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

[andrewkirk]

if it were just but if it were just hanging in space on both ends it would not break. Expansion would not affect it because it would be internally connected by forces that are MUCH stronger than dark energy.

I think it can be shown that there is a length beyond which it would break, without having to resort to reductio ad absurdam, with its accompanying problems, per my previous post.

Let F be the breaking strain of the wire, and the wire's mass be w kg/m. The impact of the cosmological constant is that there is some distance D such that l>D\Rightarrow \frac{d^2{l}}{dt^2}>\frac{F}{w} where l is the distance between two free-falling bodies.
Say the wire is longer than 2(D+1) metres and consider the one-metre length at either end. Each is being accelerated by \Lambda away from the centre of the wire with acceleration greater than \frac{F}{w} and since it has mass w that imparts a force of F outwards along the wire. These two opposite forces of F are sufficient to break the wire.

(I think I made the wire twice as long as it needs to be to break, but never mind.)

The length D is enormously more than it would need to be to break the wire if it were attached to a planet at either end, because the force from 'dark energy' is proportional to the mass at either end of the wire. However, there is some length at which an unattached straight wire would break.

Ah: I see that Nugatory has already made this point a few posts earlier. I had only read the first page of this thread.

 

[rede96]

The length D is enormously more than it would need to be to break the wire if it were attached to a planet at either end, because the force from 'dark energy' is proportional to the mass at either end of the wire. However, there is some length at which an unattached straight wire would break.

You'll have to excuse my ignorance, but what I am really struggling to understand is just what forces are interacting with the wire that would make the ends accelerate to a point where it creates enough stress for it to break.

As I understand dark energy, it doesn't directly interact with matter. I assume as the effects of dark energy are in all directions they tend to cancel out. So it is just a part of space that is persistent everywhere and it is that persistence which leads to the expansion.

So how I am understanding the relationship between gravity, which is also present everywhere,(EDIT and also part of space) and expansion is that as space expands, it requires work for objects to 'move' back to the distance they were at before the space between them started to expand. Gravity can provide that work. But there comes a point where gravity has tailed off to the point where it can no longer provide the work needed to maintain the original distance. So I don't see gravity as a force that keeps things together, more the it provides the work to bring things back together to compensate for expansion.

However for systems that are bound in some way as solid objects, expansion or dark energy has no effect on them, it produces no stress or forces on these systems. And as such they need to do no work to keep together.

So I am struggling to understand just where the stress / forces are being produced that would cause an untethered wire of any length to break. As I understand it, there is nothing to suggest that expansion would accelerate the ends of the wire.

So where does that logic breakdown?

 

[andrewkirk]

(1)As I understand dark energy, it doesn't directly interact with matter.
...
(2) So how I am understanding the relationship between gravity, which is also present everywhere,(EDIT and also part of space) and expansion is that as space expands
...
(3) However for systems that are bound in some way as solid objects, expansion or dark energy has no effect on them, it produces no stress or forces on these systems. And as such they need to do no work to keep together.

The logic breaks down where you are trying to think of 'dark energy' as a phenomenon distinct from 'gravity', rather than just being part of the same phenomenon, which is Einstein's field equation G+\Lambda g=8\pi T. 'Dark Energy' is just a colourful name chosen for the \Lambda g part.

That equation determines a geodesic for the one metre section S at the end of the wire, which is the path it would follow if it were not attached to the rest of the wire and subject to no non-gravitational forces. The geodesic of S is accelerating away from the opposite end of the wire at rate \frac{F}{w}. Let P be the tip of S that is the very end of the wire. To make P deviate from its geodesic requires applying a non-gravitational force to S. The point in space that is coincident with P at a fixed time t0 and thereafter maintains a constant distance from the other end of the wire is accelerating away from P with acceleration \frac{F}{w}. Hence, to make P follow that point - ie to accelerate rapidly away from its geodesic - requires the rest of the wire to pull S towards it with a force of F, which will break the wire.

Relating this to your questions, we get the following answers:
(1) It does interact with matter, as specified in the Einstein equation.
(2) The relationship is that it is gravity - construed as the phenomenon described by Einstein's equation - that causes the universe to expand. You can think of the dark energy as a sort of 'negative gravity' if you like, although it's not a strict negative because it's not related to mass-energy in the same way that the rest of the gravitational equation is.
(3) Dark energy doesn't have no effect on bound systems. It just has an effect that is proportional to their size, and the constant of proportionality is so tiny that its effect is generally immeasurably small. It's not zero though. The particles in your body sit an infinitesimally small distance further away from each other than they would if there were no dark energy, as an equilibrium is reached between the very strong electrostatic forces holding them together and the incredibly weak dark energy (pseudo-)forces pushing them apart. It is only when we start to consider things on an inter-galactic scale that dark energy becomes significant enough to take into account, and that's what is happening in this thought experiment of the wire.

 

[RandyD123]

I'm looking at it this way. Let's suppose you can stop time just for a moment. You in one galaxy and a friend in another. You have both found a way to meet in the middle and tie your wires together in an unbreakable knot. And you both have an infinite amount of wire on a coil.

Now start time again. Since both of you are now moving away from each other FTL from your own reference, what happens and why?

 

[phinds]

I'm looking at it this way. Let's suppose you can stop time just for a moment. You in one galaxy and a friend in another. You have both found a way to meet in the middle and tie your wires together in an unbreakable knot. And you both have an infinite amount of wire on a coil.

Now start time again. Since both of you are now moving away from each other FTL from your own reference, what happens and why?

The wire breaks because of expansion. By the way, you can't HAVE "an infinite amount of wire"

 

[RandyD123]

Thanks for pointing that out "phinds", it was very intuitive of you, however, you can't stop time either. In my scenario you can do both. It's called suspending certain facts to carry out a thought experiment.

And thanks for your simple answer "the wire breaks because of expansion"...which makes very little sense, because in most explanations of expansion, the balloon analogy is used. I'm thinking my wire will just uncoil some more to make room for expansion.

 

[jerromyjon]

I'm thinking my wire will just uncoil some more to make room for expansion.

When you and your friend are separated by about 30 billion lightyears distance, how can you justify each of you and your friend's spools releasing wire at a velocity >c relative to the spool?

 

[RandyD123]

When you and your friend are separated by about 30 billion lightyears distance, how can you justify each of you and your friend's spools releasing wire at a velocity >c relative to the spool?

I guess I can't. So can only light move faster than light? Or is it "space/time" moving faster than light? Or is it both?

 

[jerromyjon]

So can only light move faster than light?

Not in mainstream accepted physics. Nothing moves faster than light from any point A to any point B. "Spooky action at a distance" Is not very spooky and no one can scientifically prove there is an action. Neutrinos are still questionable.

Or is it "space/time" moving faster than light?

To reach the case of >300000000m/s (8 zeroes) stretched out over 15 billion light years = 1.419 × 10²⁶ meters (300000000m/14190000000000000000000000m)=2.1141649 × 10^-17m/s. Show me when you see that meter grow that much or more if it is accelerating or prove it isn't locally, only galactically.

Or is it both?

You tell me, it's your universe.

 

[RandyD123]

Not in mainstream accepted physics. Nothing moves faster than light from any point A to any point B. "Spooky action at a distance" Is not very spooky and no one can scientifically prove there is an action. Neutrinos are still questionable.

To reach the case of >300000000m/s (8 zeroes) stretched out over 15 billion light years = 1.419 × 10²⁶ meters (300000000m/14190000000000000000000000m)=2.1141649 × 10^-17m/s. Show me when you see that meter grow that much or more if it is accelerating or prove it isn't locally, only galactically.

You tell me, it's your universe.

hahaha...my head just broke, my universe just broke and my wire just broke!

 

[jerromyjon]

Neutrinos are still questionable.

And all you had to do... was make it out of these.

 

[rede96]

The logic breaks down where you are trying to think of 'dark energy' as a phenomenon distinct from 'gravity', rather than just being part of the same phenomenon, which is Einstein's field equation G+Λg=8πT G+\Lambda g=8\pi T. 'Dark Energy' is just a colourful name chosen for the Λg \Lambda g part.

I probably need to read up a bit more on this, but in essence I did consider dark (EDIT energy not matter) and gravity to be the same phenomenon. They are both part of 'space' as I saw it. But I'm interested in how it all fits together but as I said need to do some more reading!

That equation determines a geodesic for the one metre section S at the end of the wire, which is the path it would follow if it were not attached to the rest of the wire and subject to no non-gravitational forces. The geodesic of S is accelerating away from the opposite end of the wire at rate Fw

This is the bit I am having trouble understanding. Why one end wants to accelerate from the other. I know dark matter does not have zero effect, but from what I read it just changes the state of equilibrium but doesn't continue to 'expand' an object. And although space is expanding I didn't think it applied a tension to objects even on a cosmological scale. But from what you are saying the expanding space does apply a 'pressure' to objects that increases with mass. EDIT: OR size, I am not sure which!

 

[rootone]

Remember, this is the 'dark energy' which was being refferred to as intrinsically part of the 'gravity' phenomena.
Not dark matter, that's something else.

 

[rede96]

Remember, this is the 'dark energy' which was being refferred to as intrinsically part of the 'gravity' phenomena.
Not dark matter, that's something else.

Yes, sorry that was just a typo.

 

[andrewkirk]

This is the bit I am having trouble understanding. Why one end wants to accelerate from the other. I know dark matter does not have zero effect, but from what I read it just changes the state of equilibrium but doesn't continue to 'expand' an object. And although space is expanding I didn't think it applied a tension to objects even on a cosmological scale. But from what you are saying the expanding space does apply a 'pressure' to objects that increases with mass. EDIT: OR size, I am not sure which!

It certainly is difficult to conceptualise. And I think different metaphors or perspectives work better with different people, in making intuitive sense of it. It's hard, but not as hard as it is for quantum mechanics (in my opinion).

For me, it was focusing really hard on the notion of geodesic that first gave me an intuitive sense of GR. An object will follow a geodesic unless a non-gravitational force is applied to it, and the more radically you want to make an object diverge from its geodesic, the bigger the force you have to apply to it. The reason you have to apply a force to one end of the wire to stop it from rushing away (from the other end of the wire) with the nearby stars is the same as why you have to apply force to an apple on Earth to prevent it accelerating towards the core (of the Earth, not the apple). It's because you are trying to radically accelerate the object away from its geodesic. In the apple's case, the geodesic heads directly towards the centre of the Earth, and only a continuous force upwards - by your hand, or a basket, or the ground - can accelerate it away from that geodesic. Yes, the apple sitting in the fruit bowl on your table is being accelerated!

Strict language in GR does not regard gravity as a force. Only electromagnetic, strong and weak nuclear forces are real 'forces'. This is often inconvenient. People often describe it as a force when strict language is not necessary, because it's easier, and I think I may have done so above. But the reason for being strict when it counts is that it makes it easier (at least for me) to understand things. Gravity doesn't 'push things around'. Rather it just in a sense defines their 'natural' state of motion, which is the geodesic. In a loose sense, an object that is following its geodesic is 'stationary'. And any motion relative to a geodesic must be explained by the application of some (strict) force. To me, that way of looking at it simplifies things. But as I said, it's different for everyone and I can understand if for some it confuses them instead.

I suppose the mark of a really good teacher is the ability to offer a wide selection of metaphors so that most likely everybody will be able to find at least one that is intuitive for them. Unfortunately, I only have one.

 

[rede96]

For me, it was focusing really hard on the notion of geodesic that first gave me an intuitive sense of GR. An object will follow a geodesic unless a non-gravitational force is applied to it, and the more radically you want to make an object diverge from its geodesic, the bigger the force you have to apply to it. The reason you have to apply a force to one end of the wire to stop it from rushing away (from the other end of the wire) with the nearby stars is the same as why you have to apply force to an apple on Earth to prevent it accelerating towards the core (of the Earth, not the apple). It's because you are trying to radically accelerate the object away from its geodesic. In the apple's case, the geodesic heads directly towards the centre of the Earth, and only a continuous force upwards - by your hand, or a basket, or the ground - can accelerate it away from that geodesic. Yes, the apple sitting in the fruit bowl on your table is being accelerated!

Thanks, that does really help to conceptualise it. Where I am still struggling is understanding why the gravity from the local stars near the end of the wire is strong enough to overcome the forces in the wire that hold it together, so the wire snaps. We have the local mass of the milky way, but it doesn't rip anything apart. So I just was finding it difficult to see how gravity from local stars could be that strong.

Strict language in GR does not regard gravity as a force. Only electromagnetic, strong and weak nuclear forces are real 'forces'. This is often inconvenient. People often describe it as a force when strict language is not necessary, because it's easier, and I think I may have done so above. But the reason for being strict when it counts is that it makes it easier (at least for me) to understand things. Gravity doesn't 'push things around'. Rather it just in a sense defines their 'natural' state of motion, which is the geodesic. In a loose sense, an object that is following its geodesic is 'stationary'. And any motion relative to a geodesic must be explained by the application of some (strict) force. To me, that way of looking at it simplifies things. But as I said, it's different for everyone and I can understand if for some it confuses them instead.

Again, as above I would have thought the forces that hold the wire together would have been sufficient to accelerate the end away from the geodesic. The only thing I have read that would be strong enough to tear things apart is a black hole. And you'd need to be quite close as I understand it. And as there will be various local stars along the path of such a long length of wire, I thought the wire would just find its state of equilibrium and would follow various geodesics along its length to the point where the forces in the wire become too great and would hold it back.

I suppose the mark of a really good teacher is the ability to offer a wide selection of metaphors so that most likely everybody will be able to find at least one that is intuitive for them. Unfortunately, I only have one.

Well that one was good enough! I really appreciate it thanks.

 

[DrGreg]

Where I am still struggling is understanding why the gravity from the local stars near the end of the wire is strong enough to overcome the forces in the wire that hold it together, so the wire snaps.

First it's not gravity from local stars that is responsible, it's dark energy everywhere. Second, the influence isn't just at the ends of the wire, it's cumulative along all its billion light years of length. A really tiny effect multiplied by a billion light years becomes enormous.

 

[rede96]

First it's not gravity from local stars that is responsible, it's dark energy everywhere. Second, the influence isn't just at the ends of the wire, it's cumulative along all its billion light years of length. A really tiny effect multiplied by a billion light years becomes enormous.

Yes sure. But as dark energy is everywhere then I read somewhere that the effects would cancel out. So I had ruled that out previously. Was I wrong to do so?

 

[andrewkirk]

Where I am still struggling is understanding why the gravity from the local stars near the end of the wire is strong enough to overcome the forces in the wire that hold it together, so the wire snaps. We have the local mass of the milky way, but it doesn't rip anything apart. So I just was finding it difficult to see how gravity from local stars could be that strong.

It's not gravity from the local stars that does the ripping. I only mentioned them as a reference point against which to measure motion. Their gravitational influence is insignificant.

The gravitational influence that does the ripping is the 'dark energy' of the enormous distance between the two ends of the wire. The Einstein equation says that empty space has a gravitational influence, which is given by the term \Lambda g in Einstein's field equation G+\Lambda g=8\pi T - the bit that is called 'dark energy'. The scalar quantity \Lambda determines the strength of that gravitational influence per unit distance and the item g which is the metric tensor (something that measures distance, amongst other things), is what makes the gravitational influence proportional to the distance.

The reason we often read about black holes ripping things apart but not usually dark energy is that black holes do constantly rip things apart in practice, whereas the sort of ripping described in this experiment could never happen because the universe does not contain electromagnetically connected objects like wires that are long enough for this to happen. Having said that, there is something called 'The Big Rip' which I think is something to do with Dark Energy. I think it's a name for one of the possible ultimate fates of the universe. But I've never looked into it.

EDIT: Oh I see Dr Greg already answered this. For some reason it didn't show up on my screen at first. Never mind.

 

[rede96]

It's not gravity from the local stars that does the ripping. I only mentioned them as a reference point against which to measure motion. Their gravitational influence is insignificant.

The gravitational influence that does the ripping is the 'dark energy' of the enormous distance between the two ends of the wire. The Einstein equation says that empty space has a gravitational influence, which is given by the term Λg \Lambda g in Einstein's field equation G+Λg=8πT G+\Lambda g=8\pi T - the bit that is called 'dark energy'. The scalar quantity Λ \Lambda determines the strength of that gravitational influence per unit distance and the item g g which is the metric tensor (something that measures distance, amongst other things), is what makes the gravitational influence proportional to the distance.

As I mentioned above I think the mistake I was making was that I had read that dark energy was responsible for expansion, and that expansion itself had no effect (or little) on matter. (With the exception of the change in equilibrium mentioned earlier.) I had also read that as dark energy is everywhere and in all directions that it would cancel out.

So if that isn't the case, that is were I was confusing myself.

 

[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.