Is Lorentz Contraction Indistinguishable from Standard Relativity?

[cfrogue]

Here is another way of looking at it. The two ships accelerate as per Bells's paradox, but this time the string is only connected to the front ship. The gap between the two ships stays constant according to the launch frame, but the string is length contracting. When the sting has contracted to say one hundredth of its original length, any attempt to force the string to connect the two ships, without bringing the two ships closer together (as measured in the launch frame) will snap the string. Of course, if the string is very lexible and stretching one hundred times is not sufficient to snap it, then we only have to run the experiment for a little longer until a point is reached where the string does snap, assuming that is impossible to have a string with infinite elasticity.

OK, does this imply space does not contract only rods?

Next, at any instant t in the two rocket and string frame, all three are at rest?

 

[JesseM]

Oh, the d's are measured in the moving frame and are initially known in the rest frame.

Say that the string is very weak and brittle.

Yes, but even a brittle string might have a relaxed length much greater than d...do you want to say that if we had laid out the string at rest relative to the observer with nothing pulling on either end, the distance in the observer's frame would be d? In that case, if the two ships are moving relative to the observer and the distance between them in the observer's frame is d, then since the distance between the ships in their own rest frame is greater than d, you couldn't stretch the string between the ships without breaking it.

 

[matheinste]

OK, does this imply space does not contract only rods?

Next, at any instant t in the two rocket and string frame, all three are at rest?

I think a direct answer to this in the context of standard SR would help to clarify the explanations.

Matheinste.

 

[yuiop]

Next, at any instant t in the two rocket and string frame, all three are at rest?

Nope, to one of the rocket observers the gap between the rockets is getting larger and the other rocket is getting further away, so the two rockets do not regard themselves as being at rest with respect to each other.

 

[cfrogue]

Nope, to one of the rocket observers the gap between the rockets is getting larger and the other rocket is getting further away, so the two rockets do not regard themselves as being at rest with respect to each other.

OK, so the rockets see themselves as getting further apart.

Yet, the launch frame does not see it this way. It sees the distance as constant.

How is this so?

 

[yuiop]

OK, so the rockets see themselves as getting further apart.

Yet, the launch frame does not see it this way. It sees the distance as constant.

How is this so?

The launch frame is using rulers that are not length contracted, while the rocket observers are measuring the gap using rulers that are gettting progressively shorter so to them the gap appears to be expanding.

 

[cfrogue]

The launch frame is using rulers that are not length contracted, while the rocket observers are measuring the gap using rulers that are gettting progressively shorter so to them the gap appears to be expanding.

Is there evidence that length actually contracts within a frame, I mean within the internals of a frame?

Isn't length contraction a phenomena of the "at rest" frame when viewing the moving frame?

 

[yuiop]

OK, does this imply space does not contract only rods?

Let's try a slightly modified experiment, to try and shed light on your question.

We have four rockets all with identical solid fuel propellants that burn at a fixed rate for a fixed length of time. Rocket A and B are joined by a tough cable of length d and rockets C and D are separated by a distance of d but not physically connected. All 4 rockets launch simultaneously in the launch frame. When they have exhausted their fuel rockets C and D are still a distance d apart, but rockets A and B are less than d apart. Rockets A and B have been physically pulled closer together by the length contraction of the cable.

If a fifth rocket and observer was introduced and this time only the fifth observer accelerated, then distances between the unconnected and connected rocket pairs would appear to length contract equally, but the apparent length contraction of the gap between the unconnected rockets is not physical. The changes in the clock rates and ruler lengths of the fifth observer makes the gap appear to contract.

 

[yuiop]

Is there evidence that length actually contracts within a frame, I mean within the internals of a frame?

Isn't length contraction a phenomena of the "at rest" frame when viewing the moving frame?

You can not actually observe length contraction if you are in the frame moving with the object. In the case of Bell's rocket observers they do not see the string as shrinking, but rather see it being stretched across a larger gap.

 

[cfrogue]

You can not actually observe length contraction if you are in the frame moving with the object. In the case of Bell's rocket observers they do not see the string as shrinking, but rather see it being stretched across a larger gap.

But why?

The rest frame does not see the gap getting wider.

 

[cfrogue]

Let's try a slightly modified experiment, to try and shed light on your question.

We have four rockets all with identical solid fuel propellants that burn at a fixed rate for a fixed length of time. Rocket A and B are joined by a tough cable of length d and rockets C and D are separated by a distance of d but not physically connected. All 4 rockets launch simultaneously in the launch frame. When they have exhausted their fuel rockets C and D are still a distance d apart, but rockets A and B are less than d apart. Rockets A and B have been physically pulled closer together by the length contraction of the cable.

If a fifth rocket and observer was introduced and this time only the fifth observer accelerated, then distances between the unconnected and connected rocket pairs would appear to length contract equally, but the apparent length contraction of the gap between the unconnected rockets is not physical, but brought about by changes in the clock rates and ruler lengths of the fifth observer who has undergone acceleration.
.

This thought experiment changes the game.

It should be solvable in the context we were in.

If the string contracts from the rest observer and the distance does not change, does this imply space does not contract but rods do?

 

[DrGreg]

If the string contracts from the rest observer and the distance does not change, does this imply space does not contract but rods do?

With the rapid pace of this thread, I think my post #38 may have been overlooked. I think it might be relevant to the difficulty you are having.

 

[cfrogue]

With the rapid pace of this thread, I think my post #38 may have been overlooked. I think it might be relevant to the difficulty you are having.

I read this and thought to ask you how you did those perfect graphics. I really mean this.

Assuming your post though, how do you explain this?

We have the rest frame not seeing any distance differentials. We have the accelerating frames getting further apart in their context.

How do you reconcile this?

 

[DrGreg]

I read this and thought to ask you how you did those perfect graphics. I really mean this.

I used Microsoft Powerpoint to draw the pictures. The latest version has an option to save as a PNG file.

We have the rest frame not seeing any distance differentials. We have the accelerating frames getting further apart in their context.

How do you reconcile this?

Using the notation of my diagram. "Alice" is the launch frame. "Bob" is a frame in which one of the rockets is momentarily at rest (some time later). P & Q are the two rockets.

We know y < z. That is Lorentz contraction.

We also know x = y. ("We have the rest frame not seeing any distance differentials. ")

Therefore z > x. ("We have the accelerating frames getting further apart in their context.")

 

[cfrogue]

I used Microsoft Powerpoint to draw the pictures. The latest version has an option to save as a PNG file.

Using the notation of my diagram. "Alice" is the launch frame. "Bob" is a frame in which one of the rockets is momentarily at rest (some time later). P & Q are the two rockets.

We know y < z. That is Lorentz contraction.

We also know x = y. ("We have the rest frame not seeing any distance differentials. ")

Therefore z > x. ("We have the accelerating frames getting further apart in their context.")

In order to compare these like this, you must have a uniform space.

You are depending on the trichotomy of the real numbers but the spaces are not the same in the frame to frame analysis.


Do you compare these another way I am not seeing?

 

[A.T.]

You are depending on the trichotomy of the real numbers

x,y,z are just real numbers here.

but the spaces are not the same in the frame to frame analysis.

The frame to frame part is handled by:

We know y < z. That is Lorentz contraction.

 

[A.T.]

We have the rest frame not seeing any distance differentials. We have the accelerating frames getting further apart in their context. How do you reconcile this?

Actually you answer it yourself:

but the spaces are not the same in the frame to frame analysis.

 

[DrGreg]

In order to compare these like this, you must have a uniform space.

You are depending on the trichotomy of the real numbers but the spaces are not the same in the frame to frame analysis.

I've no idea what any of that means.

 

[cfrogue]

I've no idea what any of that means.

OK, sorry, when you have some time, I am not seeing your explanation.

1) One solution suggests there exists length contraction for the string.
2) One solution suggests the ships get further apart.
3) The rest frame concludes the distance remains constant between the ships and the v and any time t is the same.

Actually, if you look from the rest frame, a reaction may be that as v increases, length contraction for the string should increase.

Yet, the SR acceleration equations do not predict this and predict a constant distance between the ships.

How is this worked out?

 

[JesseM]

Actually, if you look from the rest frame, a reaction may be that as v increases, length contraction for the string should increase.

That would be wrong, you can only use the length contraction equation for an object with a constant length in its rest frame, but the string's length in its rest frame is changing because its ends are attached to the ships.

Yet, the SR acceleration equations do not predict this and predict a constant distance between the ships.

Well, only in the launch frame, not in other frames.

 

[cfrogue]

That would be wrong, you can only use the length contraction equation for an object with a constant length in its rest frame, but the string's length in its rest frame is changing because its ends are attached to the ships..

Yes, but the ships are changing also. This would mean the space between the ships does not contract but the string does. Is this correct?

So, how would the launch frame conclude the string contracts when the launch frame concludes the distance between the ships does not change?

Well, only in the launch frame, not in other frames.

Understood

 

[JesseM]

Yes, but the ships are changing also. This would mean the space between the ships does not contract but the string does. Is this correct?

In the launch frame? No, of course the string does not contract in this frame (at least not until it breaks), how could it when it's attached to the ships? Why would you think it should? Did you read what I just said about the length contraction not applying when the rest-frame length of an object is not constant? There is no question that the rest-frame length of the string in this scenario is not constant.

 

[cfrogue]

In the launch frame? No, of course the string does not contract in this frame (at least not until it breaks), how could it when it's attached to the ships? Why would you think it should? Did you read what I just said about the length contraction not applying when the rest-frame length of an object is not constant? There is no question that the rest-frame length of the string in this scenario is not constant.

OK, then how does the string break from only the solution of the launch frame?

 

[Al68]

Yes, but the ships are changing also. This would mean the space between the ships does not contract but the string does. Is this correct?

So, how would the launch frame conclude the string contracts when the launch frame concludes the distance between the ships does not change?

In a co-moving reference frame, the ships are getting farther apart with time. In the launch frame the distance between the ships stays the same. That's length contraction, by a factor that increases with time.

In a co-moving reference frame, the ships get farther apart while the length of the string stays the same. The string breaks.

In the launch frame, the ever increasing length contraction factor results in a constant distance between the ships, while the string gets "shorter". The string breaks.

If the string stretches before it breaks, then the result of length contraction is that in the launch frame, the string's length is constant despite being stretched in its own frame.

Length contraction doesn't mean contracting with time, it means contracted relative to the proper length.

 

[JesseM]

OK, then how does the string break from only the solution of the launch frame?

You asked this question before, I gave my answer in post #42 when I said:

As I've said before, if you wanted to do the calculation solely from the perspective of the launch frame I think you would need to actually do some detailed calculation of the inter-atomic forces in this frame. Even though the average distance between atoms wouldn't change in the launch frame until the string snaps (since the length of the string and the total number of atoms remains constant in this frame), as A.T. said the way the electromagnetic field between atoms varies as a function of distance would change, and from this you could presumably show that the stress in the string was increasing. The details of such a calculation are beyond me though.

 

[JesseM]

In the launch frame, the ever increasing length contraction factor results in a constant distance between the ships, while the string gets "shorter". The string breaks.

This explanation seems confused to me...why do you say it's the "length contraction factor" that results in a constant distance? And why do you say the string gets shorter? Both the distance between ships and the length of the string are constant in the launch frame, because both ships have identical velocity as a function of time in this frame.

 

[cfrogue]

You asked this question before, I gave my answer in post #42 when I said:

Can you explain how this is consistent with the SR acceleration equations?

 

[Al68]

This explanation seems confused to me...why do you say it's the "length contraction factor" that results in a constant distance?

Because while the distance is constant in the launch frame, it is contracted by the ever increasing lorentz factor. I didn't intend to imply a causal relationship by the words "results in".

And why do you say the string gets shorter? Both the distance between ships and the length of the string are constant in the launch frame, because both ships have identical velocity as a function of time in this frame.

That part of my post was assuming the string wouldn't stretch, and therefore break. I addressed a stretchy string in the next part:

If the string stretches before it breaks, then the result of length contraction is that in the launch frame, the string's length is constant despite its proper length increasing with time.

 

[JesseM]

Can you explain how this is consistent with the SR acceleration equations?

Why would it be inconsistent with them? The SR acceleration equations say the length of the string is constant in the launch frame until the string snaps, and in my explanation I said exactly the same thing: "the average distance between atoms wouldn't change in the launch frame until the string snaps (since the length of the string and the total number of atoms remains constant in this frame)"

 

[JesseM]

Because while the distance is constant in the launch frame, it is contracted by the ever increasing lorentz factor.

Presumably you mean it's "contracted" relative to the distance in some other frame, like the instantaneous rest frame of one of the ships? But this is still a bit murky, because in every other frame the distance between the ships is changing, and you can't really compare the distance between ships at a given moment in one of these frames with the distance between them in the launch frame "at the same moment" without running into simultaneity issues. I suppose we can say that if we look at the length L' in the launch frame at any given time t, and then imagined the ships shutting off their engines simultaneously at time t in the launch frame and coasting inertially thereafter, and then we looked at the length L in the inertial frame where the ships were at rest once they had both shut off their engines, then it would make sense to say that L' is related to L by the length contraction equation L' = L/gamma.

 

[cfrogue]

Why would it be inconsistent with them? The SR acceleration equations say the length of the string is constant in the launch frame until the string snaps, and in my explanation I said exactly the same thing: "the average distance between atoms wouldn't change in the launch frame until the string snaps (since the length of the string and the total number of atoms remains constant in this frame)"

I am confused.

Where did you prove the string would snap from the POV of the launch frame?

 

[JesseM]

I am confused.

Where did you prove the string would snap from the POV of the launch frame?

I didn't prove it from the perspective of the launch frame, I just outlined how I think you would go about proving it in terms of changing electromagnetic forces between atoms, and then said "The details of such a calculation are beyond me though."

However, if we're just interested in the question of whether it snaps or not, we don't actually have to prove it from the perspective of multiple frames, proving it snaps using the perspective of anyone frame is good enough, since in relativity all frames always agree on localized physical events.

 

[Al68]

Presumably you mean it's "contracted" relative to the distance in some other frame, like the instantaneous rest frame of one of the ships? But this is still a bit murky, because in every other frame the distance between the ships is changing, and you can't really compare the distance between ships at a given moment in one of these frames with the distance between them in the launch frame "at the same moment" without running into simultaneity issues. I suppose we can say that if we look at the length L' in the launch frame at any given time t, and then imagined the ships shutting off their engines simultaneously at time t in the launch frame and coasting inertially thereafter, and then we looked at the length L in the inertial frame where the ships were at rest once they had both shut off their engines, then it would make sense to say that L' is related to L by the length contraction equation L' = L/gamma.

Yes, and gamma would depend on t, which is what I meant by the lorentz factor increasing with time.

My only point was that the constant distance between the ships in the launch frame is "contracted" distance, and that "constant distance" and "increasingly lorentz contracted distance" aren't contradictory since the proper distance between the ships is increasing with time.

 

[Al68]

Where did you prove the string would snap from the POV of the launch frame?

Are you suggesting that the string's proper length can stretch indefinitely without snapping?

 

[cfrogue]

Why would it be inconsistent with them? The SR acceleration equations say the length of the string is constant in the launch frame until the string snaps, and in my explanation I said exactly the same thing: "the average distance between atoms wouldn't change in the launch frame until the string snaps (since the length of the string and the total number of atoms remains constant in this frame)"

Oh, this is where I am confused.

Where is the string snap logic in the context of the launch frame?

I mean, where is the math? It is certainly not in the SR acceleration equations or is it?

 

[cfrogue]

Are you suggesting that the string's proper length can stretch indefinitely without snapping?

I am trying to look at the problem from the launch frame.

This is legal.

There is no distance differential in the launch frame for the ships.

 

[Al68]

I am trying to look at the problem from the launch frame.

This is legal.

There is no distance differential in the launch frame for the ships.

Right, but the length of the string is lorentz contracted. So either the string's proper length stretches or it is shorter than the (increasing) proper distance between the ships and breaks.

Are you asking for proof that a string's proper length can only increase so much before it breaks?

 

[JesseM]

Oh, this is where I am confused.

Where is the string snap logic in the context of the launch frame?

I mean, where is the math? It is certainly not in the SR acceleration equations or is it?

I already told you, you would need to calculate the changing electromagnetic force between atoms and I don't have the specific math for that, but I am totally confident this approach would show that the stress increases in the string until it snaps, because we already know that's what must happen from the analysis in other frames and it's impossible in SR for different frames to disagree in their predictions about localized events like whether a string snaps or not.

 

[atyy]

I mean, where is the math?

See the link in post #5.

 

[JesseM]

See the link in post #5.

In that chapter he does discuss how the electromagnetic field of a moving atom is altered by its velocity, but from skimming it, it doesn't look like he actually goes so far as to apply that to the case of the accelerating string to show how the stress increases until the electromagnetic forces between atoms are no longer strong enough to hold the string together and thereby show at what point it will snap.

 

[atyy]

In that chapter he does discuss how the electromagnetic field of a moving atom is altered by its velocity, but from skimming it, it doesn't look like he actually goes so far as to apply that to the case of the accelerating string to show how the stress increases until the electromagnetic forces between atoms are no longer strong enough to hold the string together and thereby show at what point it will snap.

Some crucial pages are missing to me on Google, but I think he goes far enough to show that the equilibrium state of a moving rod will be shorter - wouldn't that be enough?

 

[JesseM]

Some crucial pages are missing to me on Google, but I think he goes far enough to show that the equilibrium state of a moving rod will be shorter - wouldn't that be enough?

Ah, I didn't think of calculating the equilibrium length in the launch frame (which will get increasingly short relative to the actual length of the rod in this frame) rather than calculating the stress in the launch frame...that seems like a good approach.

 

[atyy]

Ah, I didn't think of calculating the equilibrium length in the launch frame (which will get increasingly short relative to the actual length of the rod in this frame) rather than calculating the stress in the launch frame...that seems like a good approach.

Yes, exactly the same approach you suggested earlier in the thread - but just without switching frames.

 

[yuiop]

I am surprised this thread is still going. Is there anyone here who still has any shadow of a doubt that the string connecting the rockets WILL break??

Where did you prove the string would snap from the POV of the launch frame?

Dr Greg demonstrated that the string would snap from the POV of the launch frame way back in post #33 of this thread here: https://www.physicsforums.com/showpost.php?p=2443127&postcount=33

I am confused.

Where did you prove the string would snap from the POV of the launch frame?

I mean, where is the math? It is certainly not in the SR acceleration equations or is it?

It is not in the SR equations because SR is specifically about reference frame that are NOT accelerating. However, SR does tell us that if an object is moving relative to an observer (accelerating or not) then that object should be length contracted. If the moving object is not length contracted then it MUST be under stress.

 

[yuiop]

There is a way to demonstrate that the string will break without invoking electromagnetic forces or even length contraction. Imagine rockets A and B are moving to the right with constant velocity v relative to observer C. The rockets are separated by a distance (d) and joined by an unstressed string. Rockets A and B are instructed to launch simultaneously according to the clocks in their inertial frame and thereafter accelerate to the left with constant acceleration until they come to rest in C's frame. In this scenario they are slowing down according to observer C and so length contraction of the string is not a factor according to observer C. (if anything he expects the string to expand.) When the rockets take off, C notices that the rear rocket takes of first because the clocks of A and B are not synchronised from C's point of view and the string snaps because the rear rocket slows down to a stop while the front rocket is still going to the right. In this case it is the relativity of simultaneity that snaps the string rather than length contraction, but the end result is still the same - the string snaps.

 

[matheinste]

It is not in the SR equations because SR is specifically about reference frame that are NOT accelerating. However, SR does tell us that if an object is moving relative to an observer (accelerating or not) then that object should be length contracted. If the moving object is not length contracted then it MUST be under stress.

Would it be OK to say that if an object whose unstressed rest length is known, appears to have, at a later time, the same length viewed from a frame moving relative to the one at which it is at rest, then it must at this later time be stressed.

Whatever its rest length, stressed or unstressed, it must in, real time, appear contracted when viewed from a relatively moving frame.

Matheinste.

 

[A.T.]

There is no distance differential in the launch frame for the ships.

So? Where is your problem? The condition for the string to snap is that the distance between the rockets is greater than the maximal length of the string. In the launch frame the distance between the rockets is constant, but the maximum length of the string reduces, because all elements the string is made of (fibers, chain links, down to individual atoms) are contracting as they accelerate.

 

[JesseM]

So? Where is your problem? The condition for the string to snap is that the distance between the rockets is greater than the maximal length of the string. In the launch frame the distance between the rockets is constant, but the maximum length of the string reduces, because all elements the string is made of (fibers, chain links, down to individual atoms) are contracting as they accelerate.

Well, the individual atoms contract, as do the electromagnetic fields surrounding them, but since the average distance between atoms remains constant until the string snaps in the launch frame, the fibers or chain links can't actually contract. The equilibrium length of the string (that is, the length it would be if its ends were free) does contract though, and the maximum length the string can reach without snapping is just some multiple of the equilibrium length.

 

[yuiop]

Would it be OK to say that if an object whose unstressed rest length is known, appears to have, at a later time, the same length viewed from a frame moving relative to the one at which it is at rest, then it must at this later time be stressed.

Whatever its rest length, stressed or unstressed, it must in, real time, appear contracted when viewed from a relatively moving frame.

Matheinste.

I agree. In the launch frame the (stressed) string between the accelerating rockets has length (d') and in an frame instantaneously co-moving with the average velocity of the accelerating rockets, the length of the string by their measurements (d) is greater than d' by aproximately the average gamma factor. (In the frame of one rocket the other rocket is moving so a sort of average velocity has to be used to estimate what is going on.)

 

[A.T.]

but since the average distance between atoms remains constant until the string snaps in the launch frame, the fibers or chain links can't actually contract.

If a stationary chain is already under maximal stress, then the chain links don't contract on acceleration, but simply break. But since atoms and bonding forces a difficult to grasp, it might be helpful consider a chain that still has some play when stationary, yet brakes at a certain speed, despite keeping a constant length: