Is this a way to move faster than c?

[DrGreg]

Is the "equivalence priciple" you refer to the idea that acceleration is "equivalent" no matter how you get there? In other words, was "curved spacetime" the equivalent of "gravity" provided the kinematics were the same (you know, the guy in the free falling elevator experiences no g's as does the guy who is under no gravitational force at all.) Of course, there is no place in the universe that you can find such a place.

Yes. If you zoom into a small enough region of spacetime the "acceleration due to gravity" will be near-enough constant in magnitude and direction, so you can use the "falling elevator" trick to get rid of gravity and analyse using special relativity only.

Therefore (as a crude generalisation) any statement that is true in special relativity is also "locally true" (approximately) in general relativity too.

The word "approximately" can be made rigorous using calculus limits.

 

[DrGreg]

What does "locally Minkowskian" mean? You mean that you can divide the universe into spacetime "zones" in which
ct'² - x'² - y'² - z'² = ct² - x² - y² - z² and yet there are other zones which aren't?? If so, that would really be weird, weirder than the "uncountably infinite" set of sets we discussed above!

If you have a curved surface, if you zoom in close enough it seems to be approximately flat. So you can say it's "locally flat". It doesn't mean it consists of literally flat pieces joined together with sharp corners!

This use of the word "local" is common for physicists but goes against how mathematicians tend to use the word: for mathematicians "locally flat" might mean "exactly flat throughout a small region" but relativists tend to use it to mean "approximately flat over a small scale".

(In this context, "flat" means that the Minkowskian metric applies, for a freely-falling observer.)

 

[stevmg]

If you have a curved surface, if you zoom in close enough it seems to be approximately flat. So you can say it's "locally flat". It doesn't mean it consists of literally flat pieces joined together with sharp corners!

This use of the word "local" is common for physicists but goes against how mathematicians tend to use the word: for mathematicians "locally flat" might mean "exactly flat throughout a small region" but relativists tend to use it to mean "approximately flat over a small scale".

(In this context, "flat" means that the Minkowskian metric applies, for a freely-falling observer.)

Minkowski space requires a locally flat "zone?" But the universe really isn't flat anywhere?

If so, makes sense to me, theoretically. I didn't know Minkowski diagrams, etc. required "uncurved" space or "flat" space.

Please answer as this part has been most enlightening.

 

[yossell]

This use of the word "local" is common for physicists but goes against how mathematicians tend to use the word: for mathematicians "locally flat" might mean "exactly flat throughout a small region" but relativists tend to use it to mean "approximately flat over a small scale".

Does `approximately flat' in this context mean `is flat to first order'? And is there an intuitive way of understanding this, or is it just a technical notion?

 

[stevmg]

In topology, local means "in a neighborhood" which means that no matter how close you are to a given point in an ordered set, you can always find elements in the set closer than you are and so on and so on, so a neighborhood is that point or set of points in which every point of higher or lower ordinality is closer to the original point than you are. Sets can be ordered by geometric distance.

The reason why two intersecting line have no differential at their point of intersection, be they straight or curved lines is that the point of intersection, there is no unique point for which this is true:

d(f(x))/dx = lim [f(x + h) - f(x)]/h] h \rightarrow 0

If you consider all the points \pm h from a point (x, f(x)) there is no unique quotient no matter how close you get to x

So, mathematically, by what was said above, there would be a zone which is totally flat, not near flat. Cosmologically, I guess that isn't true, so Minkowski is in the real world only an approximation to what really is.

I guess yossel is referring to the analogy in mathematics that a first derivative can be zero but a second derivative can be non trivial at the same point. If a zone on curve is locally flat, such a zone has no change in slope over a small distance and would make the change of slopes flat. I can't even think of the change of a change of slopes, so I guess a higher order derivative could be non zero. But since "approximately flat" is allowed, you don't have to worry about higher order derivatives all being zero. As you go up in order of anti-derivatives you're going to hit a non trivial answer.

My head hurts from all this. I was a math major many years ago and this is really trying my memory.

 

[stevmg]

You know Minkowski died in 1909, before General relativity came out, so I guess his geometry was "flat" in the sense of no curve in the spacetime coordinates.

 

[yossell]

You know Minkowski died in 1909, before General relativity came out, so I guess his geometry was "flat" in the sense of no curve in the spacetime coordinates.

I think it's the underlying geometry rather than the coordinates which are properly called flat. I'm not an expert but...

You can have all sorts of coordinates for a flat space-time, but only a flat space-time *can* be coordinatised in a way so that the ds^2 = -dt^2 + dx^2 + dy^2 + dz^2.

In purely intrinsic terms, I think on a curved manifold, when vectors are parallel transported around a closed curve, they do not necessarily come back pointing the same direction. In a flat manifold, they will.

But don't take my word for it

 

[nutgeb]

1) Is it possible to have an infinitely long rope or at least long enough to span the 13 b LYs of our universe?

Well first, you have to specify what part of the rope you want to be at rest in its local comoving Hubble flow.

If the center point of the rope is at rest in its local Hubble flow, then if the rope remains intact the two ends will respectively have .5c and -.5c velocities in their local comoving frame. Depending on your assumptions about the strength and flexibility of the rope (and assuming that the rope is vanishingly close to massless), there's at least a chance that the rope might remain intact. However, I tend to doubt it.

Again, the big problem is how to roll out the rope in the first place. You can have two rockets pulling the ends of the rope in opposite directions away from the center. But that still requires a lot of acceleration of the rope ends. If the rockets have very fast acceleration, and then coast to their final destination, the rope is greatly stressed even at relatively short lengths by the Born rigidity problem. If the rockets have very slow acceleration, e.g. if they accelerate constantly at a low rate throughout their journey, then a much longer length of rope can be deployed, but ultimately the great length of the rope (up to 6.74 Gly) causes it to experience increased stress, since the acceleration pressure resulting from the rope end's acceleration from the rocket is limited to moving along the rope at a local rate of < c (and in reality, the limit is probably much lower). In this latter case the great length of the rope is the cause of its demise. So in either case, I will speculate that the rope would not survive the deployment process. The tradeoff between acceleration rate and rope length is alluded to in Egan's excellent page on Rindler horizons that was linked to an earlier post.

The deployment problem is greatly increased if the rope is secured at one end (say to earth). Which means that end of the rope is at rest in its local comoving frame. That means that if the rope extended a full 13.8 Gly the far end (being pulled by the rocket) would need a local velocity of c in its distant comoving frame. It is absolutely impossible, even in theory, for a non-relativistic object to attain a speed of c in any local FRW frame, so the rope must break before that occurs. Or a more obvious way to look at it is that the rocket pulling the rope end can't attain a peculiar speed of c in any local frame, so it can't pull the rope that fast either.

As I described in an earlier post, an alternative strategy of deploying a huge number of short segments of rope end to end over the 13.8 Gly distance, and then coupling them together at a given instant in time, won't work either. The act of coupling the segments into a unified rope will impose tension (negative pressure) shocks as every part of the rope is accelerated (relative to their local comoving frame) toward whichever part of the rope is tied down in its local comoving frame. The acceleration must progressively overcome the inertia of the rope segments, which all start out at rest in their own respective local comoving frame. Those tension shocks will be initiated in all parts of the rope as they are pulled in both directions by the comoving inertia of the segments on both sides of them. The shocks will radiate lengthwise at a theoretical maximum speed limit of < c. I would expect the rope to shatter well before the shockwave reaches the far end(s), at least in the case where the rope is tied down at one end. It might shatter in many locations.

2) Is so, can the rope then be seen by many galaxies or many reference frames?

The answer to the first question was no, so maybe this question is moot. But if a rope could be stretched across some intergalactic distance (much, much less than 13.8 Gly), then in theory it could be seen from any galaxy inside our [CORRECTED] Event Horizon (which is currently believed to be at about 17 Gly.) But of course the image of the rope can travel only at the speed of light, so it could take billions of years for the image to be seen in a distant galaxy. How fast an object's peculiar velocity (its local velocity relative to its local comoving frame) is has no bearing on whether and when it will be seen by distant observers. Peculiar velocity will contribute additional red/blue shift however.

3) If the rope was attached to a planet in a galaxy that began to move away from me at speeds >c, could I observe the rope traveling at speeds >c in my reference frame?

Since we do regularly see light from galaxies whose comoving recession velocity exceeded c when the light was emitted from them, then of course you could eventually (in the far distant future) see an image of a rope end attached to such a galaxy (ignoring the practical issues of magnification and light intensity, of course; and assuming the galaxy is within our observable universe, i.e. closer than our Event Horizon). The observed image of the rope end would be redshifted by exactly the same amount as light from the galaxy itself is.

You could not observe the other (loose) end of the rope passing nearby you, because as described above, it is impossible for a rope to be deployed such that any part of it has a local speed > c.

 

[stevmg]

Is space itself expanding and therefore "carrying along" galaxies and other matter with it or is the universe an infinite vast empty void with our small piece of it expanding outward into this vast emptiness? What's the lastest take?

Also, you don't have to qualify with real world entities when you are postulating constructs such as ropes or rope segments or whatever. No need to worry about tensile strength or shock waves. Your imagination is the limit.

We have examples of that in our everyday world.

"What is a line?" - A line is the shortest distance between two points.

"What is a point?" - A point has no length, width or breadth.

No need for anything here. Totally imaginary entities which do not exist in the real world but with which, we built buildings, roads, trains, airplanes, shot men to the moon, discovered Relativity, GPS satellites - whatever.

Amazing what the mind can do when unshackled by the real world!

 

[Nickelodeon]

...

I think debating about the wire or the rope spanning galaxies is a bit of a red herring (not a criticism of your good postings just a general observation). As I read the original post the underlying meaning is "is it possible to go faster than light?".
If it is true for distant galaxies it is true for you and me whatever light compensating formulas are applied to disprove it.
The reason is that the speed of light is dictated but the relative Hubble flow at the particular point in space that you are referring to. Presumably, after you pass the Hubble horizon your light goes out (for those folk back home). That's why there is a lot of black up there.

 

[DrGreg]

Minkowski space requires a locally flat "zone?" But the universe really isn't flat anywhere?

If so, makes sense to me, theoretically. I didn't know Minkowski diagrams, etc. required "uncurved" space or "flat" space.

Please answer as this part has been most enlightening.

Minkowski space is exactly flat everywhere. It applies only to special relativity where we assume there is no gravity and no expansion of the universe. In the more general case when these assumptions aren't true we refer to "spacetime" without referring to Minkowski. We call Minkowski space "flat" because (in two dimensions, 1 space + 1 time) we can draw accurate diagrams on flat paper. When gravity is added, you need curved paper.

Minkowski space is characterised by the spacetime metric

ds^2 = c^2\,dt^2 - dx^2 - dy^2 - dz^2​

In curved spacetime we have a different equation for the metric, possibly involving "cross-terms" such as dx.dt, dx.dy, etc, and with all the coefficients variables instead of constants.

This is an extension of the same technique that can be used in 3D Euclidean geometry. For example the surface of a sphere can be described by the metric

ds^2 = r^2 (d\theta^2 + \cos^2 \theta \, d\phi^2)​

where \theta is latitude and \phi is longitude.

 

[DrGreg]

Does `approximately flat' in this context mean `is flat to first order'? And is there an intuitive way of understanding this, or is it just a technical notion?

Yes. If it's approximately flat, it's possible to perform a change of coordinates to use "approximately Minkowski" coordinates. What this means is that if the metric is given, in these coordinates, by

ds^2 = g_{\alpha\beta}\,dx^\alpha \, dx^\beta​

then, at the one event in question, all the metric coefficients g_{\alpha\beta} equal the Minkowski metric coefficients, and the first order derivatives \partial g_{\alpha\beta} / \partial x^\gamma are all zero.

So by an application of Taylor's theorem, the "deviation from flatness" depends (approximately) only on square-distance rather than distance.

 

[nutgeb]

Presumably, after you pass the Hubble horizon your light goes out (for those folk back home). That's why there is a lot of black up there.

That's the intuitive answer but it's actually not correct. It turns out that the lightcone is curved in FRW coordinates. For a photon emitted from a distance beyond the Hubble Radius, initially the photon's proper distance from the observer will increase (due to the superluminal recession velocity of the local frames the photon is moving through), but eventually the photon will arrive at the Hubble radius and cross it. Once the photon crosses the Hubble radius, its proper distance from the observer will progressively decrease until the photon arrives at the observer.

One way to describe why this happens is that the Hubble Radius itself is always moving outward as a function of time. That happens primarily because the Hubble Rate (H) naturally tends to decrease as a function of time -- the Hubble rate is measured in terms of absolute recession velocity divided by absolute distance (e.g. Km/sec/Mparsec), so if the recession velocity (H*D) between any two comoving galaxies remains constant as the distance D between them increases, then H must necessarily decline. (This equation is further complicated by the effects of gravity and Dark Energy which affect the Hubble rate). So really it's not so much that the photon crosses the Hubble Radius, as that the Hubble Radius expands until it encompasses the photon (because the photon is moving away from the observer until that time, but the Hubble Radius is moving outward faster).

The distance beyond which light emitted now from distant galaxies will never be visible to us is called the Event Horizon. (I mistakenly referred to the Particle Horizon in my last post but I corrected it). It's currently thought to be at a radius of about 17 Gly, farther than the current Hubble Radius. Because of the acceleration of the expansion caused by Dark Energy, our Event Horizon is asymptotically approaching a zero growth rate, and our Hubble Radius will stop increasing when it asymptotically approaches our Event Horizon, around 10-15 Gy in the future.

The idea that the Hubble Radius acts as a horizon is a common misconception about the expansion of the universe. I highly recommend you read http://arxiv.org/abs/astro-ph/0310808" by Davis & Lineweaver on the subject.

 

[stevmg]

Minkowski space is exactly flat everywhere. It applies only to special relativity where we assume there is no gravity and no expansion of the universe. In the more general case when these assumptions aren't true we refer to "spacetime" without referring to Minkowski. We call Minkowski space "flat" because (in two dimensions, 1 space + 1 time) we can draw accurate diagrams on flat paper. When gravity is added, you need curved paper.

Minkowski space is characterised by the spacetime metric

ds^2 = c^2\,dt^2 - dx^2 - dy^2 - dz^2​

In curved spacetime we have a different equation for the metric, possibly involving "cross-terms" such as dx.dt, dx.dy, etc, and with all the coefficients variables instead of constants.

This is an extension of the same technique that can be used in 3D Euclidean geometry. For example the surface of a sphere can be described by the metric

ds^2 = r^2 (d\theta^2 + \cos^2 \theta \, d\phi^2)​

where \theta is latitude and \phi is longitude.

As I had mentioned somewhere that Minkowski died in 1909 and probably avoided all this fun. If this were a 2D world + time, then the Minkowski lines would be "hyperboloids" of two sheets rather than hyperbolas, I suppose. Never got into that in analytic geometry. This extra dimension caused by gravity really puts a major wrinkle into this, but I got the idea.

About space itself -

Is it expanding (the "void" itself) or is everything just rushing out into this infinite void that's already there? Makes a difference. If stuff were just rushing out into an existing infinite void, then the c restriction is more likely to apply. If space itself were expanding like the surface of an expanding balloon or beachball, then there may be no restriction on any speed assuming the units still remained the same size they were originally.

 

[nutgeb]

About space itself -

Is it expanding (the "void" itself) or is everything just rushing out into this infinite void that's already there? Makes a difference. If stuff were just rushing out into an existing infinite void, then the c restriction is more likely to apply. If space itself were expanding like the surface of an expanding balloon or beachball, then there may be no restriction on any speed assuming the units still remained the same size they were originally.

Nobody knows the answer to your question. As I've expained in other posts, both the 'expanding space' paradigm and the 'kinematic' paradigm yield precisely equal mathematical calculations of what the observations would be (such as redshift). And if the galaxy-filled universe is infinite, then even in the kinematic paradigm, the galaxies are not rushing into some region of empty space, because the region containing galaxies fills all of space. An infinite kinematic universe just gets bigger without encroaching on something outside of itself. In that sense the two models tend to blend together.

I've struggled with the issue of how the proper-coordinates speed of a photon can exceed c at a distance in the kinematic model. I have seen no real explanation of that. Two approaches to the problem occur to me:

1. One can just acknowledge that superluminal speeds are somewhat unique to FRW coordinates, and therefore they are just a mathematically different way of looking at speeds, not necessarily caused by some specific physical cause. Proper Distance and Proper Time are directly observed as Proper values in the frame in which the observer is at rest relative to the spacetime events being measured.

In SR using Minkowski coordinates, when an observer at rest in one inertial reference frame interacts with an object that it is in motion, he can never directly observe the Proper Length and Proper Time of the moving object, and light received from the moving object is interpreted to indicate that the object is time dilated and Lorentz contracted, and Proper Velocities must be added with the relativistic formula and cannot exceed c.

But in GR using FRW coordinates, an observer at rest in one comoving frame can treat the comoving observers in all other comoving frames as being at rest in their frame, because their frame is equally privileged with his own, because FRW coordinates choose Proper Time and Proper Distance as the common coordinate axes for all comoving frames. The light the observer receives from a receding galaxy is redshifted due to the galaxy's recession motion, but it can be interpreted to mean that the distant galaxy is not time dilated or Lorentz contracted, and Proper Velocities can be added directly and can exceed c. In other words, in FRW coordinates we progressively shift our calculation to be at rest in each successive local frame along the photon's path, such that every segment time and distance measurement is a Proper measurement, instead of defining a single end-to-end reference frame. FRW calculations require such a frame-hopping calculation, while SR Minkowski calculations essentially disallow it.

The distinctions between a Minkowski and FRW observer do not change the characteristics of the light they see, only their interpretation. So maybe this just represents a change in perspective, not a physical change in what is "really" happening. The limit on the speed of light at a distant location is a matter of interpretation, not an absolute fact. And since the Proper Velocity of a photon (measured in its own infinitely time dilated frame) is infinite, or undefined, we can't pick a preferred interpretation by adopting Proper Velocity as our tiebreaker.

I don't find this approach to be very satisfying, because it begs the question of how two quite different interpretations of a single physical process can both be correct. But I think there's a certain truth to it and once it is accepted, all the issues fall away.

2. A second potential interpretation is that there is some physical cause as to why the speed of light at a distant location can exceed c, even in the kinematic model. One such cause could (speculatively) be attributed to Mach's Principle. In a universe with gravity, the gravitational potential exerted at anyone location (such as earth) from very distant objects is, in aggregate, far greater than the aggregate gravitational potential exerted by nearby objects. Considering each concentric thin 'shell' at each successive radial distance from earth, the number (and therefore the mass) of idealized gravitating objects in each successive shell increases due to the increase in volume at a faster rate than the gravitational force of each such object decays due to the increase in distance.

So in theory very distant masses exert a very large gravitational potential toward each point in space. The gravitational accelerations cancel out because the distant masses are isometrically arranged in a sphere around each point. But the gravitational potential is still there, just as Earth's gravitational potential is still there at the center of the earth. From the perspective of earth, a distant galaxy's sphere of distant matter (which extends out to that galaxy's Particle Horizon) is different from Earth's such sphere (which extends out to Earth's Particle Horizon), with an overlapping portion.

An Earth observer could interpret that a distant galaxy's sphere of distant matter is causing linear frame dragging, in effect curving spacetime toward the distant galaxy and away from earth. This concept has been used to offer theoretical explanations for inertia. But it seems to me that it could also be extended to suggest that in distant local frames, space is, in effect, flowing away from earth. (One could describe this as a form of spacetime curvature, but one can also analogize to the 'river model' of flowing space in Painleve-Gullstrand coordinates.) If so, then a photon moving radially away in that distant local frame would need to have a velocity of c relative to that 'flowing' local frame, rather than relative to Earth's 'stationary' local frame. (Just as photons have a velocity of c relative to their inflowing local frame in P-G coordinates inside a black hole Event Horizon.) Meaning that the proper-coordinates speed of light would increase with distance.

Of course this interpretation would work only if distant frames were "flowing" away at exactly the same recession velocity as the galaxy located at the center of the sphere of matter that is dragging them. In effect the local space near that central galaxy is gravitationally "locked" to the radial motion of its sphere of matter, relative to distant earth. I haven't tried to calculate that, and I don't know if frame dragging could even theoretically occur at 100% of the velocity of the moving 'object' (the matter sphere) doing the dragging, if the gravitational potential is less than infinite. I don't know the math of linear frame dragging. It occurs to me that only the 'leading' 1/2 (hemisphere) of the matter sphere contributes to the dragging effect. I'm not sure if the 'trailing' hemisphere works against the effect or not; my guess is not.

This interpretation turns the 'expanding space' paradigm on its head. In the frame dragging interpretation matter is dragging local space along with it, whereas in the 'expanding space' paradigm (at least in its basic form) spontaneously expanding space is what drags massive galaxies apart.

I'd be interested in discussing either of these ideas. Perhaps one or the other can be ruled out.

 

[Nickelodeon]

That's the intuitive answer but it's actually not correct. ...

The idea that the Hubble Radius acts as a horizon is a common misconception about the expansion of the universe. I highly recommend you read http://arxiv.org/abs/astro-ph/0310808" by Davis & Lineweaver on the subject.

Thanks for the link and your explanation. I thought that the 'lights would go out' for the reason that although the photon reaches you its wavelength has been red shifted to such an extent that it can no longer be considered a wave for practical 'viewing' purposes.

 

[bcrowell]

I've struggled with the issue of how the proper-coordinates speed of a photon can exceed c at a distance in the kinematic model. I have seen no real explanation of that. Two approaches to the problem occur to me:

1. One can just acknowledge that superluminal speeds are somewhat unique to FRW coordinates, and therefore they are just a mathematically different way of looking at speeds, not necessarily caused by some specific physical cause. [...]

It's not unique to FRW coordinates. It's a generic fact about GR. Coordinates are arbitrary. Coordinate velocities don't have any direct physical significance. Relative velocities of distant objects are not uniquely defined.

The light the observer receives from a receding galaxy is redshifted due to the galaxy's recession motion, but it can be interpreted to mean that the distant galaxy is not time dilated or Lorentz contracted, and Proper Velocities can be added directly and can exceed c.

This is one possible way of interpreting cosmological redshifts, but they can also be interpreted as occurring because the space occupied by the photon expands, which stretches the photon's wavelength. Since relative velocities of distant objects are not well defined, you can't unambiguously interpret cosmological redshifts as Doppler shifts.

The distinctions between a Minkowski and FRW observer do not change the characteristics of the light they see, only their interpretation.

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.

A second potential interpretation is that there is some physical cause as to why the speed of light at a distant location can exceed c, even in the kinematic model. One such cause could (speculatively) be attributed to Mach's Principle. In a universe with gravity, the gravitational potential exerted at anyone location (such as earth) from very distant objects is, in aggregate, far greater than the aggregate gravitational potential exerted by nearby objects. Considering each concentric thin 'shell' at each successive radial distance from earth, the number (and therefore the mass) of idealized gravitating objects in each successive shell increases due to the increase in volume at a faster rate than the gravitational force of each such object decays due to the increase in distance.

So in theory very distant masses exert a very large gravitational potential toward each point in space. The gravitational accelerations cancel out because the distant masses are isometrically arranged in a sphere around each point. But the gravitational potential is still there, just as Earth's gravitational potential is still there at the center of the earth. From the perspective of earth, a distant galaxy's sphere of distant matter (which extends out to that galaxy's Particle Horizon) is different from Earth's such sphere (which extends out to Earth's Particle Horizon), with an overlapping portion.

Cosmological solutions are homogeneous, so there's no way they can have a varying gravitational potential. Also, cosmological solutions are time-varying, and in general time-varying solutions don't have well-defined gravitational potentials. In a static spacetime, the difference in gravitational potential between points A and B parametrizes the log of their relative time dilation. In a cosmological spacetime, rate-matching of clocks isn't transitive among clocks A, B, and C, so you define such a potential.

 

[stevmg]

Keep going, folks...

I am not the person who will be able to contribute on iota to this. All Ilearned from this is that uncountable iterations of an iterative equation which has an asymptotic limit does not have such a limit in the uncountable.

But, what you really are saying is that SR & GR may be a small microcosm of a greater Reality which is touched on in the above discussions.

 

[bcrowell]

a symmetric case

The question as originally posed is messy to analyze because the tension in the rope is going to be a function of both position and time. That's why I'd prefer to analyze simpler cases. I'm satisfied with my own analysis of the Milne universe case, although I gather that I haven't convinced everyone here.

Here is another case that's simple. Take a cosmological solution that's spatially closed, and let t be the FRW time coordinate. At some initial time t, construct a straight rope that is long enough to close back on itself. This is dynamically possible in principle; there are none of the issues you get with having to reel out the rope as in the OP's original scenario. Construct it so that the initial tension is zero everywhere. By symmetry, the tension will always be constant throughout the rope at any later FRW time t. That means that we can use the simpler treatment of an elastic rope given in [Egan1], rather than the more complicated one in [Egan2] where the tension isn't constant.

There are several things the rope can do: (1) it can expand while continuing to be straight, (2) it can become curved, and (3) it can break. I suspect that it would actually be dynamically unstable with respect to 2, but let's assume that that's prevented by some externally applied constraint. If it does 1, then its length increases uniformly. As its length increases, the tension goes up, and the speed of sound in the rope increases.

What's nice about this example is that due to its symmetry, we can reduce the GR problem to an SR problem. Anything that happens to the rope as a whole is observable by looking at any segment of it. Therefore the dynamics are exactly the same as if we simply took a one-meter piece of rope and stretched it at the same rate as the Hubble expansion. From the analysis in [Egan1], we know that there is a maximum amount of stretch that any rope can sustain without breaking, which is Q/\sqrt{3}K, where Q and K are related to the rope's density and spring constant. This maximum stretch occurs the point at which the speed of sound exceeds the speed of light (and it's less than the bound imposed by the weak energy condition).

We conclude that within a certain amount of time, the rope has to break. Once that happens, we have a question that's analogous to the OP's question: will the end of the rope snap forward like a whip at a velocity greater than c? The answer is no, because the end of the rope travels at less than the speed of sound, which in turn is less than the speed of light.

This case may appear trivial, but I think it demonstrates some nontrivial things: (1) Cosmological expansion can produce tension in a rope, even when no external force is being applied to the rope. This is a nontrivial point, since cosmological expansion doesn't normally produce significant expansion of bound systems like nuclei, meter sticks, and solar systems. (2) There are no horizons involved in the explanation, so I don't think the generic answer to the OP's question for FRW cosmologies has anything to do with horizons.

This case is also not completely unrelated to the OP's case. When the rope snaps, it has to snap at some specific point, so it spontaneously breaks the perfect azimuthal symmetry of the problem. However, the problem still has symmetry with respect to reflection across the break. Therefore there is a point on the rope, exactly on the opposite side of the universe from the break, where the rope remains at rest relative to the local galaxies. That's exactly like the OP's idea of hitching the rope to a particular galaxy. So in fact, I think this argument actually answers the OP's question in one special case, where (a) the universe is closed, and (b) the initial conditions are set such that the rope has constant tension throughout. As others here have pointed out, the choice of initial conditions constitutes an ambiguity in the OP's scenario (e.g., do you deploy the rope by reeling it out,...?).

[Egan1] http://gregegan.customer.netspace.net.au/SCIENCE/Rindler/SimpleElasticity.html
[Egan2] http://gregegan.customer.netspace.net.au/SCIENCE/Rindler/RindlerHorizon.html

 

[nutgeb]

This is one possible way of interpreting cosmological redshifts, but they can also be interpreted as occurring because the space occupied by the photon expands, which stretches the photon's wavelength.

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

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.

The Milne model provides a satisfactory description of a universe without gravity, using Minkowski coordinates and GR. The advantage of FRW coordinates is that they provide a more homogeneous view of the universe, such that distances at high recession velocities are not inhomogeneously Lorentz contracted and time dilated as they must inevitably be in Minkowski coordinates.

Cosmological solutions are homogeneous, so there's no way they can have a varying gravitational potential.

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.

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.

 

[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.