Are Geodesics in Outer Space Curved or Straight Paths?

[TrickyDicky]

Let's imagine a test particle in outer space not being subjected to any significant force, gravitational(far enough from any massive object) or any other. Its path would be describing a geodesic that follows the universe curvature, right? Would that be an euclidean straight path, or would it follow a curved path, like an ellipse or a hyperbola?

 

[zhermes]

In space it would be euclidean straight, but not in space-time -- due to cosmological expansion.

 

[TrickyDicky]

In space it would be euclidean straight, but not in space-time -- due to cosmological expansion.

Aha, in spacetime would be ,then?
Or there is no easy way to answer this?

 

[zhermes]

It would depend sensitively on the particular cosmological model. I'm not sure what kinds of paths are characteristic or common.

 

[bcrowell]

In space it would be euclidean straight, but not in space-time -- due to cosmological expansion.

I don't think this is right.

In spacetime, the definition of straightness is a geodesic. So if the OP's question is interpreted as referring to spacetime, then the answer is definitely that it's straight.

In space, there is no way to answer the question, because you can't separate out spatial curvature from spacetime curvature in any unique way; it depends on your coordinates. In rotating coordinates, flat spacetime is spatially curved. To describe the path of a particle over cosmological distances you need to construct a coordinate system that covers cosmological distance scales, and such a coordinate system is highly arbitrary. An observer moving with the particle will describe the particle as being at rest, so its path is a point. If the particle is in motion relative to the Hubble flow, and it passes through galaxy G, which is at rest relative to the Hubble flow, then by symmetry an observer in G always sees the particle is moving in a straight line. A distant observer in galaxy Y, which is moving with its own local Hubble flow, will see both G and the particle as accelerating, and in the geometrically general case the will see the particle's path as some kind of curve (to the extent that it even makes sense to talk about a cosmologically distant observer's view of an object's motion -- in general it really doesn't make sense to do so).

 

[Hurkyl]

Let's imagine a test particle in outer space not being subjected to any significant force, gravitational(far enough from any massive object) or any other. Its path would be describing a geodesic that follows the universe curvature, right? Would that be an euclidean straight path, or would it follow a curved path, like an ellipse or a hyperbola?

If you really mean "Euclidean", then you really haven't understood the idea of "space-time" at all -- you are still thinking in terms of "space" and "time" being separate notions.

Assuming that you do understand the idea of space-time and just lack the words -- you probably meant "Minkowski straight".

Of course, space-time being curved, Minkowski only applies on small scales.



The moon orbits the Earth in a, more or less, straight path through space-time. (Assuming you are thinking in terms of space-time) the only reason you think otherwise is because you are measuring using the wrong metric.

An analogous error is thinking that latitude lines are straight lines on the Earth's surface.

 

[zhermes]

In spacetime, the definition of straightness is a geodesic. So if the OP's question is interpreted as referring to spacetime, then the answer is definitely that it's straight.

Very astute, and true. But if this was the case for the OP, he would be (effectively) asking "would a straight thing be straight," so i think its fair to assume he means solely in space (see below)

An observer moving with the particle will describe the particle as being at rest, so its path is a point. If the particle is in motion relative to the Hubble flow, and it passes through galaxy G, which is at rest relative to the Hubble flow, then by symmetry an observer in G always sees the particle is moving in a straight line. A distant observer in galaxy Y, which is moving with its own local Hubble flow, will see both G and the particle as accelerating, and in the geometrically general case the will see the particle's path as some kind of curve

Again, this is very accurate and very true; but (because of the above) a comoving frame is uninteresting, and the G perspective is shortlived; thus the 'Y' case is the most informative to OP's question--in my opinion.

If you really mean "Euclidean", then you really haven't understood the idea of "space-time" at all -- you are still thinking in terms of "space" and "time" being separate notions.

Space and time are separate notions, that's why they're different words. Space and time are intrinsically inseparable, but are none-the-less distinct (hence the somewhat common 3+1 terminology). We, as observers in reference frames non-relativistic to one-another, perceive time and space differently, and thus a discussing one while not the other can be informative and elucidating.

Assuming that you do understand the idea of space-time and just lack the words -- you probably meant "Minkowski straight".

This is a good point to note, but consider the above.

 

[TrickyDicky]

I don't think this is right.

In spacetime, the definition of straightness is a geodesic. So if the OP's question is interpreted as referring to spacetime, then the answer is definitely that it's straight.

I should have seen that one coming. My question is a bit naive and too sloppy in the choice of words. So yes, by definition a geodesic is the straightest path in a curved spacetime so just by talking about them I'm implying a curved spacetime and a straight path. I guess what I really meant was what kind of curvature does our spacetime have.

A distant observer in galaxy Y, which is moving with its own local Hubble flow, will see both G and the particle as accelerating, and in the geometrically general case the will see the particle's path as some kind of curve

My question was exactly that,what kind of curve would that be? But I see that this doesn't seem to have a straightforward answer, according to zhermes due to the fact of cosmological expansion. In a nonexpanding spacetime I guess the curvature would be that of the spatial part of the line element, right?

If you really mean "Euclidean", then you really haven't understood the idea of "space-time" at all -- you are still thinking in terms of "space" and "time" being separate notions.

Assuming that you do understand the idea of space-time and just lack the words -- you probably meant "Minkowski straight".

Of course, space-time being curved, Minkowski only applies on small scales.

Another example of sloppiness on my part, I probably should have said Minkowskian, still when I've read descriptions of the Einsten model of the universe, IIRC they talk about a hypersphere embedded in Euclidean ambient space. Perhaps someone can clarify this for me.

 

[atyy]

In a nonexpanding spacetime I guess the curvature would be that of the spatial part of the line element, right?

Spacetime is not expanding.

In general relativity, a test particle falling under the influence of only gravity traverses a spacetime geodesic.

The spacetime curvature of our universe is well approximated by the FRW solution on large scales, and by the Schwarzschild solution near our solar system.

There are many ways to divide the FRW spacetime into "space" and "time". In one of them, "space" is expanding. In another "space" is not expanding. Regardless of how the FRW spacetime is divided into "space" and "time", a test particle falling under the influence of only gravity traverses a geodesic of the FRW spacetime.

 

[Hurkyl]

There are many ways to divide the FRW spacetime into "space" and "time". In one of them, "space" is expanding. In another "space" is not expanding.

As I'm familiar with the phrase, "space is expanding" is not a coordinate-dependent phenomenon. If you compute, making use one coordinate chart, that space is expanding near a point, then you will get the same result if you repeat the calculation using any other coordinate chart.

 

[Hurkyl]

Another example of sloppiness on my part, I probably should have said Minkowskian, still when I've read descriptions of the Einsten model of the universe, IIRC they talk about a hypersphere embedded in Euclidean ambient space. Perhaps someone can clarify this for me.

The relevant mathematics was originally developed for purposes like studying the properties of Euclidean surfaces that do not depend on how the surface is embedded into Euclidean space. Also, surfaces in 3-space are the most complex examples that can be straightforwardly visualized. I think these descriptions are mainly just to convey the basic ideas of how differential geometry works, and to motivate terms like "curved".

The (IMHO) misleading thing is that they are purely spatial. If you have some surface, you could start drawing a straight line on it, tracing out one centimeter per second. One might get the idea that this is what is meant in general relativity by particles traveling along a geodesic. However, it's not.

 

[atyy]

As I'm familiar with the phrase, "space is expanding" is not a coordinate-dependent phenomenon. If you compute, making use one coordinate chart, that space is expanding near a point, then you will get the same result if you repeat the calculation using any other coordinate chart.

Really? I just meant da/dt > 0 in the usual FRW coordinates.

 

[bcrowell]

As I'm familiar with the phrase, "space is expanding" is not a coordinate-dependent phenomenon. If you compute, making use one coordinate chart, that space is expanding near a point, then you will get the same result if you repeat the calculation using any other coordinate chart.

The issue isn't so much coordinate-independence as definition-dependence. Fundamentally, there is no unique definition for the velocity of a distant object in GR, because vectors at distant points can only be compared by parallel transport, which is path-dependent. If you want to describe cosmological redshifts as kinematic Doppler shifts, you can [Bunn and Hogg 2008]. If you want to describe them as gravitational Doppler shifts, you can [Francis 2007].

Bunn and Hogg 2008, http://arxiv.org/abs/0808.1081v2
Francis 2007, http://arxiv.org/abs/0707.0380v1

 

[George Jones]

The expansion scalar (thus coordinate-independent) for the congruence of fundamental FRW observers is positive.

 

[atyy]

The expansion scalar (thus coordinate-independent) for the congruence of fundamental FRW observers is positive.

Yes, how about observer dependent then, since one presumably need not choose fundamental FRW observers?

 

[George Jones]

Yes, how about observer dependent then, since one presumably need not choose fundamental FRW observers?

Right, but FRW spacetimes arise by demanding spatial isotropy and homogeneity, and the fundamental observers are the observers for which space is homogeneous and isotropic.

 

[bcrowell]

The expansion scalar (thus coordinate-independent) for the congruence of fundamental FRW observers is positive.

Nobody disputes this, just as nobody disputes the existence of cosmological redshifts. The question is whether to describe these facts as expansion of space. Of the two references I gave in #13, one argues the point of view that they should be described as an expansion space, the other that they shouldn't. The arguments they present are arguments about pedagogy and about which interpretation is more natural. There is no objective basis on which to say that one is correct and one incorrect.

 

[TrickyDicky]

My question was exactly that,what kind of curve would that be? But I see that this doesn't seem to have a straightforward answer, according to zhermes due to the fact of cosmological expansion. In a nonexpanding "space" I guess the curvature would be that of the spatial part of the line element, right?

I rephrased it, I swear I meant space, if spacetime expanded we wouldn't notice, would we? but can someone address the quoted question now? Maybe in this less complex scenario I could understand it better.

 

[TrickyDicky]

anybody? to clear up this simple doubt?

 

[George Jones]

I'm away from home for a couple of days, and I have very limited computer access on a very slow connection, but I will make some posts after I get back. If you have access, read section 4.8 from General Relativity: An Introduction for Physicists by Hobson, Efstathiou, and Lasenby. You might be able to read this section from Google Books.

 

[Ich]

As I'm familiar with the phrase, "space is expanding" is not a coordinate-dependent phenomenon. If you compute, making use one coordinate chart, that space is expanding near a point, then you will get the same result if you repeat the calculation using any other coordinate chart.

No. Expansion of space is purely coordinate dependent. Or better, as George Jones puts it, "expansion" is a property of a congruence, like the one defined by the canonical observers in a FRW metric. It's not a property of spacetime.

Right, but FRW spacetimes arise by demanding spatial isotropy and homogeneity, and the fundamental observers are the observers for which space is homogeneous and isotropic.

Yes, that's why people are talking about "expanding space" at all.
Another more general set of fundamental observers, independent of FRW symmetries, is defined to be at rest in normal coordinates, maybe one could call them "Einstein observers", as they reproduce the inertial frames of SR if curvature is negligible, on which most people base their intuition. Expansion vanishes if you use these observers.
@TrickyDicky:
It don't think you want to see curves in space. You want to see a spacetime diagram of a neighbouring geodesic in some Riemann normal coordinates. They curve away from the origin for accelerated expansion, and curve inwards for decelerated expansion.
However, as bcrowell noted, the defining geodesic at the center is straight by definition.

 

[bcrowell]

Another more general set of fundamental observers, independent of FRW symmetries, is defined to be at rest in normal coordinates, maybe one could call them "Einstein observers", as they reproduce the inertial frames of SR if curvature is negligible, on which most people base their intuition. Expansion vanishes if you use these observers.

This is a very nice way of putting it!

 

[bcrowell]

I'm away from home for a couple of days, and I have very limited computer access on a very slow connection, but I will make some posts after I get back. If you have access, read section 4.8 from General Relativity: An Introduction for Physicists by Hobson, Efstathiou, and Lasenby. You might be able to read this section from Google Books.

Amazon gives access to that section through their "Look Inside" feature. I don't see what relevance it has to the present discussion. This is 4.8, "Tensors as geometrical objects?"

 

[TrickyDicky]

@TrickyDicky:
It don't think you want to see curves in space. You want to see a spacetime diagram of a neighbouring geodesic in some Riemann normal coordinates. They curve away from the origin for accelerated expansion, and curve inwards for decelerated expansion.
However, as bcrowell noted, the defining geodesic at the center is straight by definition.

Right, but in order to simplify, since I am a novice in this, I was asking for the case of a non-expanding spacetime manifold just to fix concepts before I go into the more geometrically complex FRW metric.

 

[Ich]

Right, but in order to simplify, since I am a novice in this, I was asking for the case of a non-expanding spacetime manifold just to fix concepts before I go into the more geometrically complex FRW metric.

The FRW metric is geometrically especially simple.
No matter, in normal coordinates, Newtonian mechanincs works well. Especially if you concentrate on radial motion.
If there is matter between worldlines, there is gravity, and the worldlines will converge.
If there is repulsion, like Dark Energy, worldlines will diverge.
If there is both, well, it depends on which is stronger.

All this works in principle like Newtonian mechanics, no matter which metric you choose to setup spacetime.
In normal coordinates, a FRW spacetime is just another ball of dust.

 

[George Jones]

Amazon gives access to that section through their "Look Inside" feature. I don't see what relevance it has to the present discussion. This is 4.8, "Tensors as geometrical objects?"

Oops; I meant section 14.8.

 

[bcrowell]

Oops; I meant section 14.8.

"Geodesics in the FRW metric"? Still not sure what point you're making.

 

[TrickyDicky]

No. Expansion of space is purely coordinate dependent. Or better, as George Jones puts it, "expansion" is a property of a congruence, like the one defined by the canonical observers in a FRW metric. It's not a property of spacetime.

Yes, that's why people are talking about "expanding space" at all.
Another more general set of fundamental observers, independent of FRW symmetries, is defined to be at rest in normal coordinates, maybe one could call them "Einstein observers", as they reproduce the inertial frames of SR if curvature is negligible, on which most people base their intuition. Expansion vanishes if you use these observers.

This is a very nice way of putting it!

But this seems to be at odds with General Covariance, according to which only those properties that are invariant under changes of coordinates are physically real, so if expansion vanishes just by a change of coordinates as youare claiming, then expansion is a coordinate artifact rather than a physical fact.
As I would hope you don't mean that, would you clarify this, perhaps I'm wrong about general covariance?

 

[Mentz114]

But this seems to be at odds with General Covariance, according to which only those properties that are invariant under changes of coordinates are physically real, so if expansion vanishes just by a change of coordinates as youare claiming, then expansion is a coordinate artifact rather than a physical fact.
As I would hope you don't mean that, would you clarify this, perhaps I'm wrong about general covariance?

From what I've been learning from this thread, we have grounds for selecting a 'canonical' coordinate system - i.e. a point of view where the universe appears homogenous and isotropic.

 

[TrickyDicky]

From what I've been learning from this thread, we have grounds for selecting a 'canonical' coordinate system - i.e. a point of view where the universe appears homogenous and isotropic.

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?

 

[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.