Is this a way to move faster than c?

[rede96]

Please excuse the silliness of this but...

As I understand it, the further away a galaxy is the faster it is moving away due to the expansion of the universe.

I think I read that the expansion is something like 77km/sec per 3.26 million light years.

Anyway, that means that there must be (or will be) some galaxies that are moving away from us faster than c.

So here's the silly question.

Imagine I was able to make (or keep adding to) an almost infinitely long wire, fly out to the nearest galaxy and attach one end of the wire to a planet, then fly back to earth. Then wait until that galaxy was moving away from Earth faster than c and grab hold of the wire and let it take me with it. I would be moving away from Earth faster than c.

If the direction was right, I could jump off at say Pluto and pass the latest football scores on to the locals, which would mean that they received a message from Earth faster than c.

Ignoring the obvious 'impracticalities', time factors, g-forces etc., what laws of physics would stop this from happening?

 

[causalset]

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". When they say that you can't exceed the speed of light, what they mean is that *IF* you somehow manage to bring the galaxies into your room, they will no longer be moving faster than the speed of light. THAT is true. Why is it true, it is a different question and it is up to the geometry of the universe -- that same curvature would somehow "slow down" the galaxies in the process of your bringing them into your room. The specific way in which that would happen is up to the specifics of geometry.

 

[rede96]

Hi causalset, thanks for the reply.

Your logic would have been correct if our spacetime was flat. But the fact is that our spacetime is curved.

Ok, I'll need to do some reading on that as I don't understand the implications.

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".

My logic would say that as I can compare velocities of things near to me, if the above statement is true, then there must be a distance threshold, some point or relative distance in spacetime where this becomes true. Is that right?

Also, we do know velocities of galaxies relative to earth. So I guess by the fact we know this, these galaxies have not yet crossed this 'threshold' yet?

When they say that you can't exceed the speed of light, what they mean is that *IF* you somehow manage to bring the galaxies into your room, they will no longer be moving faster than the speed of light. THAT is true. Why is it true, it is a different question and it is up to the geometry of the universe -- that same curvature would somehow "slow down" the galaxies in the process of your bringing them into your room. The specific way in which that would happen is up to the specifics of geometry.

I'm sure you are correct, but I have no comprehension of the logic that leads to this conclusion. More reading I guess!

 

[causalset]

Ok, I'll need to do some reading on that as I don't understand the implications.

Let me explain the implication. Suppose you have a globe, and two people are going "south". One is near the North pole, the other is near the south pole. Suppose their velocity is 5 miles per hour. So if they go with the same velocity in the same direction (namely, south), then their relative velocity is 0, right? Wrong. Their relative velocity is 10 miles per hour because meridean does 180 degree turn as you go from North Pole to South Pole.

Now, how do we know it is 180 degrees, rather than, say, 125 degrees? We use the geometry of space EXTERNAL to the globe. Now, if we had ONLY the globe, without any space external to it, we would not have that information. Therefore, we would not be able to compare velocities since we would not know the angle.

Well, the situation with curved spacetime is similar to a "globe" without any outside space. You see, our spacetime is "curved". So, this means that we need some LARGER dimensional space in which our spacetime is curved (with dimensions LARGER than 4), and then use that, larger, space to compare velocities. But we don't have the "larger" space, and that is a problem! This is precisely why we don't have the notion of comparison of velocites.

My logic would say that as I can compare velocities of things near to me, if the above statement is true, then there must be a distance threshold, some point or relative distance in spacetime where this becomes true. Is that right?

First of all, the comparison of velociites of the objects near to you are not exact. It is an approximation. You ASSUME that inside of your room the spacetime is flat. If so, the velociites cna be compared. But that assumption is wrong. Even inside your room it is curved. But the curvature can't be felt on such a small scale (similarly to the curvature of the Earth not being felt on the scale of your local town). That is why you can say it is APPROXIMATELY flat and based on this assumption define APPROXIMATE comparison of velocities.

Now, as far as "distance threashold", it depends on two things:

1) The curvature of our spacetime near a given location

2) The level of approximation you can accept

The curvature is simply a gravitational field. Thus, if you have a very massive object then the curvature will be so high, that even on a level of millimeter it would have very significant effect. Thus, you can't have a universal threshold. Any "threshold" you use depends on a specific situation you are in, and the strength of gravitational fields (or, equivalently, curvature).

On the other hand, even if gravitational fields are small, you might want to do very exact calculations and from that point of view the curvature effects in your room are not acceptable. So, you can't say "for a given field the threshold is this size of a room", because you have to also take into account the level of approximation you are willing to accept.

From strictly mathematical point of view, the "threshold" is the infinitesimal scale. Within that scale everyone agrees things are flat, regardless of curvatures.

Also, we do know velocities of galaxies relative to earth. So I guess by the fact we know this, these galaxies have not yet crossed this 'threshold' yet?

We know the velocities between galaxies and Earth because we can draw a "geodesic curve" that connects Earth and galaxies and measure its length. Now, the "geodesic curve" is a generalization of the notion of "straight line" for the curve space. You see, in curved space the notion of "straight line" is not defined -- after all the meridians on a globe are not "straight"; but the notion of "geodesics" is, in fact, well defined (you can read into it).

But, here is a catch. While the "distance" (or the length of geodesic) has a physical meaning, taking its DERIVATIVE (and identifying it with velocity) does not. The only physically meaningful "velocity" is the one taken inside a region where curvature can not be felt.



I'm sure you are correct, but I have no comprehension of the logic that leads to this conclusion. More reading I guess![/QUOTE]

 

[rede96]

Let me explain the implication. Suppose you have a globe, and two people are going "south". One is near the North pole, the other is near the south pole. Suppose their velocity is 5 miles per hour. So if they go with the same velocity in the same direction (namely, south), then their relative velocity is 0, right? Wrong. Their relative velocity is 10 miles per hour because meridean does 180 degree turn as you go from North Pole to South Pole.

Now, how do we know it is 180 degrees, rather than, say, 125 degrees? We use the geometry of space EXTERNAL to the globe. Now, if we had ONLY the globe, without any space external to it, we would not have that information. Therefore, we would not be able to compare velocities since we would not know the angle.

I think I get what you mean. I can have two people on the opposite side of a globe traveling 'south'. Knowing that they are on a globe of 360 degrees and traveling on the same line of longitude, I can say that they are moving towards each other or apart from each other, each with a velocity of 5 mph.

So if I take the case where they are moving apart from each other then I can add their velocities and say that they are moving apart by 10 mph relative to each other.

Is that right?

So, what if they are both on the same longitude, only separated by a small distance and both traveling in the same direction along the same longitude?

Then can't I say that there relative velocity is 0 and as long as they keep traveling along the same line of longitude, then their relative velocity will always be 0 and that they are at rest with respect to each other?

Thus, even if I didn’t know the curvature of the globe, when they are moving along the same line of longitude, I could still say that their relative velocity is 0.

 

[causalset]

I think I get what you mean. I can have two people on the opposite side of a globe traveling 'south'. Knowing that they are on a globe of 360 degrees and traveling on the same line of longitude, I can say that they are moving towards each other or apart from each other, each with a velocity of 5 mph

This is a tricky question. If you define distance in terms of regular space, rather than geodesic, then the distance between them is 2R, where R is the radius of the earth. The distance will NOT be the length of geodesic but rather the length of the line passing through the centre of the earth. In terms of THIS distance, its derivative is 0 regardless of direction of their velocities, so they are neither moving towards each other nor away from each other.

HOWEVER, that particular distance does not exist in our universe, since the "center" of a glob is OUTSIDE of the universe (the universe is only the surface of the globe). So, our only options is to define distance based on the length of geodesics (which, in this case, are meridians of the globe). Then the distance is pi R, instead of 2R, and then its derivative is non zero.

But, again, we have more than one way to define the distance. If we want, we can use the part of meridian that does not pass through the pole (that was the assumption I made in previous reply to you). Or, instead, we can use the other part of meridian that passes through the pole. So the sign of the derivative of the distance will be opposite depending on which you will choose.

The bottom line is that htere is no objective way of answering whether they are moving towards each other or apart from each other. When I used the "south" criteria, it was one of the MANY subjective definitions that I pulled out of the air, all of which will contradict each other. The fact that subjective definitions of velocity contradict each other is a proof that the velocity can't be defined. Again, the ultimate reason for this is the curvature of space.

So, what if they are both on the same longitude, only separated by a small distance and both traveling in the same direction along the same longitude?

In this case there is still A LITTLE BIT of ambiguity, but the ambiguity is much smaller and it is negligeable as far as most calculations are concerned.

So the "negligeable" ambiguity comes from two sources:

a) Since your "universe" is on the surface of the globe, you are forced to say the distance is the length of the arch rather than the straight line. So if $\alpha$ is the angle between the two lines comming from the center of the earth, then the distance you are FORCED to measure is $R \alpha$. But, at the same time, we know that the actual distance is defined based on straight line (which doesn't exist in our "universe") and the length of that line is $2 R sin (\alpha /2)$. Now, in the limit of $\alpha$ approaching $0$, the two are the same. Now, $\alpha$ approaching $0$ is the same thing as the two objects are closer and closer to each toher. Thats why on small scales this effect is negligeable.

b) You can either define a distance based on a "small" arc that connects the two objects, or you can use the "large" arc that circles the Earth and then comes back to the other object. Of course, you have to choose "small" arc. But this is not always the case -- if the curvature was not constant, while the difference between arc lengths was not as dramatic, it could have been desireable to take larger arc rather than smaller one. So you have to make subjective judgement that in this case it is silly to talk of the "large" arc and small one is the key.

Now, both a and b are much SMALLER issues than the ones raised on the example of two opposite poles, and that is because the two objects are very close to each other. Now, if you want to OBJECTIVELY make a and b go away, then you have to bring the two objects infinitely close. In this case, the answer to the above two questions is

a) Whatever USED to be a limit now is exact, and the straight line connecting two objects COINCIDES with the arch connecting them

b) The curve segment of INFINITESIMAL length is the only one that is legitimate, since it is the only one where the above discussed effects disappear. Therefore, since one segment is OBJECTIVELY infinitesimal while the other one is finite, we choose the infinitesimal one, and that choice is purely objective.

 

[Dale]

Hi rede96, causalset has given you some excellent replies. I will just post a link to a couple of relatively recent posts where I gave a concrete example of how there is no unique way to compare the angle of two distant vectors in a curved space.

https://www.physicsforums.com/showpost.php?p=2758350&postcount=25

https://www.physicsforums.com/showpost.php?p=2637761&postcount=79

 

[bcrowell]

Although I think everything causalset says sounds correct, to me it doesn't feel like a complete and satisfying solution to the proposed paradox. Rede96 has proposed a specific experiment involving a wire, and although he/she acknowledges that the experiment is utterly impractical, it still seems to me that a proper resolution should address the actual experiment. Either we should explain why the experiment can't (even in theory) be carried out, or we should say what the results of the experiment would be and why that is consistent with the predictions of GR.

I would conjecture that the answer is simply that in the usual cosmological models, there is a limit to the length of an inelastic wire that is laid out along a geodesic. I think the limit is simply k/H_o, where H_o is the Hubble constant and k is a unitless constant of order 1. (The exact value of k probably depends on the specific cosmological model.) Any wire longer than this must break. This has the same flavor as other arguments involving relativity and the strength of materials. For instance, there has to be a limit on the tensile strength of wires in GR, because otherwise you could use a wire to haul objects out from behind the event horizon of a black hole. In SR, a very lightweight, strong wire could be used to send information at >c using vibrations.

This is also highly reminiscent of the Bell spaceship paradox.

 

[Bussani]

My view on it might be too simple compared to what's been said, but something you should keep in mind is that the galaxy you're attaching your rope to isn't actually moving faster than light. So sure, you could grab on and the distance between you and Earth could expand faster than light. Or to make it simpler, you could just have Earth flying in one direction and you flying in the other, and you could be putting distance between you and Earth at a rate faster than c. However, you won't actually be going faster than c, and thus light from Earth would still be overtaking you. No matter how you do it, a radio message from Earth would still beat you to Pluto.

In SR, a very lightweight, strong wire could be used to send information at >c using vibrations.

Could it? I thought the vibrations couldn't travel faster than light through a medium like that. Wouldn't they be limited by the speed of sound in the wire?

 

[bcrowell]

something you should keep in mind is that the galaxy you're attaching your rope to isn't actually moving faster than light.

Depending on the definition of "moving," it may actually be moving faster than c relative to the earth. This is essentially the point of causalset's posts: that there is no well-defined way to say how fast two objects are "moving" relative to one another when they're separated by cosmological distances. Relativity doesn't forbid velocities greater than c; there are lots of well-known examples such as the searchlight, the scissors, ... What relativity forbids is certain much more specific things, like relative motion of two *nearby* objects at >c, or transmission of information at >c.

In SR, a very lightweight, strong wire could be used to send information at >c using vibrations.

Could it? I thought the vibrations couldn't travel faster than light through a medium like that. Wouldn't they be limited by the speed of sound in the wire?

That's the point. We know that information can't propagate at >c, and therefore the speed of sound in the wire must be <=c. This puts constraints on the possible properties of any material (density, Young's modulus, tensile strength,...).

 

[Bussani]

Depending on the definition of "moving," it may actually be moving faster than c relative to the earth. This is essentially the point of causalset's posts: that there is no well-defined way to say how fast two objects are "moving" relative to one another when they're separated by cosmological distances. Relativity doesn't forbid velocities greater than c; there are lots of well-known examples such as the searchlight, the scissors, ... What relativity forbids is certain much more specific things, like relative motion of two *nearby* objects at >c, or transmission of information at >c.

Hmm, so are you not allowed to leave Earth at relatively more than c? Say if Earth was going one way at 0.8c and you went the other way at the same? Although I guess that would be very hard under normal circumstances since you'd also be going 0.8c in the same direction as Earth before launching... I think I might see what you mean.

That's the point. We know that information can't propagate at >c, and therefore the speed of sound in the wire must be <=c. This puts constraints on the possible properties of any material (density, Young's modulus, tensile strength,...).

Okay, that makes sense.

 

[causalset]

Hmm, so are you not allowed to leave Earth at relatively more than c? Say if Earth was going one way at 0.8c and you went the other way at the same?

You have to remember that in relativity the velocity addition formula is NOT u+v but rather (u+v)/(1+uv/c^2). So, yes, it is possible for two objects to go with velocity 0.8c in opposite directions, but their velocity relative to each other would NOT be 1.6c. It would be (1.6c)/(1+0.8^2) which would be smaller than c.

The reason the formula u+v does not work is that when we derive it we assume that if two events are "simulteneous" in one reference frame, they are also "simulteneous" in other ones. But that is not correct. When you move, what happens is that you make a "rotation" in space-time, so the two events in space are no longer simulteneous.

Think of a rotation in ordinary Eucledian space. The x-axis is time, and y-axis is a position. Then the velocity is the TANGENT of an angle. Now, what satisfies addition formula is an angle, itself, NOT its tangent. When the angle is small, the two are approximately teh same (the limit of (tan x)/x is 1 as x approaches 0), and that's why addition formula SEEMS to work when velocities are "small" (and, yes, our everyday velocities ARE small since speed of light is equal to 1 in the TRUE units). But once velocities get large it stops working.

Now, in case of spacetime, there is one more saddlety. The geometry is NOT the same as on a plane. While in the plane we have Eucledian geometry, where distances are given by x^2+y^2, in spacetime we have Minkowskian geometry where distances are given by t^2-x^2, NOTICE THE MINUS SIGN. Due to this minus sign, the rotation has a property that, no matter how much you rotate, you can never go beyond the speed of light (that is, a diagonal line on spacetime diagram); on the other hand, were the space Eucledian, you would be able to rotate by 90 degrees and travel with infinite speed, and then rotate more and travel back in time. Furthermore, in the argument that I made, the word "tangent" should be replaced with "hyperbolic tangent", again due to metric being Minkowskian. But, on a CONCEPTUAL level what I said is still correct.

 

[Bussani]

You have to remember that in relativity the velocity addition formula is NOT u+v but rather (u+v)/(1+uv/c^2). So, yes, it is possible for two objects to go with velocity 0.8c in opposite directions, but their velocity relative to each other would NOT be 1.6c. It would be (1.6c)/(1+0.8^2) which would be smaller than c.

Oh yeah. Thanks. I should have remembered that, since it's the same as if you had two things passing each other at such speeds. I guess that means I was mistaken about the whole point of the question; it isn't that the galaxy is moving away from us and we're moving away from the galaxy thus causing the relative greater than c velocity, but rather that the spacetime between us and the galaxy itself is expanding faster than c, right? I was looking at it all wrong.

 

[bcrowell]

Oh yeah. Thanks. I should have remembered that, since it's the same as if you had two things passing each other at such speeds. I guess that means I was mistaken about the whole point of the question; it isn't that the galaxy is moving away from us and we're moving away from the galaxy thus causing the relative greater than c velocity, but rather that the spacetime between us and the galaxy itself is expanding faster than c, right? I was looking at it all wrong.

Some people prefer to speak in terms of the expansion of space, while others don't like that way of talking about cosmological expansion. Neither way is right or wrong. It's just a matter of preference.

Arguments against thinking in terms of expanding space: E.F. Bunn and D.W. Hogg, "The kinematic origin of the cosmological redshift," American Journal of Physics, Vol. 77, No. 8, pp. 694, August 2009, http://arxiv.org/abs/0808.1081v2

Arguments in favor of thinking that way: http://arxiv.org/abs/0707.0380v1

What would be incorrect would be to expect to be able to apply *any* velocity addition formula (either the linear Newtonian one or the nonlinear relativistic one) to objects that are separated by cosmological distances. The whole idea of relative velocity of distant objects is not well defined.

 

[rede96]

Sorry for the late reply, I’ve had to work away for a couple of days.

I’ve been trying to get my head around this and there seem to be a number of problems with my thought experiment, which to be honest I don’t fully understand.

Anyway, I’ve tried to go through the main ones below and apply my limited logical ability to see if I can ‘test out’ a few thoughts I had.

Again please forgive the absurdity of the idea, but I am genuinely trying use this to get a better understanding.



1) Firstly I tried to consider, is it possible for an object to be moving away at speeds greater than c, relative to earth?

For me the Hubble constant, v = H0D states that even a galaxy that is for all intents and purposes at rest with respect to ours, will at some point in time be moving away from us at speeds greater than c from the expansion point of view.

I understand that it may be the expansion of the universe that is pushing it away and that the galaxy itself may still be at ‘rest’ relative to Earth if it wasn’t for expansion. So it’s not breaking any laws by moving away from us faster than c.

But the net result is sooner or later it must be moving away at speeds greater than c relative to the earth.

(I suppose that according the Hubble law, these galaxies could be moving away from us at infinite speeds maybe?)


2) The next issue was one of measuring velocities of objects at cosmological distances.

For my silly thought experiment to work, I don’t need to measure or even know the velocity of the distant galaxy or planet. I just know that at some point it will be moving at speeds greater than c relative to the Earth due to the Hubble constant.

There was also the mention of the wire not actually traveling at speeds greater than c relative to earth. However if the wire is part of the other galaxies frame of reference and that galaxy is moving away at speeds greater than c, then the wire must also be moving >c

3) As far as the limit for the length / strength of the wire, let's say that instead of me hanging on to the wire, I just write a simple message on it that someone could read as it passed by them.

Also, if I was to start making an almost infinite length of wire here on Earth and just let it lie on the ground, then tensile strength may not matter either as I am not ‘pulling’ anything with it. I guess sooner or later gravitational effects might be a problem, but if I let that wire spread out in space then those effects are also spread out.

I suppose there would be an issue when the wire had to accelerate from one reference frame to the other (which I assume is where the relativistic effects may take place.) but let’s just say for now I could find a way around that and that the wire doesn’t break.



Thus...

I am still left with the conclusion that I ‘could’ in theory attach the message onto the wire, which is moving away from me at speeds greater than c and pass that message to someone on Pluto faster than a beam of light could do it.

A counter argument that I thought of is that as the wire would be moving great then c relative to me, then maybe the wire would be expanding and the message moving back in time. So the net result might be that the message still gets to the person on Pluto traveling faster then c, but due to relativity, would only arrive at the same time as a light beam sent, thus no info has traveled faster than c.

Sorry for the brain dump! But does that stack up?

 

[nutgeb]

rede96, your idea about communicating faster than the speed of light won't work. No communication can occur at faster than c.

But your point about the Hubble velocity exceeding the speed of light is well taken. Of course Hubble recession velocities of distant galaxies exceed the speed of light when considered in FRW (Friedmann-Robertson-Walker, also known as FLRW) coordinates, which are the standard coordinates used for cosmological analysis. The Hubble velocity is simply H*D, where D is proper distance, and as D increases without bound, and H is constant, without doubt the recession velocity will begin to exceed c at some distance (that distance actually is called the Hubble Radius).

Too much effort is made to scare people away from this conclusion by referring to the ambituities of parallel transport of 4-vectors, etc in curved spacetime. While those ambiguities clearly exist, we should not be dissuaded at all from concluding that distant recession velocities do exceed c, in appropriate coordinate systems. Non-local velocities usually will vary from one coordinate system to another, which is a basic fact of physics that we just have to accept and move on.

Hubble proper velocities in excess of c are not unique to curved spacetimes. For example, they are a general feature of FRW coordinates in 'open' model universes with vanishingly small gravity content, such as specified by the Milne model, where the spacetime curvature approximates zero.

FRW coordinates assume a homogeneous distribution of matter (or test particles, in the case of a model without gravity) and an isotropic expansion that follows Hubble's Law, H*D. The axes of an FRW chart are proper time and proper distance. That is very different from the Minkowski coordinates used for Special Relativity, for which the axes are local coordinate time and local coordinate distance. The use of proper time means that all fundamental comovers (e.g., galaxies at rest in their local Hubble flow) share the same cosmological time, and therefore no SR time dilation occurs as between them, regardless of recession velocity. The use of proper distance means that FRW measures the distance that would be obtained if a huge number of comoving observers, lined up next to each other, laid down rulers end to end at the same instant, to measure the distance between two distant galaxies. There is no (or really vanishingly small) SR Lorentz contraction of the rulers, and the SR velocity addition formula is not used: the lengths of the rulers are simply added together. The Hubble recession velocity then is simply a change in proper distance divided by a change in proper time.

Imagine using FRW coordinates in a model universe empty of gravity. We start with a huge number of comoving observers spread evenly at very small intervals in a radial line leading away from the origin. Each successive observer a little bit further from the origin has the same recession velocity relative to each of its radial neighbors. The arrangement is such that the comoving observer at the furthest end of the line has a proper recession velocity in excess of c relative to the observer. Note that unlike Minkowski coordinates, a single FRW local frame can never be extended to include the recession motion of any distant comover. Instead, distance measurements must be aggregated by adding local frame measurements together.

This same measurement could be modeled alternatively in Minkowski coordinates, where SR effects apply. In that case, the reference frame of the measurer at the origin can be extended to encompass the most distant comoving observer. Then the SR velocity addition formula would ensure that the the furthest comoving observer has a coordinate recession velocity less than c, and the distance to him would be Lorentz contracted -- but those would be measurements made relative to the local inertial frame of the origin observer. They are coordinate measurements, not proper distance and proper velocity measurements. The former is made relative to a single location; the latter is made by summing the local measurements of observers (who are all in motion relative to each other) all along the line. Yet, by transforming from one coordinate metric to another, the measurements obtained can be considered entirely consistent, even if the numbers are different. For some reason (probably to avoid confusing students learning about SR) most commentators shy away from saying plainly that, yes of course a summed series of proper velocities can exceed c.

The reason why your extended wire scenario won't work is that, even in FRW coordinates, velocities within each local comoving reference frame cannot exceed c under any circumstance. Locally FRW coordinates approximate to Minkowski coordinates. If two distant observers, who have a recession velocity relative to each other, try to hold two ends of a long wire, the wire must of course stretch or break. Over short distances the stretching force is relatively small, but at cosmological distances the strength required for the wire to resist becomes infinite. Even in theory, the fastest the wire could be passing by any distant galaxy is constrained to be less than c, in that distant galaxy's local reference frame. There is no way around that constraint. The wire must stretch or break.

Note that by comparison, the idea of laying down rulers end to end works only because the rulers are physically disconnected from each other. In reality each ruler is moving away from the next ruler at the local Hubble rate. With rulers butted against each other, that motion is vanishingly small and undetectable between any two rulers, but over a huge number of rulers it aggregates to the full Hubble recession velocity between the two galaxies. If ahead of time you instructed all the observers to lock all adjacent rulers together at a given instant, the combined structure would instantly fracture (probably at many locations) due to Borne rigidity effects resulting from the local accelerations occurring in individual local comoving frames. By definition when the rulers are suddenly coupled, some of them must begin accelerating in some local comoving frame(s), because their 'rest' inertias are already in motion relative to each other. Remember that SR effects occur within FRW local comoving reference frames (but not between them).

To summarize, a series of local proper velocities, each less than c, can be summed radially to a total proper velocity exceeding c, but never a velocity exceeding c within any single local reference frame. An FRW chart naturally conveys the summation of proper velocities of comovers who all are in motion relative to each other. Whereas it would be unnatural to portray it on a Minkowski chart, because the latter does not treat all comovers as being in equally privileged local reference frames (there is always a single privileged local reference frame, and only times measured by a clock carried by an observer at rest in that one frame, and distances measured between points at rest in that one frame, are proper times and proper distances.) Summed proper velocities can exceed c even in the absence of spacetime curvature, so the concept can be explained without reference to general relativity. However, it can be extended equally well to curved spacetimes, where it continues to work in a very similar way.

 

[Bussani]

1) Firstly I tried to consider, is it possible for an object to be moving away at speeds greater than c, relative to earth?

For me the Hubble constant, v = H0D states that even a galaxy that is for all intents and purposes at rest with respect to ours, will at some point in time be moving away from us at speeds greater than c from the expansion point of view.

I understand that it may be the expansion of the universe that is pushing it away and that the galaxy itself may still be at ‘rest’ relative to Earth if it wasn’t for expansion. So it’s not breaking any laws by moving away from us faster than c.

But the net result is sooner or later it must be moving away at speeds greater than c relative to the earth.

(I suppose that according the Hubble law, these galaxies could be moving away from us at infinite speeds maybe?)

It seems that's only one way of looking at it (late thanks for the links, bcrowell!), but yes.

There was also the mention of the wire not actually traveling at speeds greater than c relative to earth. However if the wire is part of the other galaxies frame of reference and that galaxy is moving away at speeds greater than c, then the wire must also be moving >c

But the wire must also be in Earth's frame of reference as well, right? It doesn't seem like the galaxy's one should take precedence just because the wire is tied to it.

3) As far as the limit for the length / strength of the wire, let's say that instead of me hanging on to the wire, I just write a simple message on it that someone could read as it passed by them.

I think bcrowell's point is that the wire can't possibly exist at all, not just that it can't pull something heavy. Maybe we could look at it like the wire couldn't possibly be strong enough to pull itself? But I might be wrong there.

 

[nutgeb]

Bussani, see my post #15 which went up shortly before yours.

 

[rede96]

First of all, thanks to everyone for their time and information. It is very much appreciated.

But the wire must also be in Earth's frame of reference as well, right? It doesn't seem like the galaxy's one should take precedence just because the wire is tied to it.

I don't know. This got me thinking about what exactly is the cut off for something moving from one frame of reference to another?

For example would you say that a plane flying overhead is in a different frame of reference with respect to you, or is it that we are 'all' in the Earth's frame of reference?

As I understand it, the effects of relativity are relevant between all different frames of reference, (i.e. where something is moving with respect to you or in a different gravitational field.) This would be regardless of how 'close' or they are.

For me, what brings an object into my frame of reference is when I physically interact with it.

So it would be possible for the wire to still be in the other galaxy's frame of reference until I actually touched it.

I'm guessing that probably isn't quite true, but I it seemed a good argument!

I think bcrowell's point is that the wire can't possibly exist at all, not just that it can't pull something heavy. Maybe we could look at it like the wire couldn't possibly be strong enough to pull itself? But I might be wrong there.

Again, I don't know. My thought process was that I first of all I imagined the wire to be at rest with respect to me and ignored gravity. Hence there would be no 'forces' acting upon it, so there 'shouldn't' be any limit to the length of it.

Taking into account gravity, I didn't know of any limit to the amount of mass that could exist in a given area of space-time, although I'm sure there must be. But even if there is, those effects would be spread out over the length of the wire if I let it unravel in space and for all intents and purposes, the wire is still at 'rest'


Anyway, to try and bring this back into the realm of the sensible, I guess there a number of basic questions that my silly thought experiement raised for me.

1) Can an object move faster than c relative to me - The answer seems to be yes.

2) Can I use that fact for finding a way to communicate faster than the speed of light - The answer seems to be No, although I obviously don't fully understand why. (And will do some more reading on this.)

3) Are there any relativistic effects caused by an object that is traveling at greater than c with respect to me and if so what would they be? - We've not really discussed this so again I'll need to do some study. However even if there were some effects, I don't think that they would ever have any relevance for my frame of reference.

4) When is another frame of reference not another frame of reference? - I would be very interested to know what the answer is to that one.


There were also lots of other stuff that was very useful for me, thanks to all.

One of the main ones I would really like to grasp is the effects of curved space and the coordinate systems mentioned. So I'll do some research on that too.

 

[nutgeb]

This got me thinking about what exactly is the cut off for something moving from one frame of reference to another?

For example would you say that a plane flying overhead is in a different frame of reference with respect to you, or is it that we are 'all' in the Earth's frame of reference?

As I understand it, the effects of relativity are relevant between all different frames of reference, (i.e. where something is moving with respect to you or in a different gravitational field.) This would be regardless of how 'close' or they are.

For me, what brings an object into my frame of reference is when I physically interact with it.

So it would be possible for the wire to still be in the other galaxy's frame of reference until I actually touched it.

I'm guessing that probably isn't quite true, but I it seemed a good argument!

The concept of how big a frame of reference can be is a flexible concept that depends on which coordinate system you are using and what you are trying to measure or calculate.

In flat, empty spacetime using the normal Minkowski coordinates of SR, a reference frame in effect extends to infinity -- no size limit. So I can extend my reference frame to include an object, say, 100 GLy away from me. If it has a zero velocity relative to me, then it is 'at rest' in my inertial frame of reference. If it has a nonzero velocity then its motion relative to my rest frame is governed by Special Relativity.

In the case of a massive body such as the earth, Schwarzschild coordinates are often used. In that case, the size of a local reference frame is fuzzy. It depends on how much spacetime curvature there is but also on the degree to which you are willing to approximate away small errors in calculations. For an object plunging in freefall toward the earth, it is typical to say that its local reference frame encompasses the volume around it that is small enough such that tidal effects across that distance are "negligible". (They will never be absolutely zero). If you want to measure relative gravitational accelerations or time dilations at two different heights, then it depends on how sensitive your instruments are, whether the difference in outcome between the two heights is too small for them to detect.

In the case of FRW coordinates used at cosmological distances, again it is a question of how much approximation error you want to tolerate, but the distance scales are much more vast. For example one might plausibly define local frame encompassing a radial distance of 1 Ly or more.

In FRW coordinates, if you can touch a wire, it certainly is in your local frame. But if a wire extends past a number of galaxies, you should think conceptually that the wire is passing through many, many local reference frames, and certainly you should think of the galaxies themselves as being in separate local frames. So you should not think that a given wire could be simultaneously at rest in your local frame as well as in the local frame of a distant galaxy. Definitely not. If one end of the wire is at rest in one of those frames, then all the rest of the wire must be locally in motion relative to the receding galaxies within the many other local frames the wire passes through.

The scenario breaks down entirely when you try to put the wire in place. Imagine that you tie one end of the wire to the Earth and then spool it out from a rocket which starts passing galaxies along its way (this is a figurative discussion). The rocket has to continually accelerate to match the increasing Hubble velocities (H*D), relative to earth, it enters as it moves farther and farther away. From the rocket's perspective, Earth is receding away from the rocket at faster and faster Hubble velocities, and pulling the tied end of the wire with it. Even assuming there is no friction in the unrolling spool, the increasing acceleration of the Earth end of the wire puts more and more stretching stress on the wire. Also the spool would be turning at an angular velocity approaching c. The amount of spool's inertial force that the already deployed wire would require to pull more wire out of the spool would approach infinity as the speed approaches c. All of this stress must cause the wire to break.

My point is not that you need a more creative way to put the wire in place. Rather, it simply can't be done even in theory because no matter how you try to do it, an intact wire of sufficient length would require local motion in excess of c within a local frame, which can't happen.

3) Are there any relativistic effects caused by an object that is traveling at greater than c with respect to me and if so what would they be?

In FRW coordinates, no Special Relativistic effects (time dilation, Lorentz contraction) occur as between two distant comoving galaxies. There are General Relativistic effects, in the sense that in a universe with gravity, over time gravity will seek to slow the relative recession velocity between the two galaxies, and Dark Energy will seek to speed up that recession velocity. The relationship between the Hubble velocity and the mass-energy density of the universe will also determine whether there is spatial curvature, which is a GR effect. However, if a homogeneous matter distribution is assumed, there will be no gravitational time dilation as between comovers in FRW coordinates.

 

[rede96]

In flat, empty spacetime using the normal Minkowski coordinates of SR, a reference frame in effect extends to infinity -- no size limit. So I can extend my reference frame to include an object, say, 100 GLy away from me. If it has a zero velocity relative to me, then it is 'at rest' in my inertial frame of reference. If it has a nonzero velocity then its motion relative to my rest frame is governed by Special Relativity.

That was my understanding too. In SR, a different reference frame depends on relative motion, not relative distance.

In the case of a massive body such as the earth, Schwarzschild coordinates are often used. In that case, the size of a local reference frame is fuzzy. It depends on how much spacetime curvature there is but also on the degree to which you are willing to approximate away small errors in calculations.

For an object plunging in freefall toward the earth, it is typical to say that its local reference frame encompasses the volume around it that is small enough such that tidal effects across that distance are "negligible". (They will never be absolutely zero). If you want to measure relative gravitational accelerations or time dilations at two different heights, then it depends on how sensitive your instruments are, whether the difference in outcome between the two heights is too small for them to detect.

Ok, this where I get a little lost. Are you saying that objects that share the same gravitational forces are basically in the same frame of reference?

In the case of FRW coordinates used at cosmological distances, again it is a question of how much approximation error you want to tolerate, but the distance scales are much more vast. For example one might plausibly define local frame encompassing a radial distance of 1 Ly or more.

Again, I understand this to be that in the case of FRW, objects that are at rest with respect to each other and that are less than the 1 Ly example above, then they can be said to share the frame of reference.

But just to clarify, objects that are in motion relative to each other will always be in different frames of reference, no matter how close they are in distance. Is that right?

In FRW coordinates, if you can touch a wire, it certainly is in your local frame. But if a wire extends past a number of galaxies, you should think conceptually that the wire is passing through many, many local reference frames, and certainly you should think of the galaxies themselves as being in separate local frames. So you should not think that a given wire could be simultaneously at rest in your local frame as well as in the local frame of a distant galaxy. Definitely not. If one end of the wire is at rest in one of those frames, then all the rest of the wire must be locally in motion relative to the receding galaxies within the many other local frames the wire passes through.

I agree that a wire could not be simultaneously at rest in my local frame as well as in the local frame of a distant galaxy, but that wasn't what I was trying to say.

For example, I can have long rope that is at rest wrt to me. I tie one end onto a car and the car sets off. For a period of time, part of the rope will be in motion wrt me and part of it will be at rest wrt me.

So the rope shares both frames for a time, no? But once the car is in steady motion then the rope shares the same reference frame as the car and a different reference frame to anyone else that is moving relative to the car, even if they are stood very near to the rope.

However if they then grabbed onto the rope, they would share the same frame as the car.

I see this to be the same as the wire passing galaxies.

 

[bcrowell]

The argument I'd have liked to make, but that doesn't quite work, is by analogy with a black hole. A black hole has an event horizon. The future light cone of any event inside the horizon lies inside the event horizon, not outside it, so it's not possible for an event inside to be the cause of something that happens outside. If you could hang a wire through the event horizon, this would be a counterexample, because you could send signals using vibrations in the wire, or use the wire to haul things out. We therefore conclude that no wire can be strong enough to withstand the spaghettifying forces created when we hang the wire down through the horizon and prevent it from falling. It's not surprising that there is an absolute relativistic limit on the tensile strength of materials. If not, then vibrations in materials could go faster than c, but matter is held together by electromagnetic forces, and electromagnetic disturbances propagate at c.

Unfortunately this doesn't carry through 100% to the case of cosmic expansion. Cosmological solutions can have horizons, but it's not correct to say that if point A and point B are expanding away from one another at >c, then B must be behind an event horizon as seen by A. The whole notion of the relative velocity of cosmologically distant points isn't even well defined.

I need to think about this some more, but this makes me suspect that there is a hidden problem of definition involved here. We've been assuming that a statement like "There is an unbroken rope of length L connecting points A and B" is a well-defined physical statement that can be checked by observation. Now, it's not totally obvious to me that it is well defined. For one thing, the fact that simultaneity is not well defined in relativity means that different observers will not necessarily agree on whether the rope is broken "right now" at some distant point.

 

[nutgeb]

Ok, this where I get a little lost. Are you saying that objects that share the same gravitational forces are basically in the same frame of reference?

That's a good enough description for Schwarzschild coordinates.

Again, I understand this to be that in the case of FRW, objects that are at rest with respect to each other and that are less than the 1 Ly example above, then they can be said to share the frame of reference.

But just to clarify, objects that are in motion relative to each other will always be in different frames of reference, no matter how close they are in distance. Is that right?

To make it perfectly clear: Yes but...

Objects that are close to each other -- say Earth and a rocket that has just blasted off from it, can be considered to be in the same FRW local frame, for the purposes of FRW coordinates.

But remember that an FRW local frame approximates to Minkowski (flat spacetime, Special Relativity (SR)) coordinates. So within that local space, we can apply standard SR to the relative motion between the Earth and the spaceship. But once the rocket travels outside of the Earth's local area (loosely defined), we can no longer apply SR alone to the rocket's motion. Then we can use FRW coordinates to calculate the relationship between Earth's local frame and the spaceship's comoving frame; and combine that with an SR effects that result from the rocket's peculiar motion within that distant FRW frame.

In FRW it is important to distinguish between comoving recession motion (where galaxies or other objects are at rest in their local Hubble flow) and peculiar motion (where galaxies or other objects have a locally-measured velocity with respect to their local Hubble flow. In reality, most galaxies have some peculiar motion which differs from the FRW-calculated Hubble recession velocity (H*D). The real universe is not a perfect toy model, and there are significant inhomogeneities in the matter distribution locally (such as galaxies), whose gravity induces local peculiar motions.

I agree that a wire could not be simultaneously at rest in my local frame as well as in the local frame of a distant galaxy, but that wasn't what I was trying to say.

For example, I can have long rope that is at rest wrt to me. I tie one end onto a car and the car sets off. For a period of time, part of the rope will be in motion wrt me and part of it will be at rest wrt me.

So the rope shares both frames for a time, no? But once the car is in steady motion then the rope shares the same reference frame as the car and a different reference frame to anyone else that is moving relative to the car, even if they are stood very near to the rope.

However if they then grabbed onto the rope, they would share the same frame as the car.

I see this to be the same as the wire passing galaxies.

Again, your description isn't meaningful unless you specify which coordinate system you are working with.

In Minkowski coordinates, if two objects have zero velocity relative to each other, then they can both be considered to be at rest in the same reference frame. That would be true no matter how long the rope is. In this case, the passing rope's velocity is always less than c.

In FRW coordinates, if the rope is long, then the car (let's say rocket) and the person grabbing the rope are NOT in the same FRW local frame, because they are far apart. Their respective comoving local frames have a built-in recession velocity relative to each other.

You should forget about the grabber and just focus on the rope itself. There is no way to deploy the rope such that it is being pulled by a rocket in a distant comoving frame and have its trailing end passing through a local frame that has a recession velocity greater than c (relative to the distant frame the rocket is in). It can't be done. The rope will break, for the reasons I described. Regardless whether the rope is spooled, or coiled, or whatever, such a scenario would require local motion in one local frame or another that exceeds c, which is not allowed by relativity. The rope must break before that happens. As the local speed of some part of the rope starts to approach c, the rope will experience infinite stress.

If the rope can't be deployed as required by your scenario, then it won't be there for anyone to grab!

 

[nutgeb]

The argument I'd have liked to make, but that doesn't quite work, is by analogy with a black hole. A black hole has an event horizon. The future light cone of any event inside the horizon lies inside the event horizon, not outside it, so it's not possible for an event inside to be the cause of something that happens outside. If you could hang a wire through the event horizon, this would be a counterexample, because you could send signals using vibrations in the wire, or use the wire to haul things out. We therefore conclude that no wire can be strong enough to withstand the spaghettifying forces created when we hang the wire down through the horizon and prevent it from falling. It's not surprising that there is an absolute relativistic limit on the tensile strength of materials. If not, then vibrations in materials could go faster than c, but matter is held together by electromagnetic forces, and electromagnetic disturbances propagate at c.

Agreed, if a stationary observer outside a BH event horizon dangles a rope below the event horizon, the rope will immediately break. The gravitational stress on the rope is infinite (unless the rope is being hurled downward at close to c) and no material can have infinite strength.

Unfortunately this doesn't carry through 100% to the case of cosmic expansion. Cosmological solutions can have horizons, but it's not correct to say that if point A and point B are expanding away from one another at >c, then B must be behind an event horizon as seen by A.

I sympathize with your attempt to draw the analogy, I've tried to think along the same lines. I note that in the BH case, the rope is stressed by gravity, while in the FRW case, the rope is stressed by acceleration overcoming inertia. The equivalence principle should allow some analogy between the two, just as it is possible to compare Rindler and BH event horizons. However, I agree that there is no event horizon in FRW at the Hubble Radius.

Part of the difficulty in constructing an analogy arises simply because in FRW coordinates the difference between one local frame and its neighbor is not a different acceleration rate (2nd derivative of distance) as with a BH; it is merely a different recession velocity (1st derivative of distance). You can't build an event horizon out of velocity differentials alone (other than the trivial case that a photon chasing another photon cannot gain ground on it, all other factors being equal). Inside a BH horizon there is no analogous situation where two radially separated observers have relative acceleration = 0 but relative velocity > c.

I suggest modeling the acceleration of the rope itself in Rindler coordinates. If this is possible, an event horizon should make its appearance. In this scenario the acceleration occurs not because the comoving recession velocities are accelerating, but because the rocket must continually accelerate (relative to earth/origin) in order to be able to continue passing successive galaxies with ever faster Hubble rates. I'm not sure if there is a problem superimposing Rindler motion coordinates over an FRW coordinate background. Whatever that means -- in other words, would the results consistently comply with the required behaviors of both coordinate systems (e.g. no coordinate velocity > c in any single reference frame)?

The whole notion of the relative velocity of cosmologically distant points isn't even well defined.

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.

For one thing, the fact that simultaneity is not well defined in relativity means that different observers will not necessarily agree on whether the rope is broken "right now" at some distant point.

As I said, if one sticks to FRW proper radial distance coordinates, the answer is clear. If the two observers are both at rest in their respective comoving local frames, then they share a common cosmological time; their clocks tick at the same rate; so there is no failure of simultaneity. But if one of the observers has a relativistic peculiar velocity within his FRW comoving frame, then SR is simply applied to that peculiar velocity within that one local frame, to determine how simultaneity differs between the two observers; the comoving recession velocity element adds nothing to the equation.

 

[bcrowell]

The gravitational stress on the rope is infinite (unless the rope is being hurled downward at close to c) and no material can have infinite strength.

I don't think it's infinite. It's finite, but greater than the finite limit that relativity places on the strength of materials.

Velocity is well defined as long as one sticks to a single coordinate system.

Sure. What I mean when I say relative velocity of cosmologically distant points is not well defined is that the velocity is only defined up to a choice of coordinates -- which is completely arbitrary.

 

[stevmg]

I thought that iterations of the velocity addition equation would allow the relativistic velocity with respect to, say, Earth to asymptotically approach c but never exceed it.

But, if the actual framework of spacetime is expanding and light travels in that "medium" so I can see how that would exceed c. In otherwords, in a 2D world on a balloon, say an object is moving at 0.99c across the surface of this balloon. Now, increase the radius of the balloon which in itself increases space on the surface of the balloon and one can see how the 0.99c could easily become 1.1 c or whatever.

Am I in the "ballpark?"

 

[nutgeb]

I thought that iterations of the velocity addition equation would allow the relativistic velocity with respect to, say, Earth to asymptotically approach c but never exceed it.

That depends on the coordinate system you use. In the Minkowski coordinates normally used for SR analysis, the relativistic velocity addition formula applies. But in FRW coordinates, it does not apply to comoving recession velocities. For example, Hubble velocities are simply added straight up.

Keep in mind that when the SR velocity addition formula is used, the velocities involved are not both proper velocities. At least one of them is 'merely' a coordinate velocity (i.e a velocity calculated by someone moving at a different relative velocity). If you take the two actual proper velocities (as measured in two separate inertial frames) and just add them together, it is perfectly possible to get a total proper velocity that exceeds c. But special relativity does not recognize such a result, because SR demands that a single preferred reference frame be used as the starting point to judge all velocities.

That is contrary to the case of FRW coordinates, where every comoving frame is an equally preferred frame, so one can (and must) simply add up individual proper velocities straight away to get a total proper velocity. Neither Minkowski/SR nor FRW is 'better' or 'more correct' than the other, they are just different ways of looking at the same problem. It just so happens that FRW proper distance coordinates are much more useful and natural for cosmological analysis than Minkowski/SR coordinates are. Minkowski coordinates cannot be used at vast cosmological distances in curved spacetimes, i.e. to model a universe that includes gravity. In that case one's other primary choice is to turn to Schwarzschild coordinates, but those do not respect the gravitational homogeneity that the universe is believed to have, so they provide only a fragmented and inhomogenous portrayal of individual parts of the universe.

The lesson to be learned here is that different coordinate systems can be applied to model the same physical environment, and they bring to bear very, very different mathematical and heuristic approaches. Nevertheless, they should predict the same observations.

But, if the actual framework of spacetime is expanding and light travels in that "medium" so I can see how that would exceed c. In otherwords, in a 2D world on a balloon, say an object is moving at 0.99c across the surface of this balloon. Now, increase the radius of the balloon which in itself increases space on the surface of the balloon and one can see how the 0.99c could easily become 1.1 c or whatever.

That is one explanation for the behavior of FRW coordinates: that empty space is expanding, thereby dragging the galaxies apart. (And that SR applies as a reasonable approximation to local spaces but not to the much vaster intervening space that is pushing the local spaces apart). It's by no means ruled out as a theory, but it has some problems. If you are aware of the tethered galaxy exercise, it demonstrates that expanding space does not push objects around like a flowing stream. The balloon analogy still works quite well but you have to interpret it carefully, in a way that is somewhat nonintuitive.

The other explanation for the behavior of FRW coordinates is that galaxies are actually moving through space (the kinematic model). The increasing separation of the galaxies could be thought of as creating new space (i.e. the added space is the effect not the cause), or the idea of empty space expanding can be ignored entirely (other than perhaps with respect to the action of Dark Energy). The universe expands simply because the galaxies retain pre-existing recessionary momentum left over from the Big Bang/inflation, subsequently decelerated by gravity and more recently accelerated by Dark Energy.

There is no way to tell through observations or math whether the 'expanding space' or the 'kinematic' explanation should be preferred. For example, they each calculate precisely the same cosmological redshift, but using different equations. The expanding space model was more widely embraced in the 80's and 90's, but in the past few years the kinematic model has gained a lot of support.

 

[nutgeb]

I don't think it's infinite. It's finite, but greater than the finite limit that relativity places on the strength of materials.

This is not the thread to discuss the point in detail, but, no, although the proper gravitational acceleration experienced by a freefalling observer is finite at the event horizon, for a hovering observer at the event horizon it is infinite. See http://www.mathpages.com/rr/s7-03/7-03.htm" . That's why it is impossible to hover at the event horizon by applying a finite acceleration force. I was corrected on this point myself along with an avalanche of argument in an earlier thread.

Sure. What I mean when I say relative velocity of cosmologically distant points is not well defined is that the velocity is only defined up to a choice of coordinates -- which is completely arbitrary.

Yes the choice of coordinates is arbitrary, but FRW proper radial distance coordinates happen to be almost uniquely natural and powerful for cosmological radial distance and velocity analysis. The coordinates used -- proper distance, proper time, and proper velocity -- are invariant and have unambiguous meaning.

 

[Dale]

Agreed, if a stationary observer outside a BH event horizon dangles a rope below the event horizon, the rope will immediately break. The gravitational stress on the rope is infinite (unless the rope is being hurled downward at close to c) and no material can have infinite strength.
...
I suggest modeling the acceleration of the rope itself in Rindler coordinates.

I don't think it's infinite. It's finite, but greater than the finite limit that relativity places on the strength of materials.

Here is my favorite page on the topic which includes a very detailed discussion of dangled ropes:
http://gregegan.customer.netspace.net.au/SCIENCE/Rindler/RindlerHorizon.html

 

[stevmg]

This is not the thread to discuss the point in detail, but, no, although the proper gravitational acceleration experienced by a freefalling observer is finite at the event horizon, for a hovering observer at the event horizon it is infinite. See http://www.mathpages.com/rr/s7-03/7-03.htm" . That's why it is impossible to hover at the event horizon by applying a finite acceleration force. I was corrected on this point myself along with an avalanche of argument in an earlier thread.

Yes the choice of coordinates is arbitrary, but FRW proper radial distance coordinates happen to be almost uniquely natural and powerful for cosmological radial distance and velocity analysis. The coordinates used -- proper distance, proper time, and proper velocity -- are invariant and have unambiguous meaning.

Here is my favorite page on the topic which includes a very detailed discussion of dangled ropes:
http://gregegan.customer.netspace.net.au/SCIENCE/Rindler/RindlerHorizon.html

That depends on the coordinate system you use. In the Minkowski coordinates normally used for SR analysis, the relativistic velocity addition formula applies. But in FRW coordinates, it does not apply to comoving recession velocities. For example, Hubble velocities are simply added straight up.

Keep in mind that when the SR velocity addition formula is used, the velocities involved are not both proper velocities. At least one of them is 'merely' a coordinate velocity (i.e a velocity calculated by someone moving at a different relative velocity). If you take the two actual proper velocities (as measured in two separate inertial frames) and just add them together, it is perfectly possible to get a total proper velocity that exceeds c. But special relativity does not recognize such a result, because SR demands that a single preferred reference frame be used as the starting point to judge all velocities.

That is contrary to the case of FRW coordinates, where every comoving frame is an equally preferred frame, so one can (and must) simply add up individual proper velocities straight away to get a total proper velocity. Neither Minkowski/SR nor FRW is 'better' or 'more correct' than the other, they are just different ways of looking at the same problem. It just so happens that FRW proper distance coordinates are much more useful and natural for cosmological analysis than Minkowski/SR coordinates are. Minkowski coordinates cannot be used at vast cosmological distances in curved spacetimes, i.e. to model a universe that includes gravity. In that case one's other primary choice is to turn to Schwarzschild coordinates, but those do not respect the gravitational homogeneity that the universe is believed to have, so they provide only a fragmented and inhomogenous portrayal of individual parts of the universe.

The lesson to be learned here is that different coordinate systems can be applied to model the same physical environment, and they bring to bear very, very different mathematical and heuristic approaches. Nevertheless, they should predict the same observations.

That is one explanation for the behavior of FRW coordinates: that empty space is expanding, thereby dragging the galaxies apart. (And that SR applies as a reasonable approximation to local spaces but not to the much vaster intervening space that is pushing the local spaces apart). It's by no means ruled out as a theory, but it has some problems. If you are aware of the tethered galaxy exercise, it demonstrates that expanding space does not push objects around like a flowing stream. The balloon analogy still works quite well but you have to interpret it carefully, in a way that is somewhat nonintuitive.

The other explanation for the behavior of FRW coordinates is that galaxies are actually moving through space (the kinematic model). The increasing separation of the galaxies could be thought of as creating new space (i.e. the added space is the effect not the cause), or the idea of empty space expanding can be ignored entirely (other than perhaps with respect to the action of Dark Energy). The universe expands simply because the galaxies retain pre-existing recessionary momentum left over from the Big Bang/inflation, subsequently decelerated by gravity and more recently accelerated by Dark Energy.

There is no way to tell through observations or math whether the 'expanding space' or the 'kinematic' explanation should be preferred. For example, they each calculate precisely the same cosmological redshift, but using different equations. The expanding space model was more widely embraced in the 80's and 90's, but in the past few years the kinematic model has gained a lot of support.

As DaleSpam can tell you I am very new to all of this, and FRLW, Hubble, et al are way out of my league - so, in that regard, I am not even in the same zip code as the "ballpark." It took me long enough to understand Minkowski or relativistic coordinates, proper time, etc.

We are going to leave this one alone for a while...

Hard enough to understand why x' = xcosh (\phi) - tsinh (\phi)
and
t' = -xsinh (\phi) +tcosh (\phi) where \phi = tanh^-1(\phi) = \beta = v/c


stevmg

 

[bcrowell]

Nutgeb, thanks for the correction about the infiniteness of the tension. I was wrong.

Yes the choice of coordinates is arbitrary, but FRW proper radial distance coordinates happen to be almost uniquely natural and powerful for cosmological radial distance and velocity analysis. The coordinates used -- proper distance, proper time, and proper velocity -- are invariant and have unambiguous meaning.

I don't think this is right. If something is invariant, that means that it's coordinate-independent. I don't see how coordinates can be invariant, since that would mean that coordinates were coordinate-independent.

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

The connection with the Bell spaceship paradox can be seen very clearly in Egan's discussion. "We can imagine a flotilla of spaceships, each remaining at a fixed value of s by accelerating at 1/s. In principle, these ships could be physically connected together by ladders, allowing passengers to move between them. Although each ship would have a different proper acceleration, the spacing between them would remain constant as far as each of them was concerned." Egan's spaceships avoid breaking the ropes because their accelerations are unequal. The unequal accelerations are sufficient to compensate for Lorentz contraction, so Lorentz contraction doesn't break the ropes.

Suppose observers aboard Egan's flotilla observe an ambient dust-cloud of test particles, all of which are at rest relative to one another in their own (inertial) frame. What the astronauts observe is in some ways similar to cosmological expansion. Particles near the back of the flotilla accelerate more rapidly (as judged in the flotilla's frame), particles near the front less rapidly. Therefore the flotilla sees the dust-cloud as expanding in a manner that is reminiscent of Hubble expansion. There are some ways in which it's not like a cosmological model, though: it appears nonisotropic in the flotilla's coordinates, and the motion of the ships is noninertial.

I think we can get at some of the interesting issues using the Milne model. The logic would be very similar to the logic of Egan's treatment of ropes in the Rindler metric, since in both cases we're just talking about Minkowski space with a change of coordinates. The difference is that, unlike the Rindler-metric observer, an observer in the Milne model sees everything as being isotropic, and the motion of the galaxies in the Milne model is inertial.

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. Let K be a coordinate system (t,r) in which the Milne universe is described by a finite, spherical cloud of test particles expanding into a surrounding vacuum. Let K' be the coordinate system (\tau,\rho), where \tau is proper time, and \rho is defined in the customary way, so that space is infinite, isotropic, and scaling linearly with time. We can have a chain that's straight and infinitely long according to K at a given time t. This is a description of the simultaneous positions of all the links in the chain. But an observer who prefers K' will disagree that this set of positions was taken simultaneously. According to K', the list of positions includes links that were very far away at some earlier time. "Hmph," says K', "that's old data. Those distant parts of the chain are probably broken by now."

I wonder if Egan's analysis can be easily extended to the Milne universe, which is static, unlike realistic cosmological models.

 

[Dale]

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

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.

 

[Altabeh]

Please excuse the silliness of this but...

As I understand it, the further away a galaxy is the faster it is moving away due to the expansion of the universe.

I think I read that the expansion is something like 77km/sec per 3.26 million light years.

Anyway, that means that there must be (or will be) some galaxies that are moving away from us faster than c.

So here's the silly question.

Imagine I was able to make (or keep adding to) an almost infinitely long wire, fly out to the nearest galaxy and attach one end of the wire to a planet, then fly back to earth. Then wait until that galaxy was moving away from Earth faster than c and grab hold of the wire and let it take me with it. I would be moving away from Earth faster than c.

If the direction was right, I could jump off at say Pluto and pass the latest football scores on to the locals, which would mean that they received a message from Earth faster than c.

Ignoring the obvious 'impracticalities', time factors, g-forces etc., what laws of physics would stop this from happening?

Actually this "story" is so idealistic as you're not taking into account many physical factors that would stop this from happening. For example, the wire must be something like a really thin string with an infinitely large elasticity so that if I just flipped one end of the wire, the other end would be "swinging" at least after L/c seconds where L is the length of the wire that attaches two distant galaxies together. This is not about electromagnetic waves but transverse mechanical waves (because the string or wire are not really massless) which automatically invalidates the example. The reason behind me telling that the string must have an infinite elasticity is that the other galaxy pulling the wire does apply a force to one end and this force must in turn generate a pulse along the wire as the frequency f of this pulse has to be satisfy, in the weakest possible state, f=c/\lambda rather than satisfying f=v/\lambda. Well this is only possible for a massless wire and if this was the case, the pulse couldn't apply a force to your body because it would no longer be a material. (The pulse is generally just an "agitated" part of the wire carrying mechanical energy which only imposes a force when being in a physical contact such as hitting a wall.)

AB

 

[bcrowell]

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.

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

 

[nutgeb]

I don't think this is right. If something is invariant, that means that it's coordinate-independent. I don't see how coordinates can be invariant, since that would mean that coordinates were coordinate-independent.

Well I think that's just casual terminology on my part. Technically, "proper time" is called the "timelike spacetime interval" and "proper distance" is the "spacelike spacetime interval." All freefalling observers will agree on the value of these quantities, regardless of their coordinate system, so they are considered to be invariants.

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.

 

[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 \geqc 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.