Is Lorentz Contraction Indistinguishable from Standard Relativity?

[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.
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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.