Is this a way to move faster than c?

[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.