Is this a way to move faster than c?

[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.

 

[yuiop]

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.

Born rigid bodies are not useless for linear acceleration and could in principle exist and be accelerated with Born rigid motion. In the case of a Born rigid disk, we could give it Born rigid angular acceleration, if we relax the constraint that the radius must remain constant. I guess a Born rigid ring would be more appropriate in that case, which could be spun up to a given angular velocity while the distance between adjacent points on the ring remains constant from the point of view of observers on the ring, if the radius of the ring is allowed to shrink by the appropriate amount as the ring is spun up. In the case of a ring that circles the closed universe, there is no requirement to give it angular velocity. Sure the distance between adjacent observers on the ring is increasing as the universe expands, so that the observers appear to be moving relative to each other, but there is no overall angular motion imparted to the ring.

 

[yuiop]

This is an attempt at an counter argument to the "we can ignore all the mass in the universe that is not inside the sphere under consideration" argument, when calculating effective relative gravitational potentials in a homogeneous universe. Let's call this second argument the "effective sphere" argument for brevity.

Consider two stationary observers, A and B a distance 2r apart in a static(not expanding) infinite, homogeneous density, isotropic universe. To calculate the redshift of a signal from B as observed by A, we consider a sphere of radius r centred on A and ignore all the mass outside this hypothetical sphere. Now let us say that a star near B is visible from A. B measures the characteristic emission spectrum of excited hydrogen locally as w. The effective sphere argument predicts that A at the centre of his effective sphere, sees B at a higher gravitational potential and A measures the received wavelength to be blue shifted relative to the emitted wavelength measured by B locally. Similarly B measures the received light from A's star to be blue shifted relative the emitted wavelength measured by A.

Now a third observer exactly half way between A and B observes that both A and B are at the surface of the sphere centred on C and that the wavelength of the signal received by A will have exactly the same wavelength as the emission signal measured by B when it was emitted. This is a direct contradiction to the earlier statement that A measures the signal sent from B to blue shifted. Therefore, it would seem that the effective sphere argument and being able to ignore the rest of the universe argument is flawed.

My argument is based on a non expanding universe and I guess it is possible that some effect due to expansion causes some sort of cancellation of terms and allows us to ignore material external to the effecitve sphere, but this would be very coincidental and precise expansion (or maybe it is not coincidental and the two things are related)?

Counter-counter-arguments to my counter-argument are welcome

 

[bcrowell]

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?

Absolutely not. For example, you could use this method to get a nonzero answer for the Milky Way's velocity relative to itself, if the wire passed all the way around a closed universe. That seems like a clear ambiguity to me. There are all kinds of issues with the dynamics of the wire, the initial conditions, etc., which make it unclear whether the wire can be constructed and put in place, whether it transmits useful information, etc. These aren't just difficulties with knowing how to do certain calculations; they're signs that the rope's intended purpose in the OP's thought experiment is fundamentally meaningless. It's similar to describing a thought experiment designed to determine an observer's velocity with respect to the ether. That velocity is meaningless to talk about.

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.

No, that doesn't work either. There is no unambiguous way to resolve the redshift into gravitational and kinematic parts. Two good papers on this are [Bunn] and [Francis]. If you prefer to call it 100% kinematic, then Bunn shows you can do that. If you prefer to say that part of it is gravitational, and space is expanding, then Francis shows you can do that.

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.

It would be very interesting to figure out if there is a unique, self-consistent way of extending the idea of Born-rigidity from SR to GR. In SR there are various types of limitations on Born-rigidity, such as the inability to perform angular accelerations, and it took decades after Born's initial definition for these limitations to be clearly understood. If one doesn't understand those limitations, one can use Born-rigidity to prove all kinds of paradoxes in SR. Personally, I strongly doubt that there is any useful or interesting way to generalize Born-rigidity to GR. If you think such a generalization might be helpful in the current thread, then probably the first thing to do would be to search the literature. Maybe it's been shown to be impossible to generalize, or it's been shown that the generalization is non-unique, or every proposal for generalizing it has been shown to be non-self-consistent. But I think that if anyone wants to use Born-rigid objects in the present discussion, the burden of proof should be on them to demonstrate that the idea of Born-rigidity in GR has been studied and found to be meaningful. Without any such evidence, I'm not willing to accept any argument based on Born-rigidity, because it's just too easy to come up with obviously paradoxical examples, such as wrapping a Born-rigid ring around a closed spacelike geodesic in a closed cosmology.

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?

Hmm...you're trying to substitute notions like constant-ruler-distance and constant-radar-distance for the idea of a wire, but I don't think that helps. A ruler is just a wire by another name, or possibly a wire with a slightly different set of dynamical properties. The notion of constant radar distance is frame-dependent. Suppose A and B determine themselves to be at constant radar distance from one another. Observer C, at a cosmologically distant location, says that A and B are both accelerating, and therefore their time-dilation factors are changing over time. C says that the round-trip radar signals between A and B are actually taking different amounts of time.

[Bunn] http://arxiv.org/abs/0808.1081v2
[Francis] http://arxiv.org/abs/0707.0380v1

 

[bcrowell]

If it makes your head hurt, start with simpler things, get your head around them, and then advance to higher levels of complexity.

If you had given a self-consistent definition of what you meant by the generalization of Born-rigidy to GR, then it would have been legitimate to challenge me to wrap my head around your definition. But you haven't given a valid definition. I've shown that your definition is not self-consistent, and I've said that trying to fix your definition makes my head hurt. If you want to come up with a definition that is self-consistent, then it's up to you to do that, not me.

 

[bcrowell]

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.

The paradoxes are just easier to pose in the case of a closed universe. I've given lots of examples in this thread that don't involve a closed universe. The underlying issue is that relative velocities of cosmologically distant objects are not well defined. This is a standard part of the interpretation of GR, and it applies to both closed and open universes.

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.

Einstein did formulate the equation for gravitational attraction. That's what GR is.

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.

I think the observations you have in mind were something different. There's the possibility that the universe has a nontrivial topology, and there have been searches for evidence of that. For instance, you can have a universe that looks like the flat FLRW solution locally, but has the topology of a strangely connected soccer ball or something. Even in closed universes with a zero cosmological constant, I believe you can only see the back of your own head through a telescope after a certain point in time.

 

[yuiop]

I mentioned the internal Schwarzschild non-vacuum solution earlier. In its simple form (no linear or angular motion) it can be expressed as:

d\tau_r^2 = dt^2 \left(\frac{3}{2} \sqrt{1-\frac{2GM}{c^2 a}} - \frac{1}{2}\sqrt{1-\frac{2GMr^2}{c^2 a^3}} \right)

where a is the surface radius of the gravitational body and r is location of a stationary clock that measures proper time d\tau_r inside the body (r<=a) and M is the total mass of the gravitational body. Outside the gravitational body is considered to be a vacuum and is covered by the external Schwarzschild solution.

For uniform density (p) using the simple formula of total mass divided by total volume so that:

p=\frac{3M}{4\pi a^3}

and using a factor K defined as:

K = \frac{8\pi G}{3c^2}

the internal solution can now be expressed as:

dt_r^2 = dt^2 \left((3/2)\sqrt{1-pKa^2} - (1/2)\sqrt{1-pKr^2}\right)

If we consider the special case of a clock located at the centre of the body the proper time of this clock is given by:

d\tau_0^2 = dt^2\left( (3/2)\sqrt{1-pKa^2} - 1/2\right)

The ratio between clock rates of a clock located at r and another clock located at the centre of the body is then given by:

\frac{dt_r}{dt_0} = \left(\frac{3\sqrt{1-pKa^2} - 2\sqrt{1-pKr^2}}{-1+3\sqrt{1-pKa^2}}}\right)^{1/2}

Now the "effective sphere" argument for an infinite universe of constant density means we can ignore all mass at a greater radius than a from the centre and so we can treat the external as a vacuum and declare the internal Schwarzschild metric as a valid way to anlayse this effective sphere.

Now if we carry out the calculate the ratio of the clock rate ratios for a>r and then repeat the calculation for a=r, (effectively removing the shell with internal radius r and external radius a), we see that the ratio between the clock rates at the centre and r increases. This suggesting that the red or blue shift measured between points at the centre and at r is dependent on the mass in spherical shells at radii greater than r, suggesting that our initial assumption that we can treat mass external to the effective sphere as vacuum, is flawed. In fact if a is sufficiently large the clock at the centre stops and after that the clock at the centre starts running backwards or becomes imaginary and the region presumably collapses in on itself and becomes a black hole. To me, it is more satisfactory to consider all points in a homogeneous infinite universe to be at equal gravitational potential at any given time, and in this case, the only way effective gravitational potentials can arise between points, is by by temporal changes in potential in an expanding universe.

 

[Ich]

If you had given a self-consistent definition of what you meant by the generalization of Born-rigidy to GR

What I meant with a generalizsation of Born-rigidity to GR? That's completely your idea.
What I meant to do, and I explicitly said so, is to give the definition of a rope in the context of this thread some rigor. Which I did.
Born rigidity was merely an analogy, because you flamed against pre-arranged motion.
And I applied the definition explicitly to open topologies only.
And I said

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.

Which means: I have no idea how to generalize the notion to arbitrary topologies, and it makes my head hurt if I think about it. And I'll think about it later.

If one doesn't understand those limitations, one can use Born-rigidity to prove all kinds of paradoxes in SR. Personally, I strongly doubt that there is any useful or interesting way to generalize Born-rigidity to GR.

etc. etc.
Calm down. You are talking about generalizations of Born rigidity, not I. I'm talking about the OP's rope. In an open topology.

If you're interested in that rope thing, as you indicated, then I'm sure it's worthwile for you to follow the definition I gave - even if you could call it original research, as I never bothered to find sources for it. Because the principles are not too complicated for a forum discussion, and the results are easy to check analytically with the Milne/de Sitter models.

 

[Ich]

The effective sphere argument predicts that A at the centre of his effective sphere, sees B at a higher gravitational potential and A measures the received wavelength to be blue shifted relative to the emitted wavelength measured by B locally. Similarly B measures the received light from A's star to be blue shifted relative the emitted wavelength measured by A.

No.
If all observers are to stay at fixed distances, at least some of them have to undergo proper acceleration. Either A, or B, or both. It's your say.
If you choose one which is free falling, this one is the hub of the world. All red- or blueshifts are directed towards it.
Check it with Newtonian gravity, interior Schwarzschild is definitely an overkill.

 

[Ich]

I am not quite sure I understand you here. The stationary observer low down sees a gravitational redshift of the signal from higher up (basically because his clock is running slower)

Blueshift, of course.

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

Yep, I was talking about comoving observers. Only they are interchangeable.

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.

From what I can tell, your definition is exactly the one I used, as long as you define "radar distance" to be the back and forth light travel time.
There's only one thing you omit, and it throws you off the curve in your later analysis: You have to pick an origin, the preferred point without proper acceleration.

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.

Don't use it.

In other words, non-zero redshift does not imply non-zero relative motion using this idea.

This is definitely true, and it's the reason why I defined "constant distance" by a two-way measurement.
But the redshift has nothing to do with your alleged "temporal potential". It's a simple spatial potential, centered at the origin. All those redshifts either point toward it, or away from it. There's nothing reciprocal, and in fact it's irrelevant whether there is expansion or contraction. The only thing that counts is the local matter (and pressure) density.

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.

Yes, I understood. This doesn't work. The redshift indeed comes from a PLACE where the potential is lower.

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.

I already said that there are two (or three, depending on how you count) different possibilities for the end of the chain:
1) It goes asymptotically to v=c in an infinite distance. That's the Milne model with its coordinate singularity.
2) It ends in a horizon, where the chain breaks. That's the de Sitter model, which is like an inverse black hole.
3) It ends abruptly in the Big Bang. These are the matter-containing models, where at one time everything is ok - if a bit frantic, and in the very next moment everything falls toward the end of the chain at light speed.
This moment is, of course, the Big Bang singularity, and not part of the manifold we're considering.

One thing is ubiquitous: The "rope simultaneity" goes further and further back in time with increasing distance, if compared with FRW time. If there is a Big Bang (all models except de Sitter), the end of the rope will be there, either in a finite (models with matter) or infinite (Milne/empty model) comoving distance.

 

[Ich]

BTW, I'm away for a week now. Maybe we can contiue this discussion later.

 

[bcrowell]

Now the "effective sphere" argument for an infinite universe of constant density means we can ignore all mass at a greater radius than a from the centre and so we can treat the external as a vacuum and declare the internal Schwarzschild metric as a valid way to anlayse this effective sphere.

I don't think this works. Birkhoff's theorem implies that if you have a spherical cavity in a spherically symmetric universe, you can ignore the external mass, and the spacetime inside the cavity has to be Minkowski. It doesn't tell you anything about the case where there is no spherical cavity. It's different from the Newtonian-gravity shell theorem, because GR isn't linear like Newtonian gravity. Therefore you can't take just any old spherically symmetric mass distribution, break it down into concentric shells, and sum the fields made by the shells.

To me, it is more satisfactory to consider all points in a homogeneous infinite universe to be at equal gravitational potential at any given time, and in this case, the only way effective gravitational potentials can arise between points, is by by temporal changes in potential in an expanding universe.

As I've been pointing out since #67, you can't analyze cosmological solutions using a gravitational potential. You need a static spacetime in order to define a gravitational potential. There is a good discussion of this in Rindler, Essential Relativity, 2nd ed., section 7.6. If cosmological solutions could be described by a gravitational potential, then you would be able to resolve cosmological red-shifts into unambiguously defined kinematic and gravitational terms. But this is impossible, as discussed in the references in #14.

 

[yuiop]

Blueshift, of course.

Yes, of course. I have corrected my typo in the original post. Thanks.

From what I can tell, your definition is exactly the one I used, as long as you define "radar distance" to be the back and forth light travel time.

Yep, that is what I meant by radar distance.

There's only one thing you omit, and it throws you off the curve in your later analysis: You have to pick an origin, the preferred point without proper acceleration.

I find this an odd statement. In a FLRW universe, most significant objects are at rest with the Hubble flow and nothing has proper acceleration in the cosmological sense. When we see a distant galaxy moving away from us at some great velocity and even if we acknowledge dark energy or the cosmological constant, neither the distant galaxy or ourselves have proper acceleration with respect to each other, not the kind you can measure with an accelerometer anyway. Both the distant galaxy and ourselves will appear to be approximately at rest with respect to the CMBR.

Don't use it.

I was hoping for a more detailed counter argument than "don't use it" and "forget about it" to my argument.

But the redshift has nothing to do with your alleged "temporal potential". It's a simple spatial potential, centered at the origin. All those redshifts either point toward it, or away from it. There's nothing reciprocal, and in fact it's irrelevant whether there is expansion or contraction. The only thing that counts is the local matter (and pressure) density.

Is it relevant if there is neither expansion nor contraction? (i.e a static universe). I can see a spatial potential in a finite universe where clearly objects "near the edge" will have a different potential to objects near the centre, but in an infinite universe, there is no such thing as a centre or a near the edge. You have not made it clear (to me anyway) whether you are talking baout finite or infinite models.

Yes, I understood. This doesn't work. The redshift indeed comes from a PLACE where the potential is lower.

I still don't get this. In an infinite homogeneous universe, WHERE is this PLACE with a lower potential?

I already said that there are two (or three, depending on how you count) different possibilities for the end of the chain:
1) It goes asymptotically to v=c in an infinite distance. That's the Milne model with its coordinate singularity.

Yep, that makes sense, and if you can only construct the chain at a velocity that is less than the speed of light, the chain will never catch up with the edge of the visible universe. However, the Milne model has obvious limitations because it does into take into account the GR effects of all that moving matter and energy in the universe.

2) It ends in a horizon, where the chain breaks. That's the de Sitter model, which is like an inverse black hole.

One thing we have not really addressed in this thread is the physical stress that a rope would be subjected to, when it joins two distant galaxies that are at rest with respect to each other. Just how much drag does the Hubble flow apply to an object that is not at rest with the Hubble flow? I would suggest none or very little. It is our experience that an object with a peculiar local velocity continues to move with velocity and is not subjected to any drag bringing it to rest with the CMBR.

3) It ends abruptly in the Big Bang. These are the matter-containing models, where at one time everything is ok - if a bit frantic, and in the very next moment everything falls toward the end of the chain at light speed.
This moment is, of course, the Big Bang singularity, and not part of the manifold we're considering.

One thing is ubiquitous: The "rope simultaneity" goes further and further back in time with increasing distance, if compared with FRW time. If there is a Big Bang (all models except de Sitter), the end of the rope will be there, either in a finite (models with matter) or infinite (Milne/empty model) co-moving distance.

I agree with this conclusion, but it makes your head hurt to think of a wire physically connecting here and now at one end and the big bang at the other end. Ouch!

BTW, I'm away for a week now. Maybe we can continue this discussion later.

Looking forward to your return. Have a pleasant trip!

 

[stevmg]

I have been gone for a few weeks...

What are you positing moves faster than light? Light itself? Does c refer to a hard 300,000,000 km/sec or just the speed of light, whatever that is. Light moves slower than 300 million m/sec traveling through media.

How does adding proper times up over long distances get you faster than light speed as presented earlier? How do non linear coordinates get you faster than light speed? Minkowski coordinates have no central frame of reference, so how is that possible in any coordinate system. After all, anything you chose as "central" would be arbitrary.

Einstein's "nothing faster than c" is a hypothesis. Has it ever been disproven? Can anyone give a mathematical example of how one could travel faster than light that makes any sense?

Two simultaneous events in the same frame of reference are spacelike separated and one cannot get from A to B "in tmie" Has that ever happened before?

This is pretty damn confusing. but I can understand that light speed does not have to be constant, now does it?

 

[Ich]

Looking forward to your return. Have a pleasant trip!

Thanks. We've been to http://www.ferienhof-rosenlehner.de/bauernhof.html" , but sadly there were not enough possibilities for my youngest son to practice his skills as a farmer - which he is determined to become. Still, a beautiful place.

There's only one thing you omit, and it throws you off the curve in your later analysis: You have to pick an origin, the preferred point without proper acceleration.

I find this an odd statement. In a FLRW universe, most significant objects are at rest with the Hubble flow and nothing has proper acceleration in the cosmological sense. When we see a distant galaxy moving away from us at some great velocity and even if we acknowledge dark energy or the cosmological constant, neither the distant galaxy or ourselves have proper acceleration with respect to each other, not the kind you can measure with an accelerometer anyway. Both the distant galaxy and ourselves will appear to be approximately at rest with respect to the CMBR.

Everything you say is true, but IIRC we've been talking about the chain of observers who are mutually at rest in this case. Such observers are not comoving, and if there is gravity, all but one will experience proper acceleration.
You'll have to define an origin for that chain. The origin will be moving inertially, all other elements generally won't.

Don't use it.

I was hoping for a more detailed counter argument than "don't use it" and "forget about it" to my argument.

I thought I did, in the subsequent paragraph.
The redshifts in a chain of "stationary" observers are not reciprocal, as they should be if we're talking about a change in time only.
Further, they are independent of expansion or contraction. They only depend on the local mass density, not its time derivative.

Is it relevant if there is neither expansion nor contraction? (i.e a static universe). I can see a spatial potential in a finite universe where clearly objects "near the edge" will have a different potential to objects near the centre, but in an infinite universe, there is no such thing as a centre or a near the edge. You have not made it clear (to me anyway) whether you are talking baout finite or infinite models.

I'm talking about infinite models, too. But if you want to use the well-known potential, you'll have to pick an origin and use quasistatic coordinates in its vicinity. In expanding FRW coordinates, there is no potential.

Yes, I understood. This doesn't work. The redshift indeed comes from a PLACE where the potential is lower.

I still don't get this. In an infinite homogeneous universe, WHERE is this PLACE with a lower potential?

The place with extreme (max or min) potential is just where you pick the origin of the chain of stationary observers. Pick another origin, and there will be a different potential. But it doesn't matter for your calculations.

But your idea of a "temporal potential" has some merit: In FRW coordinates, the scale factor is the equivalent of a potential in static coordinates.
The former is a scale of position as a function of time, while the latter is a scale of time as a function of position.
The former defines changes in momentum, the latter defines changes in energy.
That said, the word "potential" usually does not refer to somthing like a scale factor, and I'm not sure if this would be a good idea.
But ok, there are the complementary pairs: time - position, energy - momentum, potential - scale factor.

Just how much drag does the Hubble flow apply to an object that is not at rest with the Hubble flow? I would suggest none or very little.

Right. The Hubble fow in itself is a property of a family of observers, not a property of spacetime. It cannot possibly exert some kind of drag (usual disclaimer for nitpickers:, except for higher order corrections in the presence of mass).
For local physics, it is irrelevant whether there is outward, inward, or no Hubble flow at all. The Hubble flow is then nothing but the average motion of galaxies.

It is our experience that an object with a peculiar local velocity continues to move with velocity and is not subjected to any drag bringing it to rest with the CMBR.

Nope. Peculiar velocities tend to die out, but that's a sorting effect, not a force or drag. If something has peculiar velocity, it will simply move to a place where it hasn't.

I agree with this conclusion, but it makes your head hurt to think of a wire physically connecting here and now at one end and the big bang at the other end. Ouch!

Well, it's not so painful if you remember that the wire is not exactly a "physical connection". In the wire's frame, all its components are spacelike separated, and the supposed fate of its ends is really irrelevant for what's happening here and now.

 

[rede96]

I have been gone for a few weeks...

What are you positing moves faster than light? Light itself? Does c refer to a hard 300,000,000 km/sec or just the speed of light, whatever that is. Light moves slower than 300 million m/sec traveling through media.

As far as the original post my way of thinking was, I know (in my own way of understanding things) that nothing with rest mass can propagate at or faster than the speed of light in any medium. However, does that mean that nothing can move away from me faster than the speed of light?

According to GR, the universe is expanding and thus there are distant objects that are moving away from me at the speeds greater than the speed of light.

If so, is it then possible to use this fact to show that, although nothing can propagate through any medium faster than the speed of light, information could still be passed on faster than the speed of light.

Again, the thought process was although nothing can travel faster than c in any medium I know of, if the medium was moving away from me as well (or expanding as space seems to be.) then it seems to be possible, at least in theory.

Now I don't pretend to understand all of the very good and detailed answers given but as far as can make out, there has been no conclusive proof of this either way. ( I think!)