Is this a way to move faster than c?

[nutgeb]

A couple of additional thoughts about the physical and nonphysical interpretations I suggested for why photons move faster than c in FRW proper coordinates:

First, I have to acknowledge that the linear frame dragging idea requires a healthy dose of bootstrapping. It offers a potential answer as to why the proper-coordinates velocity of a photon can exceed c, but why can the recession velocities of massive objects (like galaxies) also exceed c? The answer must be that when each matter sphere (defined as the sphere extending out to the Particle Horizon of a central galaxy) drags the local space at its center along, it also drags along the galaxy at its center. But that means every galaxy is both being dragged, as well as helping to drag other galaxies. I don't know if that's a fatal defect in the idea. Mutual dragging may be permissible. For example if two equal sized black holes orbit each other, aren't each of them frame dragging the other, while also being frame dragged by the other? Do the orbital periods of both object precess?

Second, regarding the non-physical explanation: I am fairly convinced that for the empty Milne model the very different description of galaxy recession velocities in both the Minkowski and FRW coordinates really are just different ways to approach the same physical phenomenon.

FRW coordinates use non-Lorentz contracted proper distance, and non-dilated proper time as the coordinate system for all comoving local frames. So relative to these 'full size' coordinates', the distance and time values of recession events must have relatively larger values.

Minkowski coordinates use Lorentz hyperbolically contracted distance coordinates and dilated time coordinates for comoving galaxies with high recession velocities. So compared to the 'extra large' FRW coordinates, the scale of the universe as a whole is dramatically smaller, and the elapsed time since the Big Bang is dramatically shorter at the far ends of the universe. Relative to these 'shrunken' coordinates, the distance and time values of distant recession events also must have relatively smaller values.

Both coordinate systems preserve exactly the same Lorentz-proportional relationship between the event values they portray and the 'scale' of the coordinate system as a whole. So at cosmological distances the time scale, distance scale, and speed of light really are available in your choice of 'medium' and 'extra large' sizes.

However, in a gravitating universe, such as our own, Minkowski coordinates are not available at cosmological distances, so the speed of light is available only in 'extra large'.

And as said many times, in both coordinate systems the speed of light is c within every local frame.

 

[bcrowell]

To describe a cosmological solution, you need a coordinate system that is capable of covering cosmological distances. There is no Minkowski frame that can do this.

Agreed, but my comments were specifically about the kinematic paradigm, not the 'expanding space' paradigm.

The Milne model provides a satisfactory description of a universe without gravity, using Minkowski coordinates and GR.

The Milne model is only one special case of FRW. In general FRW models can't be described using Minkowski coordinates.

Agreed, as a general principle. However, it is appropriate to apply Birkhoff's Theorem to define a sphere of matter around a coordinate origin and then use Schwarzschild coordinates to model the role of gravity in relativistic effects, such as the redshift. (In Schwarzschild coordinates, cosmological redshift includes as discrete elements both gravitational and SR time dilation.) Birkhoff's Theorem, like the Newtonian Shell Theorem, says that all matter outside that sphere can be ignored for the purposes of modeling what happens gravitationally inside the sphere. So I expect that the same approach would work for linear frame dragging analysis.

Birkhoff's theorem has nothing to do with the fact that you can't describe a cosmological model using a gravitational potential.

Also, cosmological solutions are time-varying, and in general time-varying solutions don't have well-defined gravitational potentials.

Yes but it should be possible to take snapshots in time of a cosmological scenario, using Schwarzschild coordinates or the like.

This also has nothing to do with the fact that you can't describe a cosmological model using a gravitational potential.

 

[stevmg]

A couple of additional thoughts about the physical and nonphysical interpretations I suggested for why photons move faster than c in FRW proper coordinates:

First, I have to acknowledge that the linear frame dragging idea requires a healthy dose of bootstrapping. It offers a potential answer as to why the proper-coordinates velocity of a photon can exceed c, but why can the recession velocities of massive objects (like galaxies) also exceed c? The answer must be that when each matter sphere (defined as the sphere extending out to the Particle Horizon of a central galaxy) drags the local space at its center along, it also drags along the galaxy at its center. But that means every galaxy is both being dragged, as well as helping to drag other galaxies. I don't know if that's a fatal defect in the idea. Mutual dragging may be permissible. For example if two equal sized black holes orbit each other, aren't each of them frame dragging the other, while also being frame dragged by the other? Do the orbital periods of both object precess?

Second, regarding the non-physical explanation: I am fairly convinced that for the empty Milne model the very different description of galaxy recession velocities in both the Minkowski and FRW coordinates really are just different ways to approach the same physical phenomenon.

FRW coordinates use non-Lorentz contracted proper distance, and non-dilated proper time as the coordinate system for all comoving local frames. So relative to these 'full size' coordinates', the distance and time values of recession events must have relatively larger values.

Minkowski coordinates use Lorentz hyperbolically contracted distance coordinates and dilated time coordinates for comoving galaxies with high recession velocities. So compared to the 'extra large' FRW coordinates, the scale of the universe as a whole is dramatically smaller, and the elapsed time since the Big Bang is dramatically shorter at the far ends of the universe. Relative to these 'shrunken' coordinates, the distance and time values of distant recession events also must have relatively smaller values.

Both coordinate systems preserve exactly the same Lorentz-proportional relationship between the event values they portray and the 'scale' of the coordinate system as a whole. So at cosmological distances the time scale, distance scale, and speed of light really are available in your choice of 'medium' and 'extra large' sizes.

However, in a gravitating universe, such as our own, Minkowski coordinates are not available at cosmological distances, so the speed of light is available only in 'extra large'.

And as said many times, in both coordinate systems the speed of light is c within every local frame.

You mention "contraction" of distance and time...

Any idea how much such a contraction would be (a numerical "guess?") 10^-1, 10^-2, 10^-3 or what?

 

[nutgeb]

You mention "contraction" of distance and time...

Any idea how much such a contraction would be (a numerical "guess?") 10^-1, 10^-2, 10^-3 or what?

The contraction is the same kind of Lorentz contraction that applies in SR. It varies at a hyperbolic rate depending on recession velocity. At recession velocities well below c, the contraction is negligible. But in Minkowski coordinates, as the recession velocity approaches c, the Lorentz contraction approaches infinite. So while an FRW universe can have an infinite size, an empty Milne universe, in Minkowski coordinates, has a finite size. The finite size depends on the elapsed time since the Big Bang. Yet a finite Milne universe packs in an infinite number of Lorentz contracted galaxies (with recession velocities increasingly approaching c, compared to the observer). This is explained in http://world.std.com/~mmcirvin/milne.html" .

The Minkowski-coordinate size of the Milne universe is finite, but its Proper Distance size is infinite. FRW coordinates use Proper Distance and Proper time as the coordinate axes for all comoving frames. Proper Distance is a recognized concept in Minkowski coordinates, but it isn't portrayed directly on a Minkowski chart for all comoving frames.

 

[nutgeb]

Birkhoff's theorem has nothing to do with the fact that you can't describe a cosmological model using a gravitational potential.

This also has nothing to do with the fact that you can't describe a cosmological model using a gravitational potential.

I agree that a complete cosmological model can't be calculated using Birkhoff's. I was suggesting it as a way to take a snapshot (a constant time slice) of a spatial subset of a cosmological model in order to test certain characteristics.

What I'm specifically interested in is applying linear frame dragging within a cosmological model.

 

[bcrowell]

I agree that a complete cosmological model can't be calculated using Birkhoff's. I was suggesting it as a way to take a snapshot (a constant time slice) of a spatial subset of a cosmological model in order to test certain characteristics.

Birkhoff's theorem says that "any spherically symmetric solution of the vacuum field equations must be stationary and asymptotically flat. This means that the exterior solution must be given by the Schwarzschild metric." (Wikipedia) Cosmological solutions aren't vacuum solutions (except in the special case of the Milne universe), they aren't asymptotically flat, and they don't have a Schwarzschild metric.

 

[Ich]

bcrowell, I suggest you read up on Lemaître-Tolman dusts, Birkhoff's theorem and its corollaries, and how it allows one to ignore the rest of the universe when doing calculations in a finite region.
nutgeb definitely has a point here.

 

[bcrowell]

bcrowell, I suggest you read up on Lemaître-Tolman dusts, Birkhoff's theorem and its corollaries, and how it allows one to ignore the rest of the universe when doing calculations in a finite region.

I don't follow you at all. This thread is about cosmology. I don't see how Birkhoff's theorem can allow us to ignore the rest of the universe (modeled with a realistic, standard cosmological solution) when doing calculations in a finite region, when Birkhoff's theorem says that "any spherically symmetric solution of the vacuum field equations must be stationary and asymptotically flat." Except for pathological cases like the Milne model, cosmological solutions aren't vacuum solutions, they aren't stationary, and the standard ones such as FLRW aren't asymptotically flat.

Re Lemaître-Tolman dusts, I again don't understand your point. Here's what I found in a casual search on this topic:
http://en.wikipedia.org/wiki/Lemaitre–Tolman_metric
http://arxiv.org/abs/0802.1523
As far as I can tell, there has been a recent flurry of activity on this topic because inhomogeneous cosmological models may be capable of reproducing observations without invoking a cosmological constant. That's very interesting, but I don't see how it relates to the topic of this thread, which is a particular thought experiment involving a rope stretching across cosmological distances. I can see the hint of some vague connection with Birkhoff's theorem, because the Lemaître-Tolman metric is asymptotically flat. But the Lemaître-Tolman metric is not stationary, so it still seems to me that Birkhoff's theorem doesn't apply to it.

So in general I'm having trouble inferring anything about what you had in mind with #77...?

I know that you have some interest in nonstandard cosmological models, but it seems like you're expecting others to read your mind here as to what exactly you have in mind.

 

[nutgeb]

I don't see how Birkhoff's theorem can allow us to ignore the rest of the universe (modeled with a realistic, standard cosmological solution) when doing calculations in a finite region, when Birkhoff's theorem says that "any spherically symmetric solution of the vacuum field equations must be stationary and asymptotically flat." Except for pathological cases like the Milne model, cosmological solutions aren't vacuum solutions, they aren't stationary, and the standard ones such as FLRW aren't asymptotically flat.

I'll let Ich provide a longer answer, I don't have time right now.

The 'asymptotically flat' condition required by Birkhoff's means 'spatially flat', not flat spacetime. Our real gravitating universe is considered to be vanishingly close to spatially flat because it is at critical density.

The whole point of Birkhoff's (and the Newtonian Shell Theorem) is that a sphere of matter can be treated as a vacuum solution inside the sphere, just like external Schwarzschild coordinates which are also a vacuum solution can be used to model a dust sphere up to the radius of whatever object you're moving, and interior Schwarzschild coordinates can be used if the object is inside the radius of the dust sphere.

And as I've said repeatedly, a single constant time slice 'snapshot' of a non-stationary spacetime can be considered stationary spacetime for this purpose.

Birkhoff's has been used the way I'm suggesting by many prominent cosmologists in both textbooks and published papers. I don't have time to find citations for you right now.

You are wrong on this point. Please don't get offended.

Lemaitre-Bondi-Tolman (LTB) spacetimes can specifically be used for non-stationary spherically symmetrical spacetimes, but right now I'm focused on Scwarzschild snaphots instead.

 

[yuiop]

Birkhoff's theorem says that "any spherically symmetric solution of the vacuum field equations must be stationary and asymptotically flat. This means that the exterior solution must be given by the Schwarzschild metric." (Wikipedia) Cosmological solutions aren't vacuum solutions (except in the special case of the Milne universe), they aren't asymptotically flat, and they don't have a Schwarzschild metric.

I tend to agree with everything Ben is saying here. Birkhoff's theorem apllies to a vacuum. When there is matter external to the spherical shell we are considering, we can not ignore the external matter in GR. This is a direct contrast to Newton's shell theorem. The internal Schwarzschild solution demonstrates we have to take the external shells of matter into account when calculating gravitational potentials and gravitational gamma factors.

It is not difficult to use the internal Schwarzschild solution to calculate the gravitational time dilation factor for a particle inside a sphere of dust with homogenous density and then see that the gravitational time dilation factor for the particle is different if you remove the external dust shells.

 

[yuiop]

Your logic would have been correct if our spacetime was flat. But the fact is that our spacetime is curved. What that means is that you can't compare velocities of objects far away from each other. Thus, since galaxies are far away from you, strictly speaking there is no such thing as "the velocities of the galaxies relative to yourself".

The nice thing about the thought experiment in the OP is that it does provide us with a way (in principle) to define "the velocities of the galaxies relative to yourself". Attach a wire to the distant galaxy with a marker on the end of the wire, and the "real" velocity of the distant galaxy is the velocity of the wire end marker as it passes close by you. Now I know a lot of counter arguments have been advanced about requiring wires of infinite rigidy etc, but we take a less extreme example of not-so-distant galaxies moving at not so-so-extreme velocities and the "wire speed" would give a good working definition of the real relative velocity of the galaxy.

For example, we could attach a beacon to a long wire and lower the beacon to near the event horizon of a black hole and observe that the beacon signal is highly redshifted (high z) but if the end of the wire we are holding is stationary, then the beacon is stationary and the redshift is gravitational and not due to the velocity of the beacon relative to us. The lesson is that not all red shift is a result of relative velocity.

Now consider a very simplisitic cosmological model that consists of dust moving at random relativistic velocities, starting out from a very small confined region. Over time the the model settles down so that to any given observer, the velocity of any particle is roughly proportional to its distance from the observer. In this over simplisitic model the velocity of the particles is given by the relativistic Doppler factor as:

$$\frac{v}{c} = \frac{(z+1)^2-1}{(z+1)^2+1}$$

where z is the standard redshift factor used in cosmology.

It is easy to see that the simplistic equation above gives v/c <1 for any positive finite value of z.

Of course, the above equation does not take gravity into account and is a purely kinematic SR model. Now taking gravity into account makes things much more complicated, but I would like to suggest the following "big picture". It is obvious that in this model the average distance between dust particles increases over time and the average density of the universe therefore decreases over time. Now if we take the Schwarzschild solution, we can observe that the average density calculated by dividing the enclosing sphere by the enclosed mass is greater nearer the surface of the gravitational mass than further away, that there is relationship between increased gravitational gamma factor/ time dilation and increased mass density. Now if we apply this principle to the expanding universe (and it does not matter if we have an infinite universe as long as the average distance between particles is increasing) we might be able to conclude that when a light signal left a distant galaxy in the distant past, the density of the universe was higher and the gravitational potential was lower.

Now we know in the Schwarzschild solution, that a photon climbing from a low gravitational to high gravitational potential is red shifted and it is possible that a photon traveling from a time where the density of the universe was higher and arriving when the average density is lower will be red shifted due the change in density over the travel time of the photon. Now I am not saying that ALL the red shift is due to the change in average density of the universe, because it is implicit in my simple model that particles are MOVING away from each other, but I am also saying that that not all the redshift is purely due to motion away from us. Now although I can not put any equations to this, I think the overall big picture is that "real" relative velocitites are not as high as the red shifts suggest. Some of the red shift (in my simplistic mind view) is due to a temporal difference in gavitational potential during the travel time of the photon.

The nice thing about this picture is that an observer on the distant galaxy will see us redshifted by exactly the same factor for eactly the same reasons and there is no preferential viewpoint or location in this model. Another nice aspect of this model is that there is an effect gravitational curvature (or differences in gravitational potential) due to temporal reasons for the traveling photon, even when on large scales (and even in an infinite unverse) the distribution of matter is homogenous and the average density is the same everywhere at any given time. In this model the "real" relative velocities of visible distant galaxies would always be subluminal. Now I am using a rough interpretation of the Schwarzschild external metric to reach these conclusions, but a more precise answer will require analysis of the internal Schwarzschild metric (using the event/visible horizon as the surface boundary) to see how mass density affects gravitational potential/time dilation, but I think the conclusion will be braodly the same. I am not offering a "new theory" here. Just my interpretation/ mind model of how I view things, and I welcome enlightenment on how things really work cosmologically and why my ideas would not work. Basically, I am looking for someone to give me a better (but still fairly simple) mind model to visualise.

The problem I have with "galaxies embedded in 'something" flowing away from us" is the etherist overtones of the "substance" the galaxies are at rest with or embedded in. Do the FLRW type models make some form of Lorentz ether official?

It is obvious that FLRW type metrics and kinematic type models predict different results for the velocity of the end of wire marker passing us, even at sub luminal velocities, so the models are not just a difference in philosophical interpretation.

 

[Ich]

I don't see how Birkhoff's theorem can allow us to ignore the rest of the universe (modeled with a realistic, standard cosmological solution) when doing calculations in a finite region, when Birkhoff's theorem says that "any spherically symmetric solution of the vacuum field equations must be stationary and asymptotically flat."

In this context, the interesting thing about Birkhoff's theorem is the corollary that states that in every spherically symmetric spacetime (asymptotical flatness not required), the metric inside a spherical cavity is flat. Physics inside such a cavity is not affected by the rest of the universe. You can easily re-fill the cavity with the usual matter (at least perturbatively), and the result still holds.
As nutgeb said, this is used in some textbooks as a starting point to cosmology, e.g. http://books.google.com/books?id=uU...QS6SsLaq6yQTPyfypDQ&cd=1#v=onepage&q&f=false".

Re Lemaître-Tolman dusts, I again don't understand your point.

Analyze them, and you'll find that the behaviour of a shell is influenced by all the matter inside it, and not at all by all the matter outside: you can ignore the universe when dealing with a local patch.
This shows that the result I quoted before does not only hold in Newtonian or post-Newtonian approximation. Even if that'd be enough to do serious and accurate physics in a region of several Gly.

But the Lemaître-Tolman metric is not stationary, so it still seems to me that Birkhoff's theorem doesn't apply to it.

Birkhoff's theorem does not suppose staticity of the matter regions, to the contrary, it proves staticity of vacuum regions even in arbitrarily non-static surroundings. This is of high relevance.

I know that you have some interest in nonstandard cosmological models,

No, I'm only interested in standard cosmology. But I want to understand it, and this means that I try to approach it from as many viewpoints (i.e. coordinate descriptions) as possible. In my experience, that's the only way of gaining understanding in GR, because it enables one to extract the physics behind the coordinates.

it seems like you're expecting others to read your mind here as to what exactly you have in mind.

Well, I remember at least three times in the past year where I tried less psychic means to address you in this matter, like writing a post. But with no response, which I interpreted as a lack of interest on your side. So my intention here was to set the record straight and give nutgeb some support, not to explain my point again.

 

[yuiop]

In this context, the interesting thing about Birkhoff's theorem is the corollary that states that in every spherically symmetric spacetime (asymptotical flatness not required), the metric inside a spherical cavity is flat. Physics inside such a cavity is not affected by the rest of the universe. You can easily re-fill the cavity with the usual matter (at least perturbatively), and the result still holds.
As nutgeb said, this is used in some textbooks as a starting point to cosmology, e.g. http://books.google.com/books?id=uU...QS6SsLaq6yQTPyfypDQ&cd=1#v=onepage&q&f=false".

I agree that the metric inside a spherical cavity is flat (and therefore the gravitational potential is the same everywhere within the cavity), but I have this question. If we take two spheres, which have identical cavities but one sphere has a much thicker and denser shell, then the gravitational potential inside the two cavities will not be the same, no?

If the potential inside the two cavities is not the same, then the matter outside the cavity does have some influence on the gravitational potential within the cavity (at least on the magnitude, if not on the curvature of the gravitational potential).

 

[Ich]

Now although I can not put any equations to this, I think the overall big picture is that "real" relative velocitites are not as high as the red shifts suggest.

Ok, there's no "real" velocity between separated observers, and you better forget about that "temporal difference in gavitational potential".
The (approximative) equations are fairly simple then, just have a look at http://arxiv.org/abs/0809.4573" . There is a quadratic potential around the origin, causing additional time dilation. Try to do some calculations to find out how the results in this description match those of the FRW description.

It is obvious that FLRW type metrics and kinematic type models predict different results for the velocity of the end of wire marker passing us, even at sub luminal velocities, so the models are not just a difference in philosophical interpretation.

They don't. But you have to drop the assumption that the time derivative of proper cosmological distance is "the" velocity. Especially, two points of constant proper distance are not stationary wrt each other in any meaningful way. Or, the other way round, the ends of a "rigid" rope are not separated by a constant cosmological proper distance.

 

[Ich]

If the potential inside the two cavities is not the same, then the matter outside the cavity does have some influence on the gravitational potential within the cavity (at least on the magnitude, if not on the curvature of the gravitational potential).

And how does a constant offset in potential influence local physics?
You know, there is no canonical "potential" in most spacetimes. If a patch of spacetime is flat, that's all you have to know to predict physics inside that patch.
And if you introduce a potential in that patch to do Newtonian or post-Newtonian calculations, you're anyway free to choose its "zero value" arbitrarily.

 

[yuiop]

And how does a constant offset in potential influence local physics?
You know, there is no canonical "potential" in most spacetimes. If a patch of spacetime is flat, that's all you have to know to predict physics inside that patch.
And if you introduce a potential in that patch to do Newtonian or post-Newtonian calculations, you're anyway free to choose its "zero value" arbitrarily.

It does not affect local physics, but sending a signal from a very distant galaxy to here is not "local" physics. If the distant galaxy is at the centre of its own cavity and we are at the centre of our own cavity, then the gravitational potentials of the distant galaxy and ourselves is not necessarily the same. If you have two cavities, you can arbitrarily define one cavity to zero potential, but then you have to measure all other potentials relative to that zero, and it does not automatically follow that the second cavity will be at zero potential relative to the first.

I had a look at the paper you linked to (and thanks for the reference) and as I understand it in that paper they analyse the difference in gravitational potential in the two galaxies (A and B) like this. A is at the centre of a sphere of dust and B is at the surface of that sphere and using Birkoff's theorem we can ignore the matter external to the sphere centred on A. B is therefore at a higher gravitational potential than A and its signal is therefore blue shifted by the time the signal arrives at A. This means the velocity of B relative to A is much higher than the crude kinetic interpretation of the redshift suggests, and the relative velocity can even exceed the speed of light.

However, I have a problem with this interpretation. In a static analysis of the the two galaxies A and B in a homogeneous universe, they are each at the centre of their own gravitational sphere, so they have the same gravitational potentials and clocks on A and B are running at the same rate and there is no gravitational redshift between the two galaxies. Why should A be treated preferentially as being at the centre of a sphere, while B is treated as being at the edge of a sphere?

Here is a another thought experiment. Let us imagine we have another galaxy (C) exactly half way between A and B in this homogeneous universe acting as an independent observer. We treat C as being at the centre of its own gravitational sphere with A and B being opposite each other on the surface of the sphere centred on C. From C's point of view, A and B are at the same gravitational potential according to C and signals sent from A to B or vice versa are not red shifted relative to each other, which contradicts the claim that signals from B are gravitationally blue shifted as far as A is concerned.

That is a static analysis, but if A and B are moving away from each other, then by the time a signal travels from B (past C) and onto A, A will be effectively at a higher potential (further away from the centre C) when the signal arrives at A, than when the signal left B and this results in a red shift of the signal rather than a blue shift. Some cosmologists would explain it this way. The universe or gravitational sphere centred on C expands in the time the signal traveled from B to A and this "stretches" the wavelength of the signal, effectively red shifting the signal.

 

[yuiop]

They don't. But you have to drop the assumption that the time derivative of proper cosmological distance is "the" velocity. Especially, two points of constant proper distance are not stationary wrt each other in any meaningful way. Or, the other way round, the ends of a "rigid" rope are not separated by a constant cosmological proper distance.

Let us say we have a distant galaxy tethered to our galaxy by a high tensile wire. Let us further say that the tension in this wire remains constant over billions of years and the radar distance of the tethered galaxy remains constant over billions of years. To most people the tethered galaxy is at rest with respect to us in "real" terms, If the cosmologists tell us that this tethered galaxy is "really" moving relative to us, then clearly the cosmologists have a different notion of distance and velocity to the rest of us and when they tell us that distant galaxies are "really" moving away at speeds that are greater than the speed of light, then we would have to take into account that they have a strange notion of relative velocity.

 

[Ich]

Let us say we have a distant galaxy tethered to our galaxy by a high tensile wire. Let us further say that the tension in this wire remains constant over billions of years and the radar distance of the tethered galaxy remains constant over billions of years. To most people the tethered galaxy is at rest with respect to us in "real" terms

Yes, at least if the background doesn't change too much in the relevant time.

If the cosmologists tell us that this tethered galaxy is "really" moving relative to us, then clearly the cosmologists have a different notion of distance and velocity to the rest of us and when they tell us that distant galaxies are "really" moving away at speeds that are greater than the speed of light, then we would have to take into acount that they have a strange notion of relative velocity.

Yes.
There's the problem that most cosmologists don't communicate this fact, and that some cosmologists (e.g. Tamara Davis in her earlier papers) are not even aware of it.
The most general (and therefore not very enlightening) position is to simply state that there's no unambiguous definition of distance and velocity in GR.
A more helpful approach is to take the toy model where both (cosmological and standard SR) definitions can be applied, and compare them there.

 

[yuiop]

Hi Ich,

I have to admit that I find your arguments (consistent with the views of most cosmologists) almosts as complelling as my my own thoughts and arguments and at this point of time, I am a bit abivalent about what is "really" happening. Basically I am looking for a "clinching" argument that might settle the confusion in my head. On the plus side of my arguments (and of course I am biased) is that we can have an infinite, homogeneous, expanding (and possibly even accelerating) universe without requiring that observered red shifts of distant galaxies are explained by "real" super luminal relative velocities.

 

[bcrowell]

The 'asymptotically flat' condition required by Birkhoff's means 'spatially flat', not flat spacetime.

This is false.

First off, if you look at the standard definition of asymptotic flatness, it refers to flatness in the sense of a vanishing Riemann tensor, not spatial flatness. The general definition of asymptotic flatness is technically complicated, but it's pretty easy to tell that the definition does not refer to spatial flatness. For example, if you look at the introductory section of ch. 11 of Wald, where he introduces asymptotic flatness, it's very clear that he's concerned with making a definition that is coordinate-independent, whereas spatial flatness is a coordinate-dependent notion. Now if you transform the Schwarzschild metric, expressed in Schwarzschild coordinates, into a frame rotating rigidly about the origin with angular velocity $\omega$, you get a metric that, at large distances from the origin, is simply a Minkowski metric represented in rotating coordinates. On the axis, far from the origin, the Ricci scalar of the spatial metric equals $6\omega^2$. Therefore the spatial curvature of the Schwarzschild spacetime does not fall off to zero when expressed in one set of coordinates, but does when expressed in another set of coordinates. This counterexample establishes that the standard definition of asymptotic flatness, which is coordinate-independent, cannot refer to spatial curvature.

The next question is whether Birkhoff's theorem, in the formulation we've been discussing -- "Any spherically symmetric solution of the vacuum field equations must be stationary and asymptotically flat." -- refers to the standard definition of asymptotic flatness, or to the nonstandard one that you've proposed. You haven't provided any evidence for your assertion that it refers to the nonstandard definition, but in any case it's easy to show that it can't, by producing a counterexample to the theorem as construed by you. Here we have to consider the definition of "spherically symmetric." The question is whether this refers to a coordinate-dependent definition of symmetry or a coordinate-independent one. There is a proof of Birkhoff's theorem in appendix B of Hawking and Ellis, "The large scale structure of space-time." The first paragraph of this appendix defines spherical symmetry, and does it in a coordinate-independent way. Therefore the Schwarzschild metric described in a rotating frame is spherically symmetric according to the definition used in Birkhoff's theorem. If we then assume, as you've asserted, that "asymptotically flat" refers to spatial flatness, then this would constitute a counterexample to Birkhoff's theorem, and Birkhoff's theorem would be false.

Our real gravitating universe is considered to be vanishingly close to spatially flat because it is at critical density.

This is irrelevant for three reasons: (1) Birkhoff's theorem refers to spacetime flatness, not spatial flatness. (2) A restriction to the special case of a spatially flat cosmology would contradict your earlier assertions that Birkhoff's theorem can be used as a general tool in cosmology. (3) Birkhoff's theorem applies to vacuum solutions, but a spatially flat cosmological solution is not a vacuum solution (except in the special case of the FRW solution with zero matter density, but in that case Birkhoff's theorem becomes useless as a tool for doing what you have been claiming it can be used for).

The whole point of Birkhoff's (and the Newtonian Shell Theorem) is that a sphere of matter can be treated as a vacuum solution inside the sphere, just like external Schwarzschild coordinates which are also a vacuum solution can be used to model a dust sphere up to the radius of whatever object you're moving, and interior Schwarzschild coordinates can be used if the object is inside the radius of the dust sphere.

This is incorrect, for two reasons. (1) You've claimed this repeatedly in a context where it was clear you thought it applied to cosmological solutions. Birkhoff's theorem doesn't apply to cosmological solutions, except for trivial vacuum solutions, in which case the dust you're referring to doesn't exist. (2) The field equations of GR, unlike those of Newtonian gravity, are nonlinear. This is precisely why you can't do what you're claiming you can do, which is to break a symmetric mass distribution up into concentric shells and sum the contributions of the shells, as you could with the shell theorem.

And as I've said repeatedly, a single constant time slice 'snapshot' of a non-stationary spacetime can be considered stationary spacetime for this purpose.

This is incorrect. The term "stationary" is meaningless when applied to a spacelike surface in this way.

Birkhoff's has been used the way I'm suggesting by many prominent cosmologists in both textbooks and published papers. I don't have time to find citations for you right now.

You won't find such citations, because they don't exist.

 

[bcrowell]

In this context, the interesting thing about Birkhoff's theorem is the corollary that states that in every spherically symmetric spacetime (asymptotical flatness not required), the metric inside a spherical cavity is flat. Physics inside such a cavity is not affected by the rest of the universe. You can easily re-fill the cavity with the usual matter (at least perturbatively), and the result still holds.
As nutgeb said, this is used in some textbooks as a starting point to cosmology, e.g. http://books.google.com/books?id=uU...QS6SsLaq6yQTPyfypDQ&cd=1#v=onepage&q&f=false".

I have no argument with this, but this is not what nutgeb said. The first place Birkhoff's theorem was mentioned in this thread was in nutgeb's #70, where he said this:

Agreed, as a general principle. However, it is appropriate to apply Birkhoff's Theorem to define a sphere of matter around a coordinate origin and then use Schwarzschild coordinates to model the role of gravity in relativistic effects, such as the redshift. (In Schwarzschild coordinates, cosmological redshift includes as discrete elements both gravitational and SR time dilation.) Birkhoff's Theorem, like the Newtonian Shell Theorem, says that all matter outside that sphere can be ignored for the purposes of modeling what happens gravitationally inside the sphere. So I expect that the same approach would work for linear frame dragging analysis.

There is no mention of a spherical cavity here. As has become clear in later posts, nutgeb did not understand the meaning of Birkhoff's theorem, and thought it could be used for things that it can't be used for. The context of #70 was that nutgeb was claiming to be able to use a gravitational potential in order to analyze cosmological solutions.

 

[Ich]

Hi kev, I still have to answer your previous post.

If you have two cavities, you can arbitrarily define one cavity to zero potential, but then you have to measure all other potentials relative to that zero, and it does not automatically follow that the second cavity will be at zero potential relative to the first.

No, but I'm not concerned with different cavities. I want to model what's inside one cavity.

A is at the centre of a sphere of dust and B is at the surface of that sphere and using Birkoff's theorem we can ignore the matter external to the sphere centred on A. B is therefore at a higher gravitational potential than A and its signal is therefore blue shifted by the time the signal arrives at A.

Exactly.

This means the velocity of B relative to A is much higher than the crude kinetic interpretation of the redshift suggests, and the relative velocity can even exceed the speed of light.

Well, every photon in a higher potential goes faster than the speed of light in that sense. If light goes faster than the speed of light, it's time to see that we're talking more about coordinates (time dilation in that case) than physical impossibilities.

In a static analysis of the the two galaxies A and B in a homogeneous universe, they are each at the centre of their own gravitational sphere, so they have the same gravitational potentials and clocks on A and B are running at the same rate and there is no gravitational redshift between the two galaxies.

There is no potential defined at all. You can say that all clocks tick at the same rate in a coordinate system where the time coordinate is the proper time of each comoving observer. But that's trivially true.

Why should A be treated preferentially as being at the centre of a sphere, while B is treated as being at the edge of a sphere?

Because you chose A to be at the center.

...From C's point of view, A and B are at the same gravitational potential according to C and signals sent from A to B or vice versa are not red shifted relative to each other, which contradicts the claim that signals from B are gravitationally blue shifted as far as A is concerned.

Forget about the real reason for blue- and redshift. The transformation from one viewpoint to the next is equivalent to the introduction of a homogeneous gravitational field. The equivalence principle tells us that this is a mere coordinate transformation and doesn't change any results.
Try to do the calculations. One time, it's gravitational blueshift, the other time it's the observer losing speed during the light travel time, thus reducing the redshift. The effect is the same.
It's all a bit clumsy, because Newtonian calculations imply absolute velocity and acceleration. But they work, of course.

 

[Ich]

The context of #70 was that nutgeb was claiming to be able to use a gravitational potential in order to analyze cosmological solutions.

Which is perfectly legitimate. Because, as I said, you can cut out a cavity, see that it's flat spave, re-fill the cavity with what's been there before, and then do perturbative calculations. We're talking about really weak fields on the scale of some Mly or, say, a galaxy or a solar system.
Which means that you can do exact calculations in that patch (at the perturbative level) without caring about the rest of the universe. And, of course, as you're working with static coordinates then, you can define and use a gravitational potential.

 

[bcrowell]

Ich, I don't have any objection to your statements about cavities, but nutgeb never mentioned cavities, and his posts contained many mistakes, which I've pointed out.

It seems to me that quite a bit of this recent discussion has nothing to do with the (very interesting, IMO) GR paradox involving a rope proposed by the OP. If nutgeb wants to discuss linear frame dragging, for example, then I would be interested in learning more about that topic, but it seems to me that that should happen in a separate thread, because I don't see any evidence that it has any relevance at all to the rope paradox. I have started a separate thread with some questions about the technical aspects of Birkhoff's theorem, because I think that whole discussion in this thread has taken us far off the topic of the OP.

 

[bcrowell]

The nice thing about the thought experiment in the OP is that it does provide us with a way (in principle) to define "the velocities of the galaxies relative to yourself". Attach a wire to the distant galaxy with a marker on the end of the wire, and the "real" velocity of the distant galaxy is the velocity of the wire end marker as it passes close by you. Now I know a lot of counter arguments have been advanced about requiring wires of infinite rigidy etc, but we take a less extreme example of not-so-distant galaxies moving at not so-so-extreme velocities and the "wire speed" would give a good working definition of the real relative velocity of the galaxy.

I agree that the OP's thought experiment is fascinating, and I want to thank you for steering the discussion back to it.

It's true that in the limit of not-so-distant galaxies, we can define an unambiguous notion of relative speed. However, I don't think that should be taken as implying the same thing for more distant galaxies. The equivalence principle guarantees that spacetime is locally Minkowski, and nobody is disputing that there is a preferred notion of relative velocity in Minkowski space. The issue is whether that can be bootstrapped up to cosmological distances.

Now consider a very simplisitic cosmological model that consists of dust moving at random relativistic velocities, starting out from a very small confined region. Over time the the model settles down so that to any given observer, the velocity of any particle is roughly proportional to its distance from the observer. In this over simplisitic model the velocity of the particles is given by the relativistic Doppler factor as:

$$\frac{v}{c} = \frac{(z+1)^2-1}{(z+1)^2+1}$$

where z is the standard redshift factor used in cosmology.

It is easy to see that the simplistic equation above gives v/c <1 for any positive finite value of z.

What you're describing here is the Milne universe. Since the Milne universe is a flat spacetime, you could say there is a preferred notion of relative velocity, which you can get by describing it in standard Minkowski coordinates. However, there is also a set of co-moving coordinates that you could argue is also natural to use -- maybe even more natural, since we have all these dust particles that define a natural local rest frame (in the same way that the CMB defines a natural local rest frame in our own universe). You now have a problem with defining the relative velocity of a distant object, because what you have in mind is the object's velocity "now," but Minkowski observers and comoving observers disagree on simultaneity. This isn't an issue as long as the object is moving inertially, because its velocity isn't changing, so the result doesn't depend on what you mean by "now." But it is an issue if, for example, the object has ever experienced an acceleration at any time in the past.

Of course, the above equation does not take gravity into account and is a purely kinematic SR model. Now taking gravity into account makes things much more complicated, but I would like to suggest the following "big picture". It is obvious that in this model the average distance between dust particles increases over time and the average density of the universe therefore decreases over time. Now if we take the Schwarzschild solution, we can observe that the average density calculated by dividing the enclosing sphere by the enclosed mass is greater nearer the surface of the gravitational mass than further away, that there is relationship between increased gravitational gamma factor/ time dilation and increased mass density. Now if we apply this principle to the expanding universe (and it does not matter if we have an infinite universe as long as the average distance between particles is increasing) we might be able to conclude that when a light signal left a distant galaxy in the distant past, the density of the universe was higher and the gravitational potential was lower.

Here you have a problem because nontrivial cosmological solutions are time-varying, so you can't define a gravitational potential. There's a good discussion of this in Rindler, Essential Relativity, 2nd ed., section 7.6.

Now we know in the Schwarzschild solution, that a photon climbing from a low gravitational to high gravitational potential is red shifted and it is possible that a photon traveling from a time where the density of the universe was higher and arriving when the average density is lower will be red shifted due the change in density over the travel time of the photon. Now I am not saying that ALL the red shift is due to the change in average density of the universe, because it is implicit in my simple model that particles are MOVING away from each other, but I am also saying that that not all the redshift is purely due to motion away from us. Now although I can not put any equations to this, I think the overall big picture is that "real" relative velocitites are not as high as the red shifts suggest. Some of the red shift (in my simplistic mind view) is due to a temporal difference in gavitational potential during the travel time of the photon.

This is exactly the ambiguity that makes it impossible to define a gravitational potential when you have a time-varying solution. You can never establish how much of the redshift was kinematic and how much was gravitational. As an extreme example, imagine that you live in galaxy A, in a closed universe. You send out a photon, and a long time later you receive the same photon back, red-shifted. How much of this red-shift was kinematic, and how much was gravitational? If you know that it was your own photon that you received, then you could say that obviously it was 100% gravitational, and your galaxy's present velocity relative to its past velocity is zero. On the other hand, a distant observer B will say, "No, kev, I've been watching your galaxy A the whole time, and it's clearly been accelerating. It accelerated so that by the time it received the photon, you were moving toward the photon at a velocity higher than you had when you emitted it. Therefore you're seeing a combination of kinematic blueshift and gravitational redshift." Yet another observer, C, could say that your galaxy's acceleration was in the opposite direction, so they'd claim that it was a combination of kinematic redshift and gravitational redshift.

Although this scenario of galaxies A, B, and C is posed in the case of a closed universe, I think the same issues occur in open cosmologies. When a photon is emitted by D and received by E, we can't resolve the ambiguity of kinematic versus gravitational redshifts unless we can determine how fast D is moving "now," and we can't do that without figuring out how much D's velocity has changed since the time of emission. But different observers disagree on that. D says he's remained at rest during the whole time it took the photon to get to E. E says D has been accelerating.

The nice thing about this picture is that an observer on the distant galaxy will see us redshifted by exactly the same factor for eactly the same reasons and there is no preferential viewpoint or location in this model.

I think the example of A, B, and C above shows that this method actually has an observer-dependence involved.

The problem I have with "galaxies embedded in 'something" flowing away from us" is the etherist overtones of the "substance" the galaxies are at rest with or embedded in. Do the FLRW type models make some form of Lorentz ether official?

FLRW models have a preferred frame of reference, which in our universe can be interpreted as the frame of the CMB (i.e., in which the dipole variation of the CMB across the sky vanishes). This is different from an ether theory, in which the laws of physics have a preferred frame of reference. As an extreme example, consider a Milne model in which all the test particles are at rest relative to all the other test particles. There is clearly a preferred frame, but it's not a preferred frame built into the laws of physics.

It is obvious that FLRW type metrics and kinematic type models predict different results for the velocity of the end of wire marker passing us, even at sub luminal velocities, so the models are not just a difference in philosophical interpretation.

I'm not following you here. What do you mean by "kinematic type models?"

Returning to the issue of simultaneity that I raised above, I think I can see a good general way to analyze the rope paradox as initially posed by the OP.

The rope paradox has problems similar to the ones in the ABC scenario I described above. In a closed universe, you can wrap a rope all the way around the universe and determine that your own galaxy's velocity, relative to itself, right "now," is some huge number (perhaps greater than the speed of light). This conclusion is obviously absurd, so there's clearly something wrong here.

Without resorting to a closed universe, we can still produce issues of the same type. If the rope is tied to D, and E observes it going by at some speed, E can't conclude that that is D's speed "now." The information conveyed by the rope's end is at least as old as the time it takes sound waves to travel the length of the rope. If E is going to infer D's velocity "now," E has to correct for the amount of change in D's velocity during that time. Different observers say different things about that change in velocity. D says it's zero. E says it's not zero.

Since the rope end's velocity doesn't tell us anything about D's velocity relative to E right "now," there is no obvious reason to think that the rope's end would have a superluminal velocity, even if D's coordinate velocity relative to E is superluminal. If we see the rope going by at 98% of the speed of light, we can probably conclude that the rope has broken a long time ago, and furthermore the time it would have taken a vibration to travel from D to E is probably greater than the present age of the universe.

 

[Ich]

The equivalence principle guarantees that spacetime is locally Minkowski, and nobody is disputing that there is a preferred notion of relative velocity in Minkowski space. The issue is whether that can be bootstrapped up to cosmological distances.

You won't get an unambiguous notion of relative speed, but you can at least introduce a measure of speed that is compatible with what we think speed is. I.e. converges to SR speed in a flat spacetime.

You now have a problem with defining the relative velocity of a distant object, because what you have in mind is the object's velocity "now," but Minkowski observers and comoving observers disagree on simultaneity. This isn't an issue as long as the object is moving inertially, because its velocity isn't changing, so the result doesn't depend on what you mean by "now."

That's not the whole story. Even with inertially moving observers, if "Minkowski relative velocity" is zero all the time, the cosmological "recession velocity" in nonzero all the time. The problem is already in the definition of distance.

When a photon is emitted by D and received by E, we can't resolve the ambiguity of kinematic versus gravitational redshifts unless we can determine how fast D is moving "now," and we can't do that without figuring out how much D's velocity has changed since the time of emission. But different observers disagree on that. D says he's remained at rest during the whole time it took the photon to get to E. E says D has been accelerating.

As long as there is a reasonable notion of simultaneity, you can still decompose the kinematical and the gravitational part. There's a relative velocity at the time of emission (kinematic), and there's a change in relative velocity due to gravitation during the light travel time. It doesn't matter whether the photon or the observer is accelerated.
The kinematic part is linear with distance, the gravitational quadratic.

Since the rope end's velocity doesn't tell us anything about D's velocity relative to E right "now," there is no obvious reason to think that the rope's end would have a superluminal velocity, even if D's coordinate velocity relative to E is superluminal. If we see the rope going by at 98% of the speed of light, we can probably conclude that the rope has broken a long time ago, and furthermore the time it would have taken a vibration to travel from D to E is probably greater than the present age of the universe.

A superluminal recession velocity is just a number. You don't have to invoke this kind of simultaneity argument to resolve it. In the Milne universe, recession velocity is actually a rapidity, so its definition is a priori incompatible with velocity as we know it.

You can define an abstract, ideally stiff version of the rope.
Consider a non-expanding chain of observers. Operationally defined, that means that every measures a constant distance to his close neighbour (two-way light travel time or redshift).
It doesn't matter how these observers keep their position (most likely you'd have to pre-arrange their moition), as long as they have finite acceleration. One end of the chain (the origin) is comoving.

This construct is the nearest thing to a static radial coordinate centered at the origin. It also establishes a different simultaneity convention wrt the origin. Both coordinates (time and space) converge to Minkowski coordinates in the limit of zero matter content.

In an open topology, every point of the rope has a local velocity smaller than c. But the length of the rope may be constrained at a point where it would have to become c, which violates the finite acceleration condition. That's where the rope enters an event horizon.
Note that it is perfectly possible for the rope to be stable in the supposed "superluminal" region r>1/H. It just turns out that its velocity is smaller than c.

By the rope's simultaneity, it's also possible that the ends of the rope are still in the Big Bang, with their local velocity reaching c there - possibly with finite acceleration. It's rather the rest of the universe going mad then.

I'll have to think more about the closed topology case, which is more complicated. But I think the open case is interesting enough for now.

 

[bcrowell]

As long as there is a reasonable notion of simultaneity, you can still decompose the kinematical and the gravitational part.

There is no reasonable notion of simultaneity. Observers in galaxies moving away from one another disagree on simultaneity.

A superluminal recession velocity is just a number. You don't have to invoke this kind of simultaneity argument to resolve it.

The OP described a thought experiment that was imagined to produce a superluminal velocity of nearby objects (the trailing end of the rope and a galaxy that it flew by), not just distant objects.

You can define an abstract, ideally stiff version of the rope.
Consider a non-expanding chain of observers. Operationally defined, that means that every measures a constant distance to his close neighbour (two-way light travel time or redshift).
It doesn't matter how these observers keep their position (most likely you'd have to pre-arrange their moition), as long as they have finite acceleration. One end of the chain (the origin) is comoving.

Replacing a material rope with an actively maintained one doesn't affect anything of interest. You still have all the issues that make an ideally inelastic rope an impossibility in relativity. For example, to maintain perfect stiffness, the chain of observers have to communicate and keep adjusting themselves relative to their neighbors. If you want them to behave analogously to an ideally stiff rope, then they have to behave as a rope on which disturbances propagate at infinite velocity. They can't do this, because they can't communicate at infinite velocity.

 

[bcrowell]

It occurs to me that some people in this thread, including me, have been a little sloppy in our discussion of the role of a preferred set of coordinates in an FRW solution.

There are actually lots of different sets of coordinates that are commonly used to describe an FRW solution. Eric Linder lists four of them, which he calls isotropic, comoving, standard, and conformal, on p. 15 of "First Principles of Cosmology." They don't even all have the same time coordinate.

We've been referring to "coordinate velocities" as if they indicated the speeds at which distances between galaxies increased, but I believe that in isotropic, comoving, and standard coordinates, $\Gamma^r_{tt}=0$, so a galaxy that is initially at rest has a coordinate velocity dr/dt=0 forever.

I think a lot of the conceptual difficulties of this thread boil down to the issue that all of these global coordinate systems are incompatible with local coordinate systems tied to individual galaxies. For instance, if I say galaxy G is receding from us at a certain speed, then I know that my ideas of simultaneity don't agree with G's, and this isn't consistent with the cosmological t defined in one of the global coordinate systems.

 

[Ich]

Observers in galaxies moving away from one another disagree on simultaneity.

Yes, but until this effect becomes important, you have quite an area where the decomposition works unambiguously. At 100 Mpc, velocity is a mere 0.02 c.

The OP described a thought experiment that was imagined to produce a superluminal velocity of nearby objects (the trailing end of the rope and a galaxy that it flew by), not just distant objects.

Yes, because the OP was led to believe that recession velocities are velocities, and that therefore a superluminal recession velocity should be significant in one way or another. Most people believe that.

Replacing a material rope with an actively maintained one doesn't affect anything of interest.

I disagree. First, it gives you a strict definition, so everybody agrees on how the rope behaves. Then, you get rid of all these distracting engineering matters like

You still have all the issues that make an ideally inelastic rope an impossibility in relativity. For example, to maintain perfect stiffness, the chain of observers have to communicate and keep adjusting themselves relative to their neighbors.

As I said, just assume that the motion is pre-arranged in a suitable way. As long as this is physically possible, you have something which is as close to a rigid rope as it can get.

A rigid rope is something where every part is exactly at rest with its immediate neighbour. You can construct such a thing just the same way as you can construct a Born rigid body: by making every part move in an exactly defined way. You would not argue that Born rigid bodies are useless, as real bodies will contract due to acceleration stresses. You would use them as an example for accelerated motion where all the engineering matters are solved.
Same here with the rope.

 

[bcrowell]

You would not argue that Born rigid bodies are useless, as real bodies will contract due to acceleration stresses.

I would definitely argue that Born-rigid bodies are useless. For a good discussion of this, see Ø. Grøn, Relativistic description of a rotating disk, Am. J. Phys. 43 869 (1975). In section IV, he shows that giving an angular acceleration to a Born-rigid body is a kinematical impossibility.

 

[DrGreg]

I would definitely argue that Born-rigid bodies are useless. For a good discussion of this, see Ø. Grøn, Relativistic description of a rotating disk, Am. J. Phys. 43 869 (1975). In section IV, he shows that giving an angular acceleration to a Born-rigid body is a kinematical impossibility.

If we were talking about a 2-dimensional rotating disk, that would be a fair comment. But we're talking about a 1-dimensional "rod" in linear motion.

 

[bcrowell]

If we were talking about a 2-dimensional rotating disk, that would be a fair comment. But we're talking about a 1-dimensional "rod" in linear motion.

What if we're in a closed universe, and the one-dimensional rod lies along a spacelike geodesic that wraps all the way around the universe? Hmm...now it's starting to smell like a disk. Do we add an artificial constraint that says that the rod can't accelerate longitudinally? How is that constraint enforced, and how does it affect the underlying logic of the paradox proposed by the OP? Comoving observers in a cosmological solution always say that other comoving observers accelerate, so how do we forbid acceleration? It makes my head hurt.

Perhaps more fundamentally, the motivation for laying down a Born-rigid object seems to be that it allows us to define some kind of measurement system that allows us to determine observationally various things such as relative velocities of distant objects, which GR tells us fundamentally are meaningless things to talk about. But in order to carry out the choreographed program of accelerations that are required for Born-rigidity, we need some kind of prearranged marching orders from the Master Choreographer. How does the Master Choreographer know how to write these orders? Presumably because He knows what is going on everywhere in the universe. In that case, why do we need the fancy measurement apparatus? Why can't we just have Him appear as a burning bush, and reveal to us the things we want to know?

 

[yuiop]

I think a lot of the conceptual difficulties of this thread boil down to the issue that all of these global coordinate systems are incompatible with local coordinate systems tied to individual galaxies. For instance, if I say galaxy G is receding from us at a certain speed, then I know that my ideas of simultaneity don't agree with G's, and this isn't consistent with the cosmological t defined in one of the global coordinate systems.

It is true that in GR there are various coordinate systems and different ways of defining distance but do you agree that if we attach a wire to a distant galaxy (not necessarily superluminal) then there should be an unambiguous answer to the velocity of the end of the wire that passes right by us, even if we are having trouble calculating exactly what it would be at the moment? In other words if the measure the redshift of the distant galaxy to be z then we should be able to say that that the velocity of the end of the wire nearest us would be v(z) where v is a function of z. This is the most unambiguous and intuitive definition of the velocity of the distant galaxy relative to us that I can think of.

Well, every photon in a higher potential goes faster than the speed of light in that sense. If light goes faster than the speed of light, it's time to see that we're talking more about coordinates (time dilation in that case) than physical impossibilities.

This is a very good point. An observer low down in the gravitational potential well of a Schwarzschild object could measure the velocities of objects higher up to be apparently moving faster than his local measurement of the speed of light. This is definitely worth bearing in mind in these discussions.

Try to do the calculations. One time, it's gravitational blueshift, the other time it's the observer losing speed during the light travel time, thus reducing the redshift. The effect is the same.

I am not quite sure I understand you here. The stationary observer low down sees a gravitational [STRIKE]redshift[/STRIKE] blueshift of the signal from higher up (basically because his clock is running slower) Another way of looking at it is the equivalence principle. The observer lower down feels an upward proper acceleration and we can view it as if the "stationary" low down observer is accelerating towards the source during the signal travel time, so that he sees a doppler blueshift in the signal due to his effective increased velocity towards the source. Not sure why you said "observer losing speed during the light travel time" unless you meant he had an effective velocity away the source initially (equivalent to the velocity that you obtain if you treat the gravitational gamma factor as a kinematic gamma factor).

In is interesting to consider the Doppler redshift in a simple accelerating expansion model that ignores attractive gravity. When calculating the redshift of a distant galaxy with velocity v, we can treat the distant galaxy as stationary and the light signal is in effect chasing after us. During the light travel time the expansion of the universe makes our velocity greater than the initial relative velocity (v) and the measured redshift of the light from the distant galaxy is a measure of the distant galaxy's velocity relative to us "now" rather than a measure of the distant galaxies relative velocity at the time it emitted the signal.

You can define an abstract, ideally stiff version of the rope.
Consider a non-expanding chain of observers. Operationally defined, that means that every measures a constant distance to his close neighbour (two-way light travel time or redshift).
It doesn't matter how these observers keep their position (most likely you'd have to pre-arrange their motion), as long as they have finite acceleration. One end of the chain (the origin) is comoving.

I really like this idea. I think it would be a very good basis for any objective analysis of the problem at hand, but I have a slightly different operational definition to yours.

Let us consider an infinite, homogeneous, isotropic and expanding universe.

We place the observers at regular intervals in a chain and each observer is asked to maintain station with their closest neighbour. Let us say the distance between any two observers is 100 mpc so that the Hubble recession velocity is less than 0.00002c at that distance. There should be no difficulty maintaining a constant separation at that distance. In fact we could attach wires between any two neighbouring observers without worrying about needing wires of infinite rigidity. Whether or not there will be any significant tension in the wires is debatable, but for mutually stationary observers I think it would be reasonable to assume the wire tension is constant over time. I think it is also reasonable that the radar distance between any two neighbouring observers remains constant over time and I will use that as the operational definition of being stationary wrt each other over these sorts of distances.

Now if I use the argument of "temporal differences of gravitational potential" that I introduced earlier, then I have to consider how that will affect the radar distance over time. Using that argument, the mass density of an expanding universe is always reducing and so the effective gravitational potential is increasing over time and clock rates are effectively speeding up. During the radar measurement, the signal is speeding up over time, because of the reducing density that it finds itself in during its travels, but this is exactly compensated for by the increasing clock rates of the observers and so the radar distance remains constant and the apparent speed of light remains constant even with a changing density and effective gravitational potential. Therefore using a constant radar distance (or ruler distance) as the definition of being stationary wrt each other is valid whether temporal change in gravitational potential is considered or not. Where temporal change in potential does make a difference, is when you consider redshift. During the travel time, the temporal potential change means the photon is always moving from a higher density universe to a lower density universe (or a lower potential to a higher potential) during its travels, so this idea predicts that when the radar and ruler distance is constant there will be a non-zero redshift of signals sent between the stationary observers. In other words, non-zero redshift does not imply non-zero relative motion using this idea. However, I am not saying that that the universe is not expanding. The distant observers at rest wrt us, will see nearby galaxies at rest with the Hubble flow (and the CMB) whizzing past them and in fact, the temporal difference in gravitational potential requires that the universe is expanding.

Since I seem to be implying that our clock rates are increasing over time as the universe expands and the average density decreases, then wouldn't we be able to detect this in our labs? I think the answer is no. If we consider a closed lab low down in a Schwarzschild potential, they will measure the local speed of light to be c. If the lab is slowly raised to a higher potential, clocks in the lab speed up and vertical rulers expand, but the lab occupants are unable to detect this, because they always measure the speed of light in the lab to c. It is only when they send signals from a low lab to a different high lab, that these differences in clock rates reveal themselves in the form of redshift. In the Schwarzschild example, signals from a stationary source lower down, redshift because they come from a PLACE where gravitational potential is lower, while in the temporal gravitational potential example, signals from a stationary source redshift, because they come from a TIME when gravitational redshift was lower. This temporal change in potential is not detectable locally in a closed lab, just as in the Schwarzschild example, but reveals itself over cosmological distances.

The primary question is will a very distant observer at the end of very long chain of observers at rest wrt us, ever see galaxies that are near them, but at rest with the Hubble flow, moving at greater than the speed of light relative to themselves and I am pretty sure most people here would agree that the answer is no. The secondary question is, will observers that are are at constant ruler and radar distance from each other, measure a redshift in signals sent to each other, if the distances and travel times are cosmologically significant?

 

[Ich]

I would definitely argue that Born-rigid bodies are useless.

[...]Presumably because He knows what is going on everywhere in the universe. In that case, why do we need the fancy measurement apparatus? Why can't we just have Him appear as a burning bush, and reveal to us the things we want to know?

It seems that your concept of "useful" is quite different from mine - and from that of most scientists, for that matter. It also appears that you have a remarkably emotional way of looking at the concept of Born rigidity.
Of course you understand that by "Born-rigid body" I mean the mathematical concept and physical abstraction, not the real "ACME Born Rigid Body - do not turn"?

What if we're in a closed universe, and the one-dimensional rod lies along a spacelike geodesic that wraps all the way around the universe? Hmm...now it's starting to smell like a disk. Do we add an artificial constraint that says that the rod can't accelerate longitudinally? How is that constraint enforced, and how does it affect the underlying logic of the paradox proposed by the OP? Comoving observers in a cosmological solution always say that other comoving observers accelerate, so how do we forbid acceleration? It makes my head hurt.

If it makes your head hurt, start with simpler things, get your head around them, and then advance to higher levels of complexity. But I don't see how you're going to make progress if you refuse the physicist's approach of abstracting from the "general whole" to the relevant underlying principles. The "general whole" is almost always a total mess, and from what you say I gather that you'd refuse to attack a problem unless everything is considered from the start, or to use models with limited validity.

If so, I don't see a point in this thread. Except that you stated that ropes can be complicated and thus can't be modeled.

 

[yuiop]

What if we're in a closed universe...

I think closed universes are a bit messy and current cosmological observations can not rule out a flat or open universe. Personally, I hope advanced measurements will rule out the closed case and make that mess go away.

Consider two galaxies a distance (x) apart on the surface of sphere of radius (r) that represents the topology of a closed universe. Normally we would say the the gravitational attraction between the two points is proportional to GM/x^2. In the closed universe we would have to say there is an additional force that goes all the way around the universe the long way, with magnitude GM/(2*PI*r -x)^2 that acts to pull the galaxies apart. That means we would have to reformulate the equation for gravitational attraction. This of course assumes that the universe has existed long enough for the two points to become aware of their effective mirror image in the closed universe. As I said, closed universes are a bit messy and I hope they go away soon . Some cosmologists have actually looked at the patterns in the CMB to see if there are repeating patterns in opposite parts of the sky suggesting a closed universe and failed to find any evidence for the closed universe idea, using that method.