# Is this a way to move faster than c?

1. Aug 1, 2010

### 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?

Last edited: Aug 1, 2010
2. Aug 1, 2010

### 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.

3. Aug 1, 2010

### rede96

Hi causalset, thanks for the reply.

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

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?

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

4. Aug 1, 2010

### causalset

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.

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.

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]

5. Aug 1, 2010

### rede96

I think I get what you mean. I can have two people on the opposite side of a globe travelling 'south'. Knowing that they are on a globe of 360 degrees and travelling 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 travelling 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 travelling 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.

6. Aug 1, 2010

### causalset

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.

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.

7. Aug 1, 2010

### Staff: Mentor

Last edited: Aug 1, 2010
8. Aug 1, 2010

### bcrowell

Staff Emeritus
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.

9. Aug 1, 2010

### 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.

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?

10. Aug 1, 2010

### bcrowell

Staff Emeritus
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.

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,...).

11. Aug 1, 2010

### Bussani

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.

Okay, that makes sense.

12. Aug 2, 2010

### causalset

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 thats 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.

13. Aug 2, 2010

### Bussani

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.

14. Aug 2, 2010

### bcrowell

Staff Emeritus
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.

15. Aug 3, 2010

### 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 travelling 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, lets 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 travelling faster then c, but due to relativity, would only arrive at the same time as a light beam sent, thus no info has travelled faster than c.

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

16. Aug 3, 2010

### 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 occuring 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.

Last edited: Aug 4, 2010
17. Aug 3, 2010

### Bussani

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

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 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.

18. Aug 3, 2010

### nutgeb

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

19. Aug 4, 2010

### rede96

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

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!

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 travelling 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.

20. Aug 4, 2010

### nutgeb

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.
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.

Last edited: Aug 4, 2010