relative motion of a train from a station.

In the FRW models the motion is very regular and symmetric, which makes it unusual and capable of being transformed away. It's not generally possible to do this, as you point out.I'm glad you've raised this issue, because the thread has been instructive.

 

[TrickyDicky]

Exactly. The coordinates in which we don't see expansion don't correspond to reality, so they can't be justified on physical grounds.

Amongst the coordinate choices , it seems there's only one that can be thought of as the 'real' observer. George Jones says more or less this in post #16.

That's what I am trying to get right, but the problem is that what General relativity claims is that the physics shouldn't be dependent on a certain coordinates, that there shouldn't be only one choice that gives us the "right physical reality", instead that physical reality should come from what is invariant regardless the choice of coordinates. This is at least my understanding of general covariance, if I'm wrong about it I would like for someone to correct this interpretation of GR.

I'll also refer you you Mike Fontenot's posts in another thread, where he claims that from amongst the available choices, there's only one set of coords that describe an accelerating frame.
([url//]http:www.physicsforums.com/showthread.php?t=425142&page=9[/url].)

This seems unrelated to this discussion, he is talking about SR twins paradox

I'm glad you've raised this issue, because the thread has been instructive.

Well, hopefully, but for me I'm still trying to reconcile Hurkyl point of view(post #10) with Ich's(post#21) and bcrowell's(post#22) and according to GR Hurkyl is right, I think.

 

[JesseM]

What I am trying to understand here is the concept of general covariance, do you agree that according to this principle a real motion affecting the whole universe should be preserved under any coordinate change?

I don't understand what you mean by "real motion". My understanding is that general covariance says that the Einstein Field Equations work in any arbitrary coordinate system (with the metric and other coordinate systems written in terms of that coordinate system), so they are all "equally valid" in that sense. And of course, regardless of what coordinate system you use, as long as you use the correct expression for the laws of physics in that coordinate system you are guaranteed to get the same predictions about coordinate-invariant facts, like the proper time between two events along some worldline. However, all statements about velocities are inherently coordinate-dependent as I understand it, so I don't know what coordinate-independent observation could correspond to an observation of "real motion".

Well if affects all gallaxies in the universe is not a relative motion but a rather absolute motion.

What do you mean by "affects"? Again, is there some coordinate-independent observation you're thinking of? Statements about when light waves hit an observer according to his own proper time are coordinate-independent, so you can say it's a coordinate-independent fact that each galaxy sees pretty much the same thing when looking out at other galaxies, including the fact that more distant galaxies (with 'distant' defined in some visual sense, like the apparent brightness of 'standard candles') are more redshifted on average.

If it really can be made to vanish by a suitable choice of coordinates

If what can be made to vanish? The visual appearances can't be made to vanish, but the idea that the galaxies are moving apart certainly can.

then according to general covariance it can't affect all the galaxies(i.e. all reference frames in the universe if there is isotropy as we all assume)

What do you mean by "all reference frames"? The same thing as "all coordinate systems"? I don't see what it means to say that the motion of galaxies "affects" all coordinate systems.

but that is not what standard cosmology says, standard cosmology says that expansion must be perceived from any point in the universe (no special or privileged point of view must exist).

The only thing cosmology says is the same for everyone is what is "perceived" in a visual sense when one looks at the surrounding universe. It doesn't say that different coordinate systems must all "perceive" some coordinate-dependent notions like the notion that the galaxies are moving apart.

 

[JesseM]

Exactly. The coordinates in which we don't see expansion don't correspond to reality, so they can't be justified on physical grounds.

What do you mean by "justified on physical grounds"? The basic principle of diffeomorphism invariance (discussed here), which I think is closely related to general covariance although I'm not sure about the exact difference, says that the laws of general relativity work equally well in absolutely any coordinate system, so in terms of the basic law of physics there's no reason to prefer one global coordinate system over another, even though depending on the curvature of space and the arrangement of matter some coordinate systems may make the description simpler than others. Do you disagree?

Amongst the coordinate choices , it seems there's only one that can be thought of as the 'real' observer. George Jones says more or less this in post #16.

No, George Jones said that fundamental FRW observers "are the observers for which space is homogeneous and isotropic". Here, I take "fundamental FRW observers" to be the observers that are used to define the coordinate system normally used to describe the FRW metric, a coordinate system which is "simple" because both the curvature of space and the density of matter are totally uniform on any surface of simultaneity. But again, although this spacetime looks particularly simple when expressed in this coordinate system, the same basic laws of physics will still apply even in more unwieldy coordinate systems on the same spacetime.

I'll also refer you you Mike Fontenot's posts in another thread, where he claims that from amongst the available choices, there's only one set of coords that describe an accelerating frame.
(https://www.physicsforums.com/showthread.php?t=425142&page=9.)

Well, Mike hasn't yet responded to the questions I asked him in post #130 of that thread about why he thinks his method doesn't involve an arbitrary choice of how to define the accelerating observer's "measurements" and "calculations". The basis for his claims is a paper he wrote himself, we shouldn't assume that most other physicists would agree with his arguments.

 

[TrickyDicky]

I don't understand what you mean by "real motion".

By real motion I meant expansion. I understand that you say expansion is a relative motion of galaxies as perceived from each galaxy(or perhaps you adhere to the view criticized by Ich and the papers mentioned by bcrowell (Bunn and Hogg etc) about space itself really expanding.)
In case you ascribe the phenomenon of expansion to a relative motion, would you care to explain me what other relative motion is perceived by any possible reference frame? That is why I call it absolute motion.

However, all statements about velocities are inherently coordinate-dependent as I understand it, so I don't know what coordinate-independent observation could correspond to an observation of "real motion".

Then you say expansion is coordinate-dependent but not observer dependent, right?
But that is what general covariance forbids, as I understand it.

What do you mean by "affects"? Again, is there some coordinate-independent observation you're thinking of?

Yes, redshift, but redshift is not coordinate-independent.The fact that something is coordinate dependent doesn't mean is not measurable. Length contraction is coordinate-dependent and measurable just like time dilation.

If what can be made to vanish? The visual appearances can't be made to vanish, but the idea that the galaxies are moving apart certainly can.

"Visual appearances"=redshift, "the idea that the galaxies are moving apart "=expansion, I believe you can not disconnect this two just like that. Otherwise there would be no grounds to accept expansion.

The only thing cosmology says is the same for everyone is what is "perceived" in a visual sense when one looks at the surrounding universe. It doesn't say that different coordinate systems must all "perceive" some coordinate-dependent notions like the notion that the galaxies are moving apart.

It does say expansion must be detected from any galaxy. Therefore is observer-independent.

 

[Mentz114]

What do you mean by "justified on physical grounds"?

I should have said something like 'agrees with our experience and measurements'.

No, George Jones said that fundamental FRW observers "are the observers for which space is homogeneous and isotropic".

I wasn't disagreeing with that. Maybe my paraphrasing is clumsy.

Here, I take "fundamental FRW observers" to be the observers that are used to define the coordinate system normally used to describe the FRW metric,

Do you mean the coordinate frame ?

... a coordinate system which is "simple" because both the curvature of space and the density of matter are totally uniform on any surface of simultaneity. But again, although this spacetime looks particularly simple when expressed in this coordinate system, the same basic laws of physics will still apply even in more unwieldy coordinate systems on the same spacetime.

I don't think I disagree with that either. You're repeating yourself ( my bold ). We all know that.

Isn't it true that obervers like we humans don't get to choose our local coordinates, and that there's one amongst all those available that describes a particular viewpoint. So someone sitting on a lump of matter in FRW cosmos is like us. Someone traveling a spaceship between two lumps picks out another coordinate system and so on.
Maybe some of the available coordinate systems do not correspond to possible observers, even if the laws of physics are unchanged. Some observers stand out, particularly those for which spatial hyperslices are othogonal, giving non-rotating inertial-frames ( e.g. FRW coordinate frame and Painleve-Gullstrand chart in Schwarzschild )

 

[JesseM]

By real motion I meant expansion. I understand that you say expansion is a relative motion of galaxies as perceived from each galaxy(or perhaps you adhere to the view criticized by Ich and the papers mentioned by bcrowell (Bunn and Hogg etc) about space itself really expanding.)

No, I adhere to the view that both "expansion" and "motion of galaxies" is coordinate-dependent (unless you are talking purely about visual appearances), and thus both would fail to be the case in some perfectly valid coordinate systems.

In case you ascribe the phenomenon of expansion to a relative motion, would you care to explain me what other relative motion is perceived by any possible reference frame?

Coordinate systems are arbitrary ways of assigning position and time coordinates to different events (aside from the requirement that the coordinates vary continuously along any continuous path through spacetime), you could define a coordinate system where all the galaxies are rushing together, or galaxies in one region are getting closer together while galaxies in another region are getting farther apart, or galaxies are oscillating back and forth along the x-axis, or all the galaxies are moving towards the nearest point on a cosmic line drawing of Mickey Mouse, etc. According to diffeomorphism invariance (or general covariance, again I am not clear on the difference) the laws of general relativity will work in any smooth coordinate system you can come up with, no matter how crazy, so fundamentally there is no physical reason to judge one coordinate system's view of things to be "more correct" than any other, even if the description of the spacetime and matter distribution may be a lot simpler in some coordinate systems than others.

Then you say expansion is coordinate-dependent but not observer dependent, right?

I don't know what "observer dependent" means if you are not using it as a synonym for "coordinate dependent", I have only seen the first used as a synonym for the second. Are you just talking about what is seen visually by different observers? These visual appearances are coordinate-independent, but I wouldn't say that any observer sees "expansion", they just see a pattern where galaxies whose standard candles are less bright also tend to be more redshifted. If you want to say that more distant galaxies are more redshifted (or are moving away more quickly), you are no longer just talking about visual appearances, all notions of distance and speed depend on a choice of coordinate system.

Yes, redshift, but redshift is not coordinate-independent.

Yes it is, in the sense that you can define the frequency of light seen by a given observer in purely local terms, in terms of the proper time that observer experiences between successive peaks of the light wave hitting his worldline. All purely local physical facts, like the proper time on some observer's worldline when a particular signal hits him, are coordinate-independent.

The fact that something is coordinate dependent doesn't mean is not measurable.

I didn't say coordinate-dependence implies nonmeasurable.

The only thing cosmology says is the same for everyone is what is "perceived" in a visual sense when one looks at the surrounding universe. It doesn't say that different coordinate systems must all "perceive" some coordinate-dependent notions like the notion that the galaxies are moving apart.

It does say expansion must be detected from any galaxy. Therefore is observer-independent.

Nope, not if "expansion" refers to anything besides visual appearances, which again can be defined in a purely local way.

 

[JesseM]

I should have said something like 'agrees with our experience and measurements'.

OK, but does "our experience and measurements" refer to anything other than coordinate-independent local facts like the proper time between our detecting successive peaks of a light wave reaching our position? Of course all coordinate systems, even the most unwieldy and "weird" ones, will make the same predictions about local facts! Perhaps what you mean is something more like a coordinate system which we find "intuitive" in some way, and I wouldn't disagree that the standard cosmological coordinate system is more intuitive than one where surfaces of simultaneity aren't also ones where the large-scale distribution of matter is approximately homogenous.

Here, I take "fundamental FRW observers" to be the observers that are used to define the coordinate system normally used to describe the FRW metric,

Do you mean the coordinate frame ?

Yes.

... a coordinate system which is "simple" because both the curvature of space and the density of matter are totally uniform on any surface of simultaneity. But again, although this spacetime looks particularly simple when expressed in this coordinate system, the same basic laws of physics will still apply even in more unwieldy coordinate systems on the same spacetime.

I don't think I disagree with that either. You're repeating yourself ( my bold ). We all know that.

OK good, just wanted to make sure (and also I think it's important to emphasize this idea about the meaning of diffeomorphism invariance/general covariance since not everybody may be clear on what they mean, some of TrickyDicky's posts suggest confusion about the meaning for example).

Isn't it true that obervers like we humans don't get to choose our local coordinates,

By "local coordinates" do you just mean a locally inertial frame in a region of spacetime around us that's small enough so that curvature effects can be ignored? In SR any inertial observer has a unique inertial rest frame ('unique' aside from unimportant issues like the placement of the spatial origin), and similarly the equivalence principle says that in GR any free-falling observer has a unique locally inertial rest frame in a small (technically infinitesimal) region of spacetime around them. But if we're talking about a larger coordinate system covering both ourselves and other galaxies, then I would say we do get to choose the coordinate system, there is no single "correct" way to construct a non-inertial frame for any given observer in either GR or SR (though some non-inertial frames may be 'simpler' or more intuitive in any given situation and thus used more commonly).

 

[TrickyDicky]

JesseM, I think I am starting to disentangle my confusion, and I find myself agreeing with most of what you say in your last response to my concerns.
I guess you lean towards Ich and bcrowell being right about the coordinate-dependence of expansion. I can see now this would respect general covariance.

What I didn't quite get at first, and thus my confusion, was the distinction between what you call "appearances" which is what we perceive in our local phrame, IOW the redshift and the expansion or galaxy motion or coordinate stretching of space or whatever you want to call it.

I guess this distinction is hard for me to do because in cosmology when we think of expansion, inmediately we image redshift and viceversa.

 

[Mentz114]

Jesse, thanks for your considered reply to my post. I have a more observer-centric view, I suppose, but essentially I have no disagreement with what you've said.

 

[TrickyDicky]

...both "expansion" and "motion of galaxies" is coordinate-dependent (unless you are talking purely about visual appearances), and thus both would fail to be the case in some perfectly valid coordinate systems.

These visual appearances are coordinate-independent, but I wouldn't say that any observer sees "expansion", they just see a pattern where galaxies whose standard candles are less bright also tend to be more redshifted. If you want to say that more distant galaxies are more redshifted (or are moving away more quickly), you are no longer just talking about visual appearances, all notions of distance and speed depend on a choice of coordinate system.

If I understand this correctly after giving it some thought, you say that expansion or "relative motion" are clearly coordinate-dependent but "visual appearances", namely redshift, is coordinate-independent since it's a purely local physical fact, this leads to the question: how can we interpret a local coordinate-independent measure (redshift) as a sign of a coordinate-dependent motion (relative velocity of a distant galaxy) that is not in our local frame. I thought in GR we couldn't even define a relative velocity in a different inertial frame(due to path-dependent parallel transport, etc).
I mean ,for instance in the case of using the Doppler shift to determine the velocity of a car, we can do it because the car is in our local frame, we can see that it is moving wrt us. In the case of the distant galaxy, we can't confirm that motion visually and according to GR we can't assign a relative velocity to it either, so how exactly do we attribute the redshift we observe to a radial motion if we can't independently confirm that motion?
Is it ultimately because there is no other explanation to the redshift-distance relationship available?

 

[TrickyDicky]

I can't figure it out so I guess it really is decided by the elimination of alternative explanations or I'm mising something. Maybe I misunderstood JesseM.

 

[Mentz114]

In the case of the distant galaxy, we can't confirm that motion visually and according to GR we can't assign a relative velocity to it either, so how exactly do we attribute the redshift we observe to a radial motion if we can't independently confirm that motion?

According to Stephani ( "General Relativity" ( 1986)

"[formula] ... shows that the two effects can only be separated in an artificial manner depending on the coordinate system"

The 'two effects' being doppler shift and cosmological shift.

 

[TrickyDicky]

According to Stephani ( "General Relativity" ( 1986)

"[formula] ... shows that the two effects can only be separated in an artificial manner depending on the coordinate system"

The 'two effects' being doppler shift and cosmological shift.

I completely agree with this statement.

I'm not sure how it addresses my concern, I just don't know how to link the redshift we perceive (nevermind if you want to call it Doppler or Cosmological) with remote motion of galaxies, unless we proceed by elimination of alternatives that can't explain the redshift-distance relation.

 

[atyy]

I completely agree with this statement.

I'm not sure how it addresses my concern, I just don't know how to link the redshift we perceive (nevermind if you want to call it Doppler or Cosmological) with remote motion of galaxies, unless we proceed by elimination of alternatives that can't explain the redshift-distance relation.

The framework of cosmology is general relativity. A universe must be a solution of the Einstein field equations.

The red shift, plus other data, plus an assumption that the universe is almost uniform on large scales, constrain the universe to be an FRW-like solution of the Einstein field equations. The FRW solution is the "objective" thing corresponding to the "objective" red shifts that certain observers see. The description of the FRW solution as "expanding universe" is observer dependent, as there are many other descriptions of the FRW solution.

 

[TrickyDicky]

The framework of cosmology is general relativity. A universe must be a solution of the Einstein field equations.

The red shift, plus other data, plus an assumption that the universe is almost uniform on large scales, constrain the universe to be an FRW-like solution of the Einstein field equations. The FRW solution is the "objective" thing corresponding to the "objective" red shifts that certain observers see. The description of the FRW solution as "expanding universe" is observer dependent, as there are many other descriptions of the FRW solution.

Aha, in a way this is pretty much sayng that we proceed by elimination, there is no other solution than the FRW given the assumptions and constrains, and FRW solution describes redshift as due to the motion of receding galaxies.

 

[atyy]

Aha, in a way this is pretty much sayng that we proceed by elimination, there is no other solution than the FRW given the assumptions and constrains, and FRW solution describes redshift as due to the motion of receding galaxies.

Yes. Apart from experimental data like the red shift, the two major things used to eliminate are the Einstein Field Equations and the assumption of uniformity. The assumption of uniformity is called the "cosmological principle". It is possible that further observations may show this is not justified, since we apply it to the whole universe, even those parts we cannot observe (even in principle, if the Einstein equations are right, not just due to technological limitations). Some of those currently unobservable parts will become observable in the future.

http://arxiv.org/abs/1005.2165 (Solid speculation, it shows what's being worked on at the moment)

 

[Passionflower]

Aha, in a way this is pretty much sayng that we proceed by elimination, there is no other solution than the FRW given the assumptions and constrains, and FRW solution describes redshift as due to the motion of receding galaxies.

That seems the common view but for instance Penrose’s Weyl tensor hypothesis is interesting,e.g. FLRW is basically no Weyl curvature, but in time matter tends to clump due to gravitation, thus increasing Weyl, thus the model diverts from reality in time.

 

[JesseM]

If I understand this correctly after giving it some thought, you say that expansion or "relative motion" are clearly coordinate-dependent but "visual appearances", namely redshift, is coordinate-independent since it's a purely local physical fact, this leads to the question: how can we interpret a local coordinate-independent measure (redshift) as a sign of a coordinate-dependent motion (relative velocity of a distant galaxy) that is not in our local frame. I thought in GR we couldn't even define a relative velocity in a different inertial frame(due to path-dependent parallel transport, etc).

You can't define a coordinate-independent notion of relative velocity, but you can certainly talk about relative velocity in some choice of cosmological coordinate system. In fact if I understand things right you really only need a choice of simultaneity convention, then you can determine the "proper distance" to any object on a surface of simultaneity (which is just found by integrating the metric line element along a particular spacelike curve between the two objects that is entirely contained within that surface of simultaneity, just like you do to find proper time along a timelike worldline), and you can then talk about how the proper distance to some object is changing with your own proper time. The velocity given by Hubble's law can be understood in terms of change in proper distance over change in cosmological coordinate time, which is the same as the proper time of any galaxy at rest relative to the Hubble flow.

 

[TrickyDicky]

That seems the common view but for instance Penrose’s Weyl tensor hypothesis is interesting,e.g. FLRW is basically no Weyl curvature, but in time matter tends to clump due to gravitation, thus increasing Weyl, thus the model diverts from reality in time.

I also find the Weyl curvature hypothesis interesting, it seems to me it is trying to tell us something about the FRW model, not exactly sure if it is that it is diverting from reality, though.

 

[TrickyDicky]

You can't define a coordinate-independent notion of relative velocity, but you can certainly talk about relative velocity in some choice of cosmological coordinate system. In fact if I understand things right you really only need a choice of simultaneity convention, then you can determine the "proper distance" to any object on a surface of simultaneity (which is just found by integrating the metric line element along a particular spacelike curve between the two objects that is entirely contained within that surface of simultaneity, just like you do to find proper time along a timelike worldline), and you can then talk about how the proper distance to some object is changing with your own proper time. The velocity given by Hubble's law can be understood in terms of change in proper distance over change in cosmological coordinate time, which is the same as the proper time of any galaxy at rest relative to the Hubble flow.

Sure, that is what the FRW model does, give you that choice of coordinates and simultaneity convention and as we were saying you only need two things to reach that model, GR field equations and the assumption of spatial uniformity. Both are solid, the second one was at first a philosophical assumption, but within not many years it is close to be observationally settled.

 

[JesseM]

Sure, that is what the FRW model does, give you that choice of coordinates and simultaneity convention and as we were saying you only need two things to reach that model, GR field equations and the assumption of spatial uniformity. Both are solid, the second one was at first a philosophical assumption, but within not many years it is close to be observationally settled.

Yeah, but the choice of simultaneity used to describe the FRW model is still a matter of convention, you could in theory describe the metric of an FRW universe with a coordinate system that used a different choice of simultaneity (though it would be a lot less convenient), and the fundamental laws of physics don't pick out any global simultaneity convention as being "preferred" over any other.