Is this a way to move faster than c?

[bcrowell]

FRW coordinates make the 'arbitrary' but rather unique choice of calibrating their time and distance axes to these invariant quantities, for all frames of reference which are comoving in accordance with Hubble's Law.

Sure, I'll grant you that the FRW coordinates are particularly useful. However, that doesn't mean that it's valid to interpret coordinate velocities as having a definite physical meaning. They also aren't invariant, for any useful or standard definition of "invariant."

Velocity is well defined as long as one sticks to a single coordinate system. In FRW proper radial distance coordinates, there is no ambiguity about how to calculate relative velocities of distant points.

I don't think this is correct. For example, consider a closed universe, where space has the topology of a sphere. We have two galaxies, separated by 1/3 of the circumference along a particular line L. If you take the other 2/3 of the circumference, you get a different line L', which has twice the length (measured in FRW coordinates). The rate at which L' expands (expressed in FRW coordinates) is twice as great as the rate at which L expands, so you can say that either one of these is a possible relative velocity of these two galaxies. So even if you stick to FRW coordinates, there *is* an ambiguity about how to calculate relative velocities of distant points.

Here's another way of seeing that what you're talking about doesn't work. If it did work, then I could measure the velocities of all the galaxies in a closed universe relative to myself, and then I could determine things like the total mass-energy of the universe, or the total angular momentum of the universe. MTW section 19.4 has a good discussion of why this is impossible.

 

[voltin]

The speed of light in a vacuum, and distant from effects of gravity, is claimed to be a constant number.
Does that mean that it is a rational and/or finite length number?

I'm a beginner at this stuff, so I hope this is not a obvious answer.

 

[nutgeb]

Any opinion on my analysis of the easier Milne-universe case?

I think modeling the Milne case is a good idea. But it seems like a lot of steps would be required, so the analysis would be convoluted. One might start with Minkowski recession velocities and chart the rope end's velocity increase (acceleration) as a function of Minkowski time. Then transfer the acceleration to a Rindler chart, and analyze the Rindler event horizon and the parameters that determine when the rope breaks. Then go back to the Minkowski chart and convert the Minkowski recession velocity components to FRW recession velocities. Compared to the SR recession velocity in Minkowski coordinates, the velocity in FRW coordinates is increased by the factor atanh:

$$V_{FRW} = \frac{1}{2} ln\left( \frac{1 + v_{sr} }{ 1 - v_{sr} } \right)$$

 

[bcrowell]

Hi, voltin -- welcom to PF!

In general, it would be better not to post an unrelated question in a preexisting thread on a different topic. Just start a new thread, using the NEW TOPIC button.

The speed of light in a vacuum, and distant from effects of gravity, is claimed to be a constant number.
Does that mean that it is a rational and/or finite length number?

No, a constant need not be rational. Pi is a constant, but it's not rational. In the SI, c is currently a quantity with a defined value, which is rational, but that's a fact about that system of units, not a physical fact about light. In general, the distinction between rational and irrational numbers is meaningless for measured quantities in science, because measurements have finite precision.

 

[Dale]

The speed of light in a vacuum, and distant from effects of gravity, is claimed to be a constant number.
Does that mean that it is a rational and/or finite length number?

Depends on the units. It is exactly 1 light-year/year and exactly 299792458 m/s, but you could make a new unit that was an irrational multiple of a meter and then the speed of light in that unit per second would be irrational.

 

[Dale]

Any opinion on my analysis of the easier Milne-universe case?

Not really. Again, my same intuitive guess would apply, but beyond that I don't want to do the analysis required, even to assert that the Rindler results could be used.

 

[Ich]

DaleSpam, thanks for the cool link to Greg Egan's page! It would be interesting to try to extend the analysis to the cosmological case, but I don't think it would be straightforward.

If you have a local definition for "no relative motion", you can pick an origin and calculate distances based on this notion. You'll have to do it numerically, but otherwise, it's straightforward. In the distant future, the anser becomes analytical again: a rope can be ~50 GLy long, until it vanishes at both ends in the horizon.

The Rindler metric is static, but realistic cosmological models are not.

They are, at least if you wait some 100 bn years (yes, de Sitter is static. That's not an error). For the time being, I think it's enough to acknowledge that neither non-staticity nor non-emptyness are defining features of FRW spacetimes. There are static FRW models, and there are empty models. In both, there is expansion, therefore expansion has nothing to do with curvature or generic non-staticity.

I think we can get at some of the interesting issues using the Milne model.

bcrowell, I think this is the beginning of a beautiful friendship.

If the question is, "Why can't I have a rope as long as I like?," then the answer becomes very clear in the Milne universe: the length of a rope is coordinate-dependent.

No, that's not the answer. You can have a rope as long as you like. Natural simultaneity is defined by neighbouring segments.
What happens is that, following this simultaneity, but expressed in FRW coordinates, right now and 13.7bn LY away, the rope goes through the big bang. That's not a problem, though, as a big bang of test particles is nothing to worry about.
Whatever, the respective spacelike geodesic is of infinite length, but it leaves the domain of the FRW coordinate system somewhere. Its "end points" are not mapped to finite distance values, however, that's why I say that this is not the answer.

Neither do I! I therefore won't make any firm conclusions about it, but my intuitive guess is that the stresses in a cosmologically long wire would become infinite before you would get any superluminal effects.

It becomes infinite when it crosses the event horizon. In non-accelerating spacetimes, there is no horizon, so there's no problem. Except for closed or non-trivial topologies, of course.
But it's easy to have a rope in the alleged "superluminal" region of proper-distance coordinates. As nutgeb explained, you simply add the dv's to get the recession "velocity", so it's clear that its definition is that of a rapidity, not a velocity.
Even in the "superluminal" region, the rope will have a velocity<c wrt the background, as long as it stays within the horizon. "Superluminal" is just a misnomer.

 

[nutgeb]

Sure, I'll grant you that the FRW coordinates are particularly useful. However, that doesn't mean that it's valid to interpret coordinate velocities as having a definite physical meaning. They also aren't invariant, for any useful or standard definition of "invariant."

I agree with you with respect to most coordinate systems, but in the particular case of FRW proper distance coordinates you are dividing change in proper distance (an invariant) by change in proper time (another invariant) to obtain proper velocity. So it seems to me that an invariant divided by an invariant is itself an invariant.

For example, consider a closed universe, where space has the topology of a sphere. We have two galaxies, separated by 1/3 of the circumference along a particular line L. If you take the other 2/3 of the circumference, you get a different line L', which has twice the length (measured in FRW coordinates). The rate at which L' expands (expressed in FRW coordinates) is twice as great as the rate at which L expands, so you can say that either one of these is a possible relative velocity of these two galaxies. So even if you stick to FRW coordinates, there *is* an ambiguity about how to calculate relative velocities of distant points.

I agree with you that in the special case of a finite 'closed' FRW model (as distinguished from an infinite 'open' one) one can arrive at a different proper distance figure by selecting a different angle of departure. But that's kind of an exception that proves the rule. You will have a single, unambiguous proper distance figure if you also specify the angle of departure (other than the trivial case where you draw a path through the destination and then go all the way around the same circumferential path again and again, counting each expanding (and eventually contracting) lap as a separate distance figure.)

Here's another way of seeing that what you're talking about doesn't work. If it did work, then I could measure the velocities of all the galaxies in a closed universe relative to myself, and then I could determine things like the total mass-energy of the universe, or the total angular momentum of the universe. MTW section 19.4 has a good discussion of why this is impossible.

Thanks for the reference, but I don't have that book.

 

[rede96]

Hi all,

I've been following the thread with much interest and although I can get a flavour for the discussions, the ingredients are a bit out of reach for the mo!

So I was wondering of someone would kindly summarise the following for me in terms of the original thought experiment please.

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

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

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?

 

[stevmg]

Two quick questions (Minkowski coordinates) - elemental, minimally related to the above:

1) If two events are outside each other's light cone (therefore "spacelike") is it always possible to find a frame of reference in which they are simultaneous?

2) If two events are such that one is inside the other's light cone (therefore "time like") can one categoriacally state that they will never be simultaneous?

If the above is true, then would a test for "potential simultaneity" be the tau test for proper time. If the square root is of a negative number, then these events are spacelike and potentially simultaneous while if positive, these events are timelike and never ever simultaneous?

I hate to bore you folks with trivialities but your consideration is most appreciated.

 

[DrGreg]

Two quick questions (Minkowski coordinates) - elemental, minimally related to the above:

1) If two events are outside each other's light cone (therefore "spacelike") is it always possible to find a frame of reference in which they are simultaneous?

2) If two events are such that one is inside the other's light cone (therefore "time like") can one categoriacally state that they will never be simultaneous?

If the above is true, then would a test for "potential simultaneity" be the tau test for proper time. If the square root is of a negative number, then these events are spacelike and potentially simultaneous while if positive, these events are timelike and never ever simultaneous?

I hate to bore you folks with trivialities but your consideration is most appreciated.

Yes, all of the above is true in the context of special relativity (i.e. ignoring gravity and expanding universes), although I'd say "spacelike-separated" rather than "spacelike" (etc). In general relativity, things can get more complicated so I wouldn't like to offer an opinion. It would still be true locally (by the equivalence principle).

 

[stevmg]

Yes, all of the above is true in the context of special relativity (i.e. ignoring gravity and expanding universes), although I'd say "spacelike-separated" rather than "spacelike" (etc). In general relativity, things can get more complicated so I wouldn't like to offer an opinion. It would still be true locally (by the equivalence principle).

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.


Thanks,

stevmg

 

[stevmg]

Another point about infinitely iterated calculations of the velocity addition formula I proposed (under Minkowski coordinates) to establish that speeds $$\geq$$c were not achievable, no matter where you arbitrarily start - no matter where. I actually forgot the difference between a countable infinite sequence (such as the set of rational numbers) versus the set of infinite yet uncountable set of numbers, such as the real numbers, which includes all the rational numbers which can be set 1-to-1 with the set of positive integers, therefore countable, while the real numbers always has all numbers "in between" the rationals.

Z = the countable set of integers
Z^$$\omega$$ = the uncountable set (Z is countable and so is $$\omega$$, but this "superset" is uncountable

My hypothesis of asymptotic approach to c from below by infinite iterations of the velocity addition formula appears logically correct, but the universe has an uncountably infinite quantity of frames of references and therefore this proposition would not be logically valid unless proven by another method.

To wit,
(1 + 1/n)ⁿ as n $$\rightarrow$$ $$\infty$$ = $$e\ =\ 2.71828182845904523$$ but that doesn't mean that (1 + 1/r)^r [if r is the set of all real numbers, not just the countably infinite set of integers] = e. But it should be, according to my meager mind, because no matter how large you go in the real numbers, you will always find a rational number or an integer greater than what you select so approaching infinity by rational or real numbers shouldn't make a difference. But that's just me.

Therefore, I stand corrected.

The next question I have is that has there ever been an experimental or observational documented speed of anything $$\geq$$ c?

The searchlight seems intriguing in that one can document a tangential velocity at radii sufficently small that these would be less than c. However, when one gets the radii large enough, the tangential velocities are all >c and each point on the "larger" circle is 1-to-1 with each point on the inner circle but the inner circle (all points with a velocity of <c) is the set of all real numbers and therefore uncountably infinite. There are no discontinuities in the outer cricle therefore whatever it is that that you want to call it moving there is moving greater than c.

 

[bcrowell]

@stevemg: There are various logical systems for dealing with infinite quantities. The one you're talking about is the ordinal numbers from set theory. The problem is that this system isn't rich enough to do analysis with. For instance, there is no notion of division, so an expression like your 1/r, where r is the cardinality of the reals, is not well defined. For systems that are rich enough to do some or all of classical analysis, see the Wikipedia articles "Surreal number" and "Non-standard analysis." But in any case, it doesn't matter physically which system you choose. The differences between the systems don't correspond to anything that can be realized by physical measurement processes.

The next question I have is that has there ever been an experimental or observational documented speed of anything $$\geq$$ c?

Yes, if you mean observations that don't contradict relativity. No, if you mean observations that contradict relativity.

Is the "equivalence priciple" you refer to the idea that acceleration is "equivalent" no matter how you get there?

http://en.wikipedia.org/wiki/Equivalence_principle
There are many different ways of stating the equivalence principle. The formulation that is most relevant to DrGreg's statement is that space is locally Minkowskian.

 

[stevmg]

@stevemg: There are various logical systems for dealing with infinite quantities. The one you're talking about is the ordinal numbers from set theory. The problem is that this system isn't rich enough to do analysis with. For instance, there is no notion of division, so an expression like your 1/r, where r is the cardinality of the reals, is not well defined. For systems that are rich enough to do some or all of classical analysis, see the Wikipedia articles "Surreal number" and "Non-standard analysis." But in any case, it doesn't matter physically which system you choose. The differences between the systems don't correspond to anything that can be realized by physical measurement processes.

It took me thirteen forevers to understand in a very limited way the set theory we just went over. Even though Z^$$\omega$$ where Z and $$\omega$$ are countably infinite is pretty rich (to use your term) you state that the set of numbers needed for cosmology has to be richer than that. I get it, but with Z^$$\omega$$ that is a set of numbers that cannot be placed in a 1-to-1 correspondence with anything. But even then, you have to be richer than that! Wow!

I do not dispute. I am merely a pawn in the game of relativity.

Yes, if you mean observations that don't contradict relativity. No, if you mean observations that contradict relativity.

Now, what does the above mean?

http://en.wikipedia.org/wiki/Equivalence_principle

There are many different ways of stating the equivalence principle. The formulation that is most relevant to DrGreg's statement is that space is locally Minkowskian.

I read that Wikipedia article - didn't make a lick of sense. Is the "free-fall vs no-g" statement a correct one for "equivalence?"

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!

H-E-L-P-!

stevmg

PS - BTW - even with restriction to the the Z^$$\omega$$ "superset" my "induction" principle wouldn't apply as it would with the natural log base e because in the latter, it is that's an imaginary one-step calculation for an incredibly large something. My supposition would require the uncountably infinite summations of an uncountable infinite frames of reference and that's even greater than
Z^$$\omega$$... that would be R^$$\omega$$ where both R and $$\omega$$ were the set of real numbers. Induction, though infinite, is still a "one-step-at-a-time" process while the universe is everything all at once. One cannot apply topological set theory to this at all.

 

[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.