How Long Can a Wire Be? Laws of Physics and Wire Length Explained

[andrewkirk]

Yes that's right - it doesn't cancel out. Do you remember where you read that it does?

 

[rede96]

Yes that's right - it doesn't cancel out. Do you remember where you read that it does?

No sorry. It would have been on here or a link I followed from here. If I find it I'll post the link.

In any case, thanks very much for your help and to everyone else who posted.

 

[Cecil Tomlinson]

With your kind permission, I would like to replace your wire with Einstein's train. Now, of course, the infinitely long train could not accelerate because it would have infinite inertia, and no locomotive could pull it. But, replace every car with a locomotive. Now the infinitely long train has infinite thrust, and the same ratio of thrust to mass as a single locomotive. Can it accelerate? NO! Why? Any increase in velocity will shorten every locomotive, and somewhere down the line all remaining locomotives would have to exceed the speed of light for the train to stay connected. The train will suffer severe whiplash at some point. Does that make sense?

 

[rede96]

Any increase in velocity will shorten every locomotive, and somewhere down the line all remaining locomotives would have to exceed the speed of light for the train to stay connected. The train will suffer severe whiplash at some point. Does that make sense?

Hi and thanks for your post. But as I understand it velocity, even an increasing velocity (acceleration) doesn't have a physical effect on the length of objects. But your post did provoke a thought.

Imagine two space ships, each of 1 meter length, facing each other and separated by a distance of 2 meters. So there is a centre point that is 1 meter in front of each space ship. Like this: > | <

The rate that the space expands between them is very small. So if I waited a while until the space ships were say 1.1 meters away from the centre line, it would be quite easy for them to accelerate shortly and close the distance back to 1 meter.

Now Imagine behind each spaceship was another spaceship facing the same direction as the one in front like this: > > | < < so there is 1 meter between each spaceship and still 1 meter between the front space ships and the centre line.

If again I wait for the distance between the front space ships and the centre line to expand to 1.1 meters the distance between the end space ships and the spaceship in front would be slightly more than 1.1 meters as the space would have expanded slightly more between them. so they would have a little more work to do to get back to 1 meter distance.

So if we continue to add space ships to the ends and then wait for the front two space ships to move away from the centre line by 1.1 meters, at some point you would think the distance the end spaceship has to make up to get back to being 1 meter away from the spaceship in front would require a speed greater than the speed of light.

Or would it?!

The space that expands between any two space ships locally ( e.g next to each other) will always be very small and so if each space ships was constantly accelerating at the same rate as the local expansion, they would never grow apart. So in principle it is possible to have an infinitely long line of space ships separated by a 1 meter gap.

If this is not true, then which spaceship in the infinitely long line is not capable of keeping a 1 meter gap with the spaceship ahead of it?

 

[andrewkirk]

infinitely long train could not accelerate because ... any increase in velocity will shorten every locomotive, and somewhere down the line all remaining locomotives would have to exceed the speed of light for the train to stay connected. The train will suffer severe whiplash at some point. Does that make sense?

I don't think that's right. Here's why.

Say the distance from the front of one car to the front of the next is 30m while it is 'at rest'. So in a 3million km length of train we have 100 million cars. If the train accelerates to speed v the length of each car will be 30m times the Lorenz factor $\sqrt{1-\frac{v^2}{c^2}}$. Say that is 29m, a gap of 1m per car that needs to be 'closed'. I think you are thinking that, across the 3million km length of train that is a gap of 100,000km that needs to be closed.

But that gap doesn't need to be closed because the length of the 100 million cars, that was 3million km at rest is also shortened by the Lorentz factor, so that it is only 2.9million km, which is exactly the length of 100million cars. There is no gap.

If you don't believe that verbal explanation, just write out the equations for the x coordinates (relative to some arbitrary origin point) of the front and the back of the $n$th car (with $n$ ranging over the entire range of positive and negative integers) in terms of $t$. You will find that the coordinate of the front of car $n$ always remains the same as that of the coordinate of the back of car $n+1$, ignoring couplings.

 

[andrewkirk]

The space that expands between any two space ships locally ( e.g next to each other) will always be very small and so if each space ships was constantly accelerating at the same rate as the local expansion, they would never grow apart. So in principle it is possible to have an infinitely long line of space ships separated by a 1 meter gap.

If this is not true, then which spaceship in the infinitely long line is not capable of keeping a 1 meter gap with the spaceship ahead of it?

It is not true because the further away from the centre point a ship is, the faster it has to move relative to its surroundings, in order to maintain a constant distance to the centre point. Far enough away from the centre point, that will require a spacelike velocity vector (ie 'superluminal' speed), which is impossible.

The first ship on the right-hand side (the two sides are symmetrical) that is not capable of maintaining the gap is:

- from a practical point of view, the first ones whose engines cannot produce enough thrust to produce the required acceleration, which will be proportional to the distance from the centrepoint.
- if the potential thrust is unlimited it will be the first one that is destroyed by the G forces of the required acceleration
- if the ship is indestructible (infinitely stiff and infinitely strong) it will be the first one that is a distance far enough away from the centre point that the surrounding stars are moving away from the centre point superluminally. That may be the distance called the Hubble horizon, although I may be mixing up my terms there.

 

[rede96]

But that gap doesn't need to be closed because the length of the 100 million cars, that was 3million km at rest is also shortened by the Lorentz factor, so that it is only 2.9million km, which is exactly the length of 100million cars. There is no gap.

Am I right in thinking that the Lorentz factor is only for transforming between different frames of reference? E.g. there is no physical 'shorting' of any physical object to do it's speed through space time, or in other words the atoms that make up an object don't move closer together or 'shrink' due it's speed or acceleration?

It is not true because the further away from the centre point a ship is, the faster it has to move relative to its surroundings, in order to maintain a constant distance to the centre point. Far enough away from the centre point, that will require a spacelike velocity vector (ie 'superluminal' speed), which is impossible.

Thanks for the reply. I did realize after my post that I was wrong. The way I imagine it now is that if I just assume a maximum thrust for my space ships, which is the same for every space ship, then as they are all in a state of constant acceleration (to keep the same distance from the centre point) and as the further the space ships are away from the centre point, the more they need to accelerate to keep up, then there must be a critical point where the one of the space ships can't accelerate fast enough to keep up with the one in front. So that distance starts to grow. But for all the other remaining space ships, they will be able to maintain the 1 meter distance.

So the 'length' of the chain of ships left is for me analogous to the maximum length of a wire. Of course this depends on the strength of the wire, but there still will be a point where the ends of the wire have to accelerate faster than c in order to maintain the proper length of the wire, and this is the maximum length of any wire.

if the ship is indestructible (infinitely stiff and infinitely strong) it will be the first one that is a distance far enough away from the centre point that the surrounding stars are moving away from the centre point superluminally. That may be the distance called the Hubble horizon, although I may be mixing up my terms there.

There is one thing I read about light from stars that were outside our Hubble horizon would still eventually reach us as our Hubble horizon is growing. So I assume that the space ships that couldn't keep up would eventually be able to catch up with the rest of the ships some time in the future? Assuming they never ran out of fuel of course.

 

[andrewkirk]

Am I right in thinking that the Lorentz factor is only for transforming between different frames of reference? E.g. there is no physical 'shorting' of any physical object to do it's speed through space time, or in other words the atoms that make up an object don't move closer together or 'shrink' due it's speed or acceleration?

Yes, if two observers are traveling at 0.6c relative to one another, each observes a 20% shrinkage in the dimensions of the other, including the distance between the atoms making up the other, but no shrinkage in their own dimensions or the distance between their own atoms.

There is one thing I read about light from stars that were outside our Hubble horizon would still eventually reach us as our Hubble horizon is growing.

It turns out I did mix up my terms. The Hubble Horizon is not the relevant measure. I think it's more likely to be the Event Horizon. Light from outside the Event Horizon will never reach us. This page describes the different cosmological horizons and their differences.

 

[rede96]

This page describes the different cosmological horizons and their differences.

Ah ok, great. Thank you again.

 

[Rishi Gangadhar]

According to Einstein's theory, it would require infinite amount of energy to accelerate something to speeds greater than that of light. So, beyond a point, the thread would have to break, or the the other galaxy would slow down(if the thread was strong enough). If the second case somehow arises, then the velocity of the other galaxy with respect to us would never be greater than the speed of light.