Lorentz Contraction: Exploring Standard Relativity & Bell's Paradox

[JesseM]

What is the length of the rod after acceleration is done?

I assumed above that the rod's length in its new inertial rest frame S' after it had stopped accelerating was d' = d*gamma. Then I showed that if we apply the length contraction equation, this implies that its length in the launch frame S must be d'/gamma = (d*gamma)/gamma = d.

 

[cfrogue]

I assumed above that the rod's length in its new inertial rest frame S' after it had stopped accelerating was d' = d*gamma. Then I showed that if we apply the length contraction equation, this implies that its length in the launch frame S must be d'/gamma = (d*gamma)/gamma = d.

Good, you finally made it.

So, where do the length contraction of SR apply?

1) We start out in a rest frame and measure d.
2) d accelerates and stops.
3) After stopping, the length is d*gamma.
4) Now, the rest frame applies LT since we have relative motion and applies length contraction as advertised, (d*gamma)/gamma = d.
5) No real length contraction.

 

[JesseM]

Good, you finally made it.

"Finally"? This is what I have been saying all along, apparently you weren't paying attention?

So, where do the length contraction of SR apply?

1) We start out in a rest frame and measure d.
2) d accelerates and stops.
3) After stopping, the length is d*gamma.
4) Now, the rest frame applies LT since we have relative motion and applies length contraction as advertised, (d*gamma)/gamma = d.
5) No real length contraction.

"Length contraction" does not refer to the length contracting over time in one frame, it refers to a comparison between two frames, with one frame seeing the length "contracted" relative to the length in the rest frame. If we consider the rod after it's finished accelerating, the length in the launch frame S is contracted relative to the length in the rod's rest frame S', this is all that is ever meant by "length contraction" in SR.

 

[cfrogue]

"Finally"? This is what I have been saying all along, apparently you weren't paying attention?

"Length contraction" does not refer to the length contracting over time in one frame, it refers to a comparison between two frames, with one frame seeing the length "contracted" relative to the length in the rest frame. If we consider the rod after it's finished accelerating, the length in the launch frame S is contracted relative to the length in the rod's rest frame S', this is all that is ever meant by "length contraction" in SR.

The logic is simple.

If a rod starts out at d and is accelerated to v, then the actual length of the rod is dλ in the moving frame. Thus, there is no length contraction for the rest frame.

 

[JesseM]

The logic is simple.

If a rod starts out at d and is accelerated to v, then the actual length of the rod is dλ in the moving frame. Thus, there is no length contraction for the rest frame.

You are using the phrase "length contraction" incorrectly. Did you even read my post above? "Length contraction" in SR always refers to a comparison of lengths in two frames, it does not refer to a change in length in a single frame, so the fact that the length doesn't change in the launch frame doesn't mean it's correct to say "there is no length contraction for the rest frame".

 

[Al68]

Good, you finally made it.

So, where do the length contraction of SR apply?

1) We start out in a rest frame and measure d.
2) d accelerates and stops.
3) After stopping, the length is d*gamma.
4) Now, the rest frame applies LT since we have relative motion and applies length contraction as advertised, (d*gamma)/gamma = d.
5) No real length contraction.

The logic is simple.

If a rod starts out at d and is accelerated to v, then the actual length of the rod is dλ in the moving frame. Thus, there is no length contraction for the rest frame.

The logic is simple:

You just said that the length of the rod in the moving frame is dλ while its length in the rest frame is d (length contraction for the rest frame).

Then you say there is no length contraction for the rest frame.

And this after it has been repeatedly pointed out that length contraction is the difference in length between two frames (d vs dλ).

I can only assume this is some kind of bizarre practical joke.

 

[A.T.]

And this after it has been repeatedly pointed out that length contraction is the difference in length between two frames (d vs dλ).

I can only assume this is some kind of bizarre practical joke.

The more common word for this kind of practical joke is 'trolling'. His entire argument is based on using the same term for two different quantities, and wondering about the contradictions that arise from it. This was already pointed out to him hundreds of posts ago. He will continue to ignore the clarification because he doesn't want to understand anything, just to argue like a kid.

 

[cfrogue]

You are using the phrase "length contraction" incorrectly. Did you even read my post above? "Length contraction" in SR always refers to a comparison of lengths in two frames, it does not refer to a change in length in a single frame, so the fact that the length doesn't change in the launch frame doesn't mean it's correct to say "there is no length contraction for the rest frame".

I have this part figured out.

I will say it again.

Two frames are at rest.

They measure a rod length to be d.

The frame O' takes the rod and accelerates to v and stops accelerating. O stays at rest.

The length of the rod has increased in O' to d*λ because of the acceleration.

Now, O applies LT length contraction to the rod.

(d*λ)/λ = d.

Thus, the rod expands from the acceleration and LT contracts it back to d from the POV of O.

What is the problem?

Note, that the "rest" length of the rod is d and the LT moving length is d.

 

[JesseM]

I have this part figured out.

I will say it again.

Two frames are at rest.

They measure a rod length to be d.

The frame O' takes the rod and accelerates to v and stops accelerating. O stays at rest.

The length of the rod has increased in O' to d*λ because of the acceleration.

Now, O applies LT length contraction to the rod.

(d*λ)/λ = d.

Thus, the rod expands from the acceleration and LT contracts it back to d from the POV of O.

What is the problem?

I have no problem with this--it was you who seemed to say there was a problem, that somehow length contraction does not apply in this example.

 

[cfrogue]

I have no problem with this--it was you who seemed to say there was a problem, that somehow length contraction does not apply in this example.

You about have it.

Length contraction does not apply to any problem that talks about an "at rest" rod distance.

Do you know any other kind?

 

[JesseM]

You about have it.

Length contraction does not apply to any problem that talks about an "at rest" rod distance.

Why do you think it doesn't apply? You just used the length contraction equation when you said:

Now, O applies LT length contraction to the rod.

(d*λ)/λ = d.

So it applies just fine! The length in the frame where the rod is moving (the launch frame O) is smaller than the length in the rod's rest frame O' by a factor of gamma, which is exactly what the length contraction equation tells you should happen.

 

[cfrogue]

Why do you think it doesn't apply? You just used the length contraction equation when you said:

So it applies just fine! The length in the frame where the rod is moving (the launch frame O) is smaller than the length in the rod's rest frame O' by a factor of gamma, which is exactly what the length contraction equation tells you should happen.

Geez, you are not getting this.

When the rest length is d, it expands to d/gamma after acceleration for v.

Then the rest frame applies LT and there is no net length contraction in the rest frame.

What is so hard about this?

 

[JesseM]

Geez, you are not getting this.

When the rest length is d, it expands to d/gamma after acceleration for v.

Then the rest frame applies LT and there is no net length contraction in the rest frame.

What is so hard about this?

What is so hard about the statement (made many times by myself and others) that length contraction refers only to a comparison between frames, not a change in length over time in a single frame? When you say "there is no net length contraction in the rest frame" you are clearly trying to use "length contraction" in the latter sense, which is just an incorrect usage of the term.

 

[cfrogue]

What is so hard about the statement (made many times by myself and others) that length contraction refers only to a comparison between frames, not a change in length over time in a single frame? When you say "there is no net length contraction in the rest frame" you are clearly trying to use "length contraction" in the latter sense, which is just an incorrect usage of the term.

You are wrong. I am not comparing a single frame.

I will post it slowly.

1) Two observers O and O' agree the rest length of a rod is d.
2) Then, all of a sudden, O' decides to accelerate to v.
3) O' frame concludes the length of the rod is d*gamma after the acceleration.
4) O is standing around and decides to calculate the length of the rod accoriding to LT with the relative motion v.
5) Now, the "at rest" length was d and so O concludes d/gamma. Is this correct? No, the rod length expanded in O'. Normally if d is the length in O', then d/gamma is the calculated length in O.
6) But, d is not the correct length in O', it is d*gamma.
7) So, after O realizes this, O calculates the rod length d*gamma/gamma and thus there is no rod contraction from the calculations of O between the "at rest" length of the rod at d and the "contracted" rod length.

 

[JesseM]

7) So, after O realizes this, O calculates the rod length d*gamma/gamma and thus there is no rod contraction from the calculations of O between the "at rest" length of the rod at d and the "contracted" rod length.

The "at rest" length of the rod after it finishes acceleration is d*gamma, O cannot define the "at rest" length in terms of its length before acceleration since the acceleration changed the rod. A condition of using the length contraction is that the object has to be moving inertially throughout the period that both frames measure its length, so here the lengths that are plugged into the equation must be measured after the rod has finished accelerating and is moving inertially. In this case, O agrees the "at rest" length after acceleration is d*gamma, and the length of the rod in O after acceleration is d, so there certainly is rod contraction when you compare both frames.

 

[cfrogue]

The "at rest" length of the rod after it finishes acceleration is d*gamma, O cannot define the "at rest" length in terms of its length before acceleration since the acceleration changed the rod. A condition of using the length contraction is that the object has to be moving inertially throughout the period that both frames measure its length, so here the lengths that are plugged into the equation must be measured after the rod has finished accelerating and is moving inertially. In this case, O agrees the "at rest" length after acceleration is d*gamma, and the length of the rod in O after acceleration is d, so there certainly is rod contraction when you compare both frames.

OMG.

They defined the length before the acceleration. They called it the at rest rod length.

You are ignoring the facts.

 

[JesseM]

OMG.

They defined the length before the acceleration. They called it the at rest rod length.

You are ignoring the facts.

You're ignoring the fact that the length contraction has certain conditions that must be satisfied in order for it to work. One of these conditions is that the object has to be moving inertially throughout the period when both frames measure its length. These conditions are violated if you compare the rest length before an acceleration to the moving length after an acceleration, the length contraction equation was never intended to apply to such a pair of measurements.

 

[Al68]

OMG.

They defined the length before the acceleration. They called it the at rest rod length.

You are ignoring the facts.

You gave a scenario that stipulated that the "at rest" rod length increased from d to d*gamma for some unstated reason (telescoping rod maybe?). Why would you give such a scenario if you are going to later deny that the rest length of the rod increased?

Maybe this will help:

Length contraction is L' = L/gamma where L is the rest length, L' is the length in a reference frame in which an object is moving.

In your scenario after the acceleration, L = d*gamma and L' = d so L' = L/gamma exactly in accordance with SR length contraction.

 

[cfrogue]

You gave a scenario that stipulated that the "at rest" rod length increased from d to d*gamma for some unstated reason (telescoping rod maybe?). Why would you give such a scenario if you are going to later deny that the rest length of the rod increased?

Maybe this will help:

Length contraction is L' = L/gamma where L is the rest length, L' is the length in a reference frame in which an object is moving.

In your scenario after the acceleration, L = d*gamma and L' = d so L' = L/gamma exactly in accordance with SR length contraction.

Agreed, I am not seeing a problem with any of this. Never did.

I just think it is uniqie that if two are at rest and measure a rod and the rod is accelerated its actual length never changes even after the acceleration stops and relative motion exists.

This does not say much of anything except length contraction may be just restoring a rod length to what it would be if the frames suddenly became at rest.

I am OK with the end of the thread.

 

[Al68]

Agreed, I am not seeing a problem with any of this. Never did.

I just think it is uniqie that if two are at rest and measure a rod and the rod is accelerated its actual length never changes even after the acceleration stops and relative motion exists.

Well, that's unique to your scenario, but very atypical for rods in general. The rod in your scenario is not a typical rod. Rods don't typically change their rest length.

Your scenario stipulated the equivalent of a telescoping rod or bungie cord that expanded for some unstated reason during the acceleration, which is perfectly fine, just not the same type of rod normally used as an example.

In a typical scenario, like one of Einstein's, a normal rod is used with a rest length that is constant. So it's rest length would be d before and after acceleration and its length in a frame in which it is moving would be d/gamma.

 

[cfrogue]

Well, that's unique to your scenario, not length contraction in general. The rod in your scenario is not a typical rod. Rods don't typically change their rest length.

Your scenario stipulated the equivalent of a telescoping rod that expanded for some unstated reason during the acceleration, which is perfectly fine, just not the same type of rod normally used as an example.

In a typical scenario, like one of Einstein's, a normal rod is used with a rest length that is constant. So it's rest length would be d before and after acceleration and its length in a frame in which it is moving would be d/gamma.

This makes no sense. We are talking about the rest frame here.

 

[Al68]

This makes no sense. We are talking about the rest frame here.

What makes no sense? I was just pointing out that most SR scenarios typically stipulate a rod that is rigid, with a "at rest" length that doesn't change because of acceleration. A normal (rigid) rod with a rest length of d would have a rest length of d after any acceleration, and a length of d/gamma in any inertial reference frame.

 

[cfrogue]

What makes no sense? I was just pointing out that most SR scenarios typically stipulate a rod that is rigid, with a "at rest" length that doesn't change because of acceleration. A normal (rigid) rod with a rest length of d would have a rest length of d after any acceleration, and a length of d/gamma in any inertial reference frame.

This thread has proved this is false.

 

[JesseM]

This thread has proved this is false.

You can't "prove false" a condition that must be satisfied in order for an equation to work. You might as well say "if we plug time-intervals into the length contraction equation rather than lengths, then the equation gives a wrong answer, therefore I've proven that the length contraction equation is false in some circumstances". Well, obviously no you haven't, because one of the conditions of the length contraction equation is that the values you plug in for L and L' must be lengths measured in the rest frame and the moving frame. Similarly, it's one of the conditions of the length contraction equation that L and L' must both be measured during a time period when the object is rigid and moving inertially. If I measured the rest length L of John when he's a baby, and the moving length L' of John when he's an adult, the two lengths won't be related by gamma, but that doesn't prove the length contraction equation false because I didn't satisfy the condition that the object being measured was rigid throughout the period when both measurements were taken.

 

[cfrogue]

You can't "prove false" a condition that must be satisfied in order for an equation to work. You might as well say "if we plug time-intervals into the length contraction equation rather than lengths, then the equation gives a wrong answer, therefore I've proven that the length contraction equation is false in some circumstances". Well, obviously no you haven't, because one of the conditions of the length contraction equation is that the values you plug in for L and L' must be lengths measured in the rest frame and the moving frame. Similarly, it's one of the conditions of the length contraction equation that L and L' must both be measured during a time period when the object is rigid and moving inertially. If I measured the rest length L of John when he's a baby, and the moving length L' of John when he's an adult, the two lengths won't be related by gamma, but that doesn't prove the length contraction equation false because I didn't satisfy the condition that the object being measured was rigid throughout the period when both measurements were taken.

I am going to prove something false.

The method is called Reductio ad absurdum.

It is a common misconception that you cannot prove something false.

There exists a greatest integer.
I am going to prove this false.

Let n be the greatest integer.

Add 1 to n.

n + 1 > n.

Contradiction.

This is an Archimedes argument.

 

[Al68]

What makes no sense? I was just pointing out that most SR scenarios typically stipulate a rod that is rigid, with a "at rest" length that doesn't change because of acceleration. A normal (rigid) rod with a rest length of d would have a rest length of d after any acceleration, and a length of d/gamma in any inertial reference frame.

This thread has proved this is false.

No it hasn't. It's just shown that you are free to stipulate a (non typical) rod that is not rigid and can have an increasing rest length (stretch). That does not prove that all rods will automatically increase their rest length. That's just absurd.

You purposely stipulated a rod that was stretchy instead of rigid, and that increased its rest length for some unstated reason.

You can't then apply that result to a different situation in which a rigid rod is stipulated.

Are you under the false impression that acceleration causes a rigid rod to increase its rest length? That's the only thing I can think of that would explain your bizarre posts.

 

[JesseM]

I am going to prove something false.

The method is called Reductio ad absurdum.

It is a common misconception that you cannot prove something false.

There exists a greatest integer.
I am going to prove this false.

Let n be the greatest integer.

Add 1 to n.

n + 1 > n.

Contradiction.

This is an Archimedes argument.

What the hell are you blabbering about cfrogue? I didn't say you couldn't prove anything false. I said you couldn't prove the length contraction equation false using a scenario in which you violate one of the conditions that are required in order for the the length contraction equation to apply. If you want mathematical analogies, here's one--

Define the "Law of real inverses" to say: for any nonzero real number R, the number has an real inverse 1/R such that R times its inverse 1/R equals 1.

cfrogue-style argument: but look, zero doesn't have a real inverse! Therefore the "Law of real inverses" is false!

Can you see why this argument would be pretty stupid?

 

[matheinste]

I am going to prove something false.

The method is called Reductio ad absurdum.

It is a common misconception that you cannot prove something false.

There exists a greatest integer.
I am going to prove this false.

Let n be the greatest integer.

Add 1 to n.

n + 1 > n.

Contradiction.

This is an Archimedes argument.

This is not a valid mathematical argument. You are trying to prove that the statement "there exists a greatest integer" is false by demonstrating that, given any integer, you can produce one that is greater by adding 1 to it. However if there were a greatest integer you would be unable to add 1 to it to make a greater one. You are assuming result before you have proved it.

Matheinste.

 

[cfrogue]

What the hell are you blabbering about cfrogue? I didn't say you couldn't prove anything false. I said you couldn't prove the length contraction equation false using a scenario in which you violate one of the conditions that are required in order for the the length contraction equation to apply. If you want mathematical analogies, here's one--

Define the "Law of real inverses" to say: for any nonzero real number R, the number has an real inverse 1/R such that R times its inverse 1/R equals 1.

cfrogue-style argument: but look, zero doesn't have a real inverse! Therefore the "Law of real inverses" is false!

Can you see why this argument would be pretty stupid?

You are getting frustrated.

You are applying universal generalizations you know do not apply.

 

[cfrogue]

This is not a valid mathematical argument. You are trying to prove that the statement "there exists a greatest integer" is false by demonstrating that, given any integer, you can produce one that is greater by adding 1 to it. However if there were a greatest integer you would be unable to add 1 to it to make a greater one. You are assuming result before you have proved it.

Matheinste.

I suggest you look at the original Archimedes proof.

It is clear to me you do not know how to argue by Reductio ad absurdum.

I do this all the time.

 

[cfrogue]

This is not a valid mathematical argument. You are trying to prove that the statement "there exists a greatest integer" is false by demonstrating that, given any integer, you can produce one that is greater by adding 1 to it. However if there were a greatest integer you would be unable to add 1 to it to make a greater one. You are assuming result before you have proved it.

Matheinste.

I suggest you look at the original Archimedes proof.

It is clear to me you do not know how to argue by Reductio ad absurdum.

I do this all the time.

The properties of the Integers is that if n belongs to the integers then n + 1 belongs to the integers.

This is Peano arithmetic.

 

[matheinste]

I suggest you look at the original Archimedes proof.

It is clear to me you do not know how to argue by Reductio ad absurdum.

I do this all the time.

I have.

That there is no greatest integer is true. It is called the Archimedean property of numbers.

It does not use this "proof".

You are absolutely categorically wrong in your proof. It is a common mistake that many people make and is completely compatible with many of your arguments.

You may get away with wordplay and ambiguityy with verbal atguments in relativity but you cannot get away with it in mathematics.

Matheinste.

 

[JesseM]

You are getting frustrated.

Yes, frustrated by the stupidity of your arguments.

You are applying universal generalizations you know do not apply.

No idea what you mean by "universal generalizations". Here was my analogy again:

Define the "Law of real inverses" to say: for any nonzero real number R, the number has an real inverse 1/R such that R times its inverse 1/R equals 1.

cfrogue-style argument: but look, zero doesn't have a real inverse! Therefore the "Law of real inverses" is false!

Tell me whether you agree or disagree that this is a stupid argument. Now, here's why it's analogous to the length contraction example:

Define the "law of length contraction" to say: if an object's length is measured in two different inertial frames, and the object is rigid and moving inertially throughout the period that both measurements are made, and if one of the frames sees the object to be at rest while the other frame sees it to be moving at speed v, then L_moving = L_rest * sqrt(1 - v^2/c^2).

cfrogue: but look, if the two measurements are made in a period of time where the object isn't rigid and isn't moving inertially throughout, then it's not true that L_moving = L_rest * sqrt(1 - v^2/c^2)! Therefore the "law of length contraction" is false!

So if the first argument is stupid, this second argument must be equally stupid, for exactly the same reason.

 

[cfrogue]

I have.

That there is no greatest integer is true. It is called the Archimedean property of numbers.

It does not use this "proof".

You are absolutely categorically wrong in your proof. It is a common mistake that many people make and is completely compatible with many of your arguments.

You may get away with wordplay and ambiguityy with verbal atguments in relativity but you cannot get away with it in mathematics.

Matheinste.

Silly.

Prove it is wrong.

 

[matheinste]

Silly.

Prove it is wrong.

I cannot be bothered.

I have learned from experience that if you have decided you are correct, no amount of logical argument will convince you otherwise. However in this case there is no doubt, your proff is invalid.

Matheinste