Lorentz-contraction

[Winterlong]

A ten meter ship travels as close to light speed as to have all the universe compressed to 1 m length, in the direction of the movement. As the ship remains stopped for the pilot, its length remains ten meters for him. If the universe is 1 m in length, where is the ship flying?

[Strange]

5 hours ago, Winterlong said:

If the universe is 1 m in length, where is the ship flying?

The question doesn't make much sense. What does "the universe is 1 m in length" mean?

[Markus Hanke]

5 hours ago, Winterlong said:

A ten meter ship travels as close to light speed as to have all the universe compressed to 1 m length

This is not as straightforward as you might think, because the universe does not have a “length” since it doesn’t have any boundaries; it’s also not an inertial frame. At most it can have the topology of a closed manifold, in which case, if one travels long enough in one direction, one would eventually return to one’s starting position. The problem with this is that, if this is the case, then the geometry of that manifold cannot be Minkowskian, so you cannot naively apply Lorentz contraction - you’d have to find a solution to the Einstein equations which combine relativistic motion with FLRW spacetime (which probably exists, though I haven’t seen it). Note that this is necessary because you’d have the manifold curve back onto itself over a total circumference of 1m, so curvature is definitely not negligible here.

It is conceivably still possible to make the total distance travelled appear to be 1m, though calculating how the ship needs to move in order to get that effect is quite nontrivial. Even if it is possible, there still wouldn’t be a paradox, because of relativity of simultaneity (which is also non-trivial here due to the background manifold not being Minkowski). 

This whole thing is conceptually similar (albeit more complicated due to the above considerations) to the well-known ladder paradox.

[Endy0816]

Have to factor in expansion as well.

For distant destinations, the distance is growing at a rate faster than the ratio c.

 

[Markus Hanke]

1 hour ago, Endy0816 said:

Have to factor in expansion as well.

For distant destinations, the distance is growing at a rate faster than the ratio c.

 

Indeed! Good point
This essentially renders the entire scenario unphysical, because you can never make the total distance =1m, regardless of how close to c you get.

Edited August 8, 2020 by Markus Hanke

[Endy0816]

3 hours ago, Markus Hanke said:

Indeed! Good point
This essentially renders the entire scenario unphysical, because you can never make the total distance =1m, regardless of how close to c you get.

Yeah. It's interesting that the Universe could have a finite volume but still be impossible to cross.

Edited August 8, 2020 by Endy0816

[studiot]

16 hours ago, Winterlong said:

A ten meter ship travels as close to light speed as to have all the universe compressed to 1 m length, in the direction of the movement. As the ship remains stopped for the pilot, its length remains ten meters for him. If the universe is 1 m in length, where is the ship flying?

I think the OP scenario is much more simplistic than replies so far.

Winterlong, the fly in your ointment argument is that the pilot does not see a contracted universe, that is the view of some observer travelling relative to him.

As the pilot masures things, not only does the ship have its normal 10m length, but the universe has its 'normal' length, whatever that is (we don't have a good number for this).

[Strange]

Just now, studiot said:

Winterlong, the fly in your ointment argument is that the pilot does not see a contracted universe, that is the view of some observer travelling relative to him.

Someone moving relative to the pilot, will see the pilot and her ship contracted. The pilot will see that other person contracted.

This must mean that the pilot will, for example, the distance between stars contracted relative to someone who is stationary relative to those stars? (But maybe I have misunderstood the point you are making?)

[studiot]

1 hour ago, Strange said:

Someone moving relative to the pilot, will see the pilot and her ship contracted. The pilot will see that other person contracted.

This must mean that the pilot will, for example, the distance between stars contracted relative to someone who is stationary relative to those stars? (But maybe I have misunderstood the point you are making?)

Yes I think you have.

And someone who is stationary relative to a different pair of stars will come ro a different measurement.
That is relativity.

Where do you think this absolute space is that the ship is travelling so fast that all of it is 'compressed' to 1m ?

What exactly is the ship travelling so fast relative to that his gamma factor will be that large?

It is like the Muon falling to Earth.

 

[Strange]

1 hour ago, studiot said:

Where do you think this absolute space is that the ship is travelling so fast that all of it is 'compressed' to 1m ?

It would be a relative measurement. I'm not sure why you said it wouldn't happen.

[Winterlong]

Quote

Have to factor in expansion as well.

For distant destinations, the distance is growing at a rate faster than the ratio c.

Indeed, that is true for any direction due to the fact that the border of the observable universe is receding at c in any direction. No matter how fast the ship travels, it won't catch it. Actually, the situation for the ship won't be special. The border still will recede at c for it, despite of its speed. The mental experiment is an impossible one 

This solves the paradox, but at the cost of making the recession of the border of the universe at c a necessity, rather than something that could be different

An universe in contraction, static or in a expansion slower than the current one, would still be sensitive to the paradox, hence, not possible. This explanation makes the recession of the border of the universe at c a requirement if the relativity to be true, something that could not be otherwise  

[studiot]

1 hour ago, Strange said:

It would be a relative measurement. I'm not sure why you said it wouldn't happen.

 

2 hours ago, studiot said:

What exactly is the ship travelling so fast relative to that his gamma factor will be that large?

 

So I will try to answer my own question since you didn't.

Google tells us that the diameter of our visible universe is 8.8 x 10²⁶ metres.

Suppose this ship flew left to right across our visible universe, what would its relative velocity be to us to compress the diameter of our universe to 1m ?
IOW at what speed would it bypass us ?

Applying the Lorenz contraction

[math]\sqrt {\left( {1 - \frac{{{v^2}}}{{{c^2}}}} \right)} \left( {8.8*{{10}^{26}}} \right) = 1[/math]

Square both sides

[math]\left( {1 - \frac{{{v^2}}}{{{c^2}}}} \right)\left( {77.4*{{10}^{52}}} \right) = 1[/math]

Rearrange

[math]\left( {1 - \frac{{{v^2}}}{{{c^2}}}} \right) = \left( {\frac{1}{{77.4}}*{{10}^{ - 52}}} \right) = 1.3*{10^{ - 54}}[/math]

Rearrange again

[math]{v^2} = {c^2}\left( {1 - 1.3*{{10}^{ - 54}}} \right)[/math]

Take square root both sides

[math]v = \left( {\sqrt {1 - 1.3*{{10}^{ - 54}}} } \right)c[/math]

Wolfram alpha cannot give me this square root the result is so close to c

 

 

Edited August 8, 2020 by studiot

[Winterlong]

Quote

15 hours ago, Markus Hanke said:

This is not as straightforward as you might think...

 

Right, but keep it simple. Even discussing curvatures, the resulting ship encroaching itself ten times is somehow awkward. As the speed remains constant, the restricted relativity should be enough for our discussion 

Edited August 8, 2020 by Strange
Fix quote

[Strange]

37 minutes ago, studiot said:

So I will try to answer my own question since you didn't.

I answered it: it is relative to another observer.

Your calculations do not answer the question you asked ("What exactly is the ship travelling so fast relative to") although, in passing you say "its relative velocity be to us". So it sounds like we agree. So I don't really know why you denied this initially.

22 minutes ago, Winterlong said:

Right, but keep it simple. Even discussing curvatures, the resulting ship encroaching itself ten times is somehow awkward. As the speed remains constant, the restricted relativity should be enough for our discussion 

It still isn't really clear what you are asking, but it looks like studiot has got the closest trying to calculate the required speed. 

Note that even if you compressed the visible universe down to 1m (a factor of about 10²⁷) the whole universe is many, many times larger than this. So it is not as if the spaceship would "stick out" of the universe either end. But as Markus points out, it is a completely unphysical scenario.

So can you explain a bit more clearly what (and maybe why) you are asking?

[Markus Hanke]

7 hours ago, Winterlong said:

This solves the paradox, but at the cost of making the recession of the border of the universe at c a necessity, rather than something that could be different

I think you need to be careful not to confuse the cosmological horizon (i.e. the limit to how far out we can observe) with the actual size of the universe. These are not the same things. The physical universe itself does not have any borders.

7 hours ago, Winterlong said:

Even discussing curvatures, the resulting ship encroaching itself ten times is somehow awkward.

But this is not what physically happens - refer again to relativity of simultaneity. I would recommend a read of the link I gave earlier on the ladder paradox (if you haven’t already), it is conceptually quite similar to this scenario.

But whichever way you look at it, there will of course never be a physical paradox, since you can’t construct those within the axioms of special relativity. 

[studiot]

8 hours ago, Strange said:

I answered it: it is relative to another observer.

Your calculations do not answer the question you asked ("What exactly is the ship travelling so fast relative to") although, in passing you say "its relative velocity be to us". So it sounds like we agree. So I don't really know why you denied this initially.

It still isn't really clear what you are asking, but it looks like studiot has got the closest trying to calculate the required speed. 

Note that even if you compressed the visible universe down to 1m (a factor of about 10²⁷) the whole universe is many, many times larger than this. So it is not as if the spaceship would "stick out" of the universe either end. But as Markus points out, it is a completely unphysical scenario.

So can you explain a bit more clearly what (and maybe why) you are asking?

 

You are getting closer to understanding my thinking.

 

1) the OP assumes an absolute space as is simply says

On 8/8/2020 at 3:08 AM, Winterlong said:

A ten meter ship travels as close to light speed as to have all the universe compressed to 1 m length, in the direction of the movement

 

As is stands this makes no sense since the important velocity should be relative to something else, in the same universe as the ship, as you have correctly noted.

If the ship was isolated in its own universe the statement "the ship has velocity v" has no meaning at all.

 

I asked two questions, one one which you addressed in a Barnum sort of way and the other you have ignored twice, although I repeated it in my first reply to you.

Here is is again.

11 hours ago, studiot said:

What exactly is the ship travelling so fast relative to that his gamma factor will be that large?

The quote clearly shows the question mark at the end of the sentence.

 

So I picked out a part of the universe for more detailed examination, and as you again correctly point out that is only part of the known universe.

We don't know how 'big' the whole universe is and probably never will.

So you are right to encourage the OP to clarify the question, to one that can be answered.

 

[Strange]

10 hours ago, Winterlong said:

This solves the paradox, but at the cost of making the recession of the border of the universe at c a necessity, rather than something that could be different

The "border" (horizon) of the observable universe recedes at more than c (about 3c, if I remember correctly). Which highlights, again, you can't use SR in a situation where GR is required.

1 hour ago, studiot said:

I asked two questions, one one which you addressed in a Barnum sort of way and the other you have ignored twice, although I repeated it in my first reply to you.

Here is is again.

12 hours ago, studiot said:

What exactly is the ship travelling so fast relative to that his gamma factor will be that large?

The quote clearly shows the question mark at the end of the sentence.

I just realised I may have misinterpreted that oddly worded sentence. It took it to mean "What exactly is the ship travelling so fast relative to" but maybe you mean something else?

Anyway, it just occurred to me that the perfect answer may be: a neutrino: they travel at speeds indistinguishable (by current measurements) from c.

 

[Winterlong]

8 hours ago, Markus Hanke said:

I think you need to be careful not to confuse the cosmological horizon (i.e. the limit to how far out we can observe) with the actual size of the universe. These are not the same things. The physical universe itself does not have any borders.

Right, but even so, the universe has a length;  the distance to be travelled to get back to the starting point. This length being small or big doesn't matter, that will be a matter of how close to c the ship has to travel to reduce such length to 1 m

8 hours ago, Markus Hanke said:

But this is not what physically happens - refer again to relativity of simultaneity. I would recommend a read of the link I gave earlier on the ladder paradox (if you haven’t already), it is conceptually quite similar to this scenario.

But whichever way you look at it, there will of course never be a physical paradox, since you can’t construct those within the axioms of special relativity. 

I see a paradox, and I don't see how the simultaneity can solve it. For the sake of simplicity, the pilot does studiot's calculations (assuming that he knows the true length of the universe) and concludes that the length of the ship is bigger than the length of the universe, without even opening the windows of the ship to look outside. How does the simultaneity fix it?

The expansion of the universe complicates the scenario (actually, solves the paradox) But to get to the point, imagine a static universe. How is the special relativity not applicable? The SR only requires absence of acceleration, and should be applicable to any amount of space, including the total length of the universe

[swansont]

9 minutes ago, Winterlong said:

Right, but even so, the universe has a length;  the distance to be travelled to get back to the starting point. This length being small or big doesn't matter, that will be a matter of how close to c the ship has to travel to reduce such length to 1 m

Does it? Can you point to science that confirms this?

 

[Winterlong]

About the reference system:

In order to keep it simple, say that we consider an static universe. There could be movement of galaxies or starts or ships in this universe but not general expasion resulting in ligh-speed receding objects. The speed of the ship is, say, relative to Earth. It doesn't really matter in this scenario 

This universe is not the real one,  but neither is considered to be impossible in Relativity

The expansion makes things so much more interesting... but maybe deserves a different discussion

1 minute ago, swansont said:

Does it? Can you point to science that confirms this?

 

I cannot. The lengh defined as "the distance to be travelled to get back to the starting point" looks to me something not prohibited by relativity and somehow commonly accepted, but I don´t know it to be real or otherwise   

Still, this is beyond my point. The mental experiment I want to discuss applies equally to an universe without expansion, something seemingly not impossible within relativity principles 

[studiot]

11 minutes ago, Winterlong said:

About the reference system:

In order to keep it simple, say that we consider an static universe. There could be movement of galaxies or starts or ships in this universe but not general expasion resulting in ligh-speed receding objects. The speed of the ship is, say, relative to Earth. It doesn't really matter in this scenario 

This universe is not the real one,  but neither is considered to be impossible in Relativity

The expansion makes things so much more interesting... but maybe deserves a different discussion

I cannot. The lengh defined as "the distance to be travelled to get back to the starting point" looks to me something not prohibited by relativity and somehow commonly accepted, but I don´t know it to be real or otherwise   

Still, this is beyond my point. The mental experiment I want to discuss applies equally to an universe without expansion, something seemingly not impossible within relativity principles 

So you have read my comments, even though you did not respond to them.

I think you are mixing up referential systems.

You are asking from the point of view of the pilot so consider what he sees.

Although you have not answered my question as to what is his velocity in an otherwise isolated universe let us further allow one referential point as a starting point.

The pilot sees the rest of the universe (the reference starting point) receeding away from him to the left at this incredible speed.

In his frame he is stationary.

If the manifold is finite but closed he then sees the reference point approaching at incredible speed from the right a very short time later.

If (as i am inclined towards) the universe is infinite of course he never sees the reference point return and the maths has a division by zero error.

So as far as the pilot and his spaceship are concerned the rest of the universe is smaller than his ship and makes a circuit around it.

So what ? Where is the paradox ?

By his measurement the total size of the universe must be 10 m plus 1 m.

[Strange]

56 minutes ago, Winterlong said:

The mental experiment I want to discuss applies equally to an universe without expansion, something seemingly not impossible within relativity principles 

The experiment you have described so far is non-physical (so you could pretty much make up any answer you like). You seem to think you have created a paradox. But if you invent an unrealistic thought experiment, then it isn't surprising that you can make it inconsistent.

Is it possible to frame your question in a way that applies in the real world? If not, that is because the paradox you are imagining does not exist in the real world.

 

[md65536]

2 hours ago, Winterlong said:

I see a paradox, and I don't see how the simultaneity can solve it. For the sake of simplicity, the pilot does studiot's calculations (assuming that he knows the true length of the universe) and concludes that the length of the ship is bigger than the length of the universe, without even opening the windows of the ship to look outside.

There's no paradox. You're talking about a finite spacetime that seems to be flat. Or basically an object with a proper length of over ten billion light years. You're calling that "the universe", which is fine as long as you avoid attaching meaning to that label, like that everything else you specify must be within that.

Not everything in the (flat, toy) universe gets length contracted. Only the stuff that is moving relative to you does. The pilot isn't moving relative to the ship. If the ship is part of the universe, the universe won't be contracted to 1m, only the stuff moving relative to it. If the ship's not part of the universe, the universe can be like a 1m-thick wall traveling past the ship at near c.

If you want to talk about the ship being at rest inside the flat universe, and then accelerating "instantly" to near c, then simultaneity is important if the universe is not static. It sounds like the ship is implicitly Born rigid, and clocks on parts of it would become out of sync with each other (by billions of years??). I think you would see the far edge of the approaching 'wall' appearing to age over twenty billion years in the nano seconds it takes to pass you, due to the relativity of simultaneity and relativistic Doppler shift.

Actually, that idea's more complicated than I thought. Say the ship starts in the middle of a toy universe, and instantly accelerates to near c. Ignoring simultaneity, you might conclude that the universe contracts to a wall in the middle of the ship, with the ship sticking out both ends. But that's impossible because the back of the ship never travels backward. No part of the ship ever enters the "back" half of the universe. But with relativity of simultaneity, the different parts of the (Born rigid) ship travel through the front half of universe at different times... eek is that right? I think a pilot in the middle of the ship could consistently conclude, "I'm in the middle of the "universe" which is 1m wide and is smaller than my ship would be at rest (which it currently is not, it doesn't share a single inertial frame), but the back of my ship has already passed through the front edge of the universe has not yet reached the same speed as me and the universe is not yet contracted for them."??? That's confusing, I doubt I got it right. However, relativity of simultaneity does resolve this part of the paradox if you do it right.

 

22 hours ago, studiot said:

Winterlong, the fly in your ointment argument is that the pilot does not see a contracted universe, that is the view of some observer travelling relative to him.

As the pilot masures things, not only does the ship have its normal 10m length, but the universe has its 'normal' length, whatever that is (we don't have a good number for this).

It sounds like you're referring to the spacetime as 'the universe' and others are referring to all the moving stuff in it as the universe? If all the stuff was moving, the pilot would measure it as length-contracted.

Edited August 9, 2020 by md65536

[studiot]

1 hour ago, md65536 said:

23 hours ago, studiot said:

Winterlong, the fly in your ointment argument is that the pilot does not see a contracted universe, that is the view of some observer travelling relative to him.

As the pilot masures things, not only does the ship have its normal 10m length, but the universe has its 'normal' length, whatever that is (we don't have a good number for this).

It sounds like you're referring to the spacetime as 'the universe' and others are referring to all the moving stuff in it as the universe? If all the stuff was moving, the pilot would measure it as length-contracted.

Thank you, yes my text was unclear small wonder Strange was puzzled too. +1

Hopefully I have corrected this in my immediately previous post.

Edited August 9, 2020 by studiot

[md65536]

1 hour ago, md65536 said:

I think a pilot in the middle of the ship could consistently conclude, "I'm in the middle of the "universe" which is 1m wide and is smaller than my ship would be at rest (which it currently is not, it doesn't share a single inertial frame), but the back of my ship has already passed through the front edge of the universe has not yet reached the same speed as me and the universe is not yet contracted for them."??? That's confusing, I doubt I got it right. However, relativity of simultaneity does resolve this part of the paradox if you do it right.

I got this wrong. There is a paradox if the ship starts in the middle of the simplified universe and accelerates, ending up sticking out both ends. It's resolved by Born rigidity https://en.wikipedia.org/wiki/Born_rigidity : 

Quote

Already Born (1909) pointed out that a rigid body in translational motion has a maximal spatial extension depending on its acceleration, given by the relation b < c^2 / R, where b is the proper acceleration and R is the radius of a sphere in which the body is located, thus the higher the proper acceleration, the smaller the maximal extension of the rigid body.

If the ship started "inside" the universe and accelerated quickly enough that its 10m rest length would stick out both ends, that ship could not maintain a 10m length during that acceleration, and it would not stick out both ends. If the ship starts outside the simplified universe and is already inertial and 10m before the back of the ship enters the 1 m contracted universe, it could stick out both ends of the universe.

Edited August 9, 2020 by md65536