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Tags length contraction , special relativity

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Old 24th January 2015, 01:14 PM   #1
wogoga
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Reciprocity of SR Length Contraction for Dummies

Reciprocity of length contraction is a logical consequence of Special Relativity. The higher the speed and Lorentz factor, the easier to recognize the dubiousness of this SR prediction. Let us start with these premises:
  • A distance of 100 light years, e.g. from our planetary system to a target 100 LY away (rest-frame)
  • A ruler (i.e. length unit) with a nominal (or proper) length of 1 LY
  • Measuring our rest-frame distance of 100 LY with the ruler moving at v = 0.99995 c (moving-frame)
As Lorentz factor of 0.99995 c is 100, the moving ruler is contracted (from 1 LY) to 0.01 LY in the rest-frame.

Common sense would suggest that with respect to the 100-fold contracted length-unit, rest-frame distances (in direction of motion) are or seem to be increased 100-fold, in our case from 100 LY to 10 000 LY. The reason is obvious: 10 000 rulers with nominal length of 1 LY, but actually contracted to 0.01 LY, must be put one next to another in order to fill the whole distance of 100 LY.

Special Relativity however states that with respect to the moving ruler, distance from us to the target is reduced from 100 LY to 1 LY (moving-frame). This is possible only if the length of the ruler increases 100-fold (i.e. from 1 LY to 100 LY in the rest-frame), as the 1-LY-ruler must cover the whole distance of 100 LY at the same time. This then means that with respect to the moving-frame, the ruler itself (with a nominal length of 1 LY) has the same length as the rest-frame distance of 100 LY. In this way a moving-frame length of only 1 LY can be attributed to the 100-LY-distance.

Here the question arises:
How can a ruler, contracted from a nominal length of 1 light year to 0.01 LY, be on the other hand expanded to 100 LY with respect to the same rest-frame length-units?
The answer is astonishingly simple:
Actual simultaneity has to follow Poincaré synchronization.
With respect to rest-frame simultaneity, the distance between front and back end of the moving 1-LY-ruler is 0.01 LY. If from the center of the contracted ruler a light signal is sent to both ends then this signal needs tBack = 0.005 LY / (c + 0.99995 c) = 0.0025 Year to reach the back end. To reach the front end, the light signal needs tFront = 0.005 LY / (c - 0.99995 c) = 99.9975 Year. So in the rest-frame, the event of reaching the front end occurs dt = tFront - tBack = 99.995 years after having reached the back end.

These two signal-receiving events, occurring 99.995 years one after the other in the rest-frame, are declared simultaneous in the moving frame. So in order to use this ruler as a length unit with respect to moving-frame simultaneity, its front end must have moved at v = 0.99995c 99.995 years longer than its back end. Only during these 99.995 years of unchanged motion, the ruler can increase from 0.01 LY (rest-frame length, rest-frame simultaneity) to 100 LY (rest-frame length, moving-frame simultaneity) by adding 99.99 LY (equals v * dt).

Only insofar as the ruler is itself expanded to 100 LY, it covers with its own nominal length of 1 LY the whole 100-LY-distance, and thus theses 100 LY turn out be just 1 light year.

In our case, SR prediction of reciprocal length contraction would only make sense in case of unchanged (inertial) motion of the ruler front-end over at least 99.995 years!

Summary for Lorentz-Ether-Theory with SR interpretation:
A ruler (i.e. length unit) of 1 light year moving at 0.99995 c is used to measure an ether distance of 100 LY. The moving ruler is obviously contracted to 0.01 LY. If all inertial movements enjoy equal rights then with respect to the moving ruler, ether distances must also be contracted 100-fold, i.e. the ruler with nominal length of 1 LY must cover our whole ether distance of 100 LY.

Such a 10000-fold length increase can be achieved by sending a signal from the center of the contracted ruler to both ends and by declaring signal-reception as simultaneous. Because of ether wind, the signal needs 99.995 years longer to the front-end than to the back-end. And during these 99.995 years of constant inertial motion, the distance between the ruler-ends increase from 0.01 LY to 100 LY, leading to an attribution of only 1 light year to the 100-LY-ether-distance.
For better understanding see also.

Cheers, Wolfgang

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Old 24th January 2015, 06:41 PM   #2
WhatRoughBeast
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This one is really too easy. Third paragraph begins

Originally Posted by wogoga View Post
Common sense would suggest
And that's where it all goes wrong. Once you start applying non-relativistic common sense to relativistic scenarios, there's no telling what sort of trouble you can get yourself into. And it establishes that your ideas are not to be taken seriously.
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Old 24th January 2015, 06:43 PM   #3
RussDill
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This is no different than the ladder/barn paradox and is easily resolvable with SR. Each time you do a measurement with a ruler, you are relying on simultaneity and performing an action similar to the closing of the barn doors.

http://en.wikipedia.org/wiki/Ladder_paradox

In this case, the barn is the ruler.
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Old 30th January 2015, 06:42 AM   #4
wogoga
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Originally Posted by WhatRoughBeast in #2 View Post
This one is really too easy. Third paragraph begins
"Common sense would suggest…"
And that's where it all goes wrong. Once you start applying non-relativistic common sense to relativistic scenarios, there's no telling what sort of trouble you can get yourself into. And it establishes that your ideas are not to be taken seriously.

You confuse modern physics with a new religion or theology. Your argument is the old argument of theological, religious orthodoxy against philosophic and scientific progress:
"Once you start applying common-sense (i.e. consistent logical reasoning) to Holy Scriptures and theological dogmas, there's no telling what sort of trouble you can get yourself into."
It seems that you admit that you do not understand enough to seriously deal with what I've written. Nevertheless you are convinced that insofar as my argument suggests that there is a problem, the argument must be wrong. Why can you not imagine that what you believe in without understanding might be wrong or at least disputable? You have again the same certainty you had in previous lives when believing in all kinds of what you now consider complete nonsense (e.g. descent from Adam and Eve).

You can fill a distance of 100 m with 100 pieces of a ruler of 1 m length. If the 1-m-rulers are Lorentz-contracted to 1 cm then according to common sense (logical-mathematical reasoning) we need ten thousand pieces of such contracted 1-m-rulers in order to fill the original 100-m-distance. This is a simple logical fact you cannot deny in any reasonable way.

In SR however, a 1-m-ruler moving at 0.99995c is not only contracted to 1 cm on the one hand, but it is also expanded to 100 m on the other hand. Otherwise the moving ruler could not attribute its own nominal length of 1 m to a 100-m-distance at rest. As Special Relativity is a scientific theory and not some kind of magic, we can try to understand such seemingly paradoxical predictions. And this is exactly what I do.

Let as assume a ruler of 1 cm moving at 0.99995 m/s to the right. At time t = 0 its left end | coincides with the left end of a 100-m-distance at x = 0:
|====|
|----+-------- ... --------+----+---
0 . 0.01m . . . . . . . 99.99m 100m
The x-coordinate of the right end then shows x = 0.01 m. 99.995 seconds later, at t = 99.995 sec, this right end crosses the 100-m-mark of our distance (as 99.995 seconds times 0.99995 m/s results in a movement of 99.99 m, i.e. from x = 0.01 m to x = 100 m).
. . . . . . . . . . . . . .|====|
|----+-------- ... --------+----+---
0 . 0.01m . . . . . . . 99.99m 100m
If we take the position of the left end at t = 0 and the position of right end at t = 99.995 sec then a length of 100 m can be attributed to the ruler.
|============= ... =============|
|----+-------- ... --------+----+---
0 . 0.01m . . . . . . . 99.99m 100m
The important point: A phase of constant linear movement of only one ruler-end is necessary in order to guarantee in this way mutual length contraction. (Ruler-expansion is equivalent to distance-contraction.) It cannot be excluded apriori that this leads to inconsistencies in special cases (e.g. Bell's spaceship paradox).

Cheers, Wolfgang
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Old 30th January 2015, 07:38 AM   #5
ctamblyn
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Originally Posted by wogoga View Post
"Once you start applying common-sense (i.e. consistent logical reasoning) to Holy Scriptures and theological dogmas, there's no telling what sort of trouble you can get yourself into."
It seems that you admit that you do not understand enough to seriously deal with what I've written.
...
I think you are putting words into WhatRoughBeast's mouth.

They didn't say "I admit I do not understand enough Special Relativity to seriously deal with what wogoga wrote," or even imply it. They said that "non-relativistic common sense" would lead you astray when thinking about relativistic scenarios. Which is fair enough.
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Old 30th January 2015, 07:52 AM   #6
Hellbound
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Originally Posted by ctamblyn View Post
I think you are putting words into WhatRoughBeast's mouth.

They didn't say "I admit I do not understand enough Special Relativity to seriously deal with what wogoga wrote," or even imply it. They said that "non-relativistic common sense" would lead you astray when thinking about relativistic scenarios. Which is fair enough.
Just to add to this, the relationship between "common sense" and "consistent logical reasoning" is akin to the relationship between "Detroit Lions" and "Superbowl Champions". They might, theoretically, coincide, but they are by no means synonymous.
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Old 30th January 2015, 12:40 PM   #7
RussDill
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Originally Posted by wogoga View Post
You confuse modern physics with a new religion or theology. Your argument is the old argument of theological, religious orthodoxy against philosophic and scientific progress:
"Once you start applying common-sense (i.e. consistent logical reasoning) to Holy Scriptures and theological dogmas, there's no telling what sort of trouble you can get yourself into."
It seems that you admit that you do not understand enough to seriously deal with what I've written. Nevertheless you are convinced that insofar as my argument suggests that there is a problem, the argument must be wrong. Why can you not imagine that what you believe in without understanding might be wrong or at least disputable? You have again the same certainty you had in previous lives when believing in all kinds of what you now consider complete nonsense (e.g. descent from Adam and Eve).

You can fill a distance of 100 m with 100 pieces of a ruler of 1 m length. If the 1-m-rulers are Lorentz-contracted to 1 cm then according to common sense (logical-mathematical reasoning) we need ten thousand pieces of such contracted 1-m-rulers in order to fill the original 100-m-distance. This is a simple logical fact you cannot deny in any reasonable way.

In SR however, a 1-m-ruler moving at 0.99995c is not only contracted to 1 cm on the one hand, but it is also expanded to 100 m on the other hand. Otherwise the moving ruler could not attribute its own nominal length of 1 m to a 100-m-distance at rest. As Special Relativity is a scientific theory and not some kind of magic, we can try to understand such seemingly paradoxical predictions. And this is exactly what I do.

Let as assume a ruler of 1 cm moving at 0.99995 m/s to the right. At time t = 0 its left end | coincides with the left end of a 100-m-distance at x = 0:
|====|
|----+-------- ... --------+----+---
0 . 0.01m . . . . . . . 99.99m 100m
The x-coordinate of the right end then shows x = 0.01 m. 99.995 seconds later, at t = 99.995 sec, this right end crosses the 100-m-mark of our distance (as 99.995 seconds times 0.99995 m/s results in a movement of 99.99 m, i.e. from x = 0.01 m to x = 100 m).
. . . . . . . . . . . . . .|====|
|----+-------- ... --------+----+---
0 . 0.01m . . . . . . . 99.99m 100m
If we take the position of the left end at t = 0 and the position of right end at t = 99.995 sec then a length of 100 m can be attributed to the ruler.
|============= ... =============|
|----+-------- ... --------+----+---
0 . 0.01m . . . . . . . 99.99m 100m
The important point: A phase of constant linear movement of only one ruler-end is necessary in order to guarantee in this way mutual length contraction. (Ruler-expansion is equivalent to distance-contraction.) It cannot be excluded apriori that this leads to inconsistencies in special cases (e.g. Bell's spaceship paradox).

Cheers, Wolfgang
Congrats, you've discovered the consequences of simultaneity in special relativity. I'm not sure what point you are trying to make.
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Old 3rd February 2015, 08:52 PM   #8
WhatRoughBeast
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Originally Posted by wogoga View Post
You confuse modern physics with a new religion or theology. Your argument is the old argument of theological, religious orthodoxy against philosophic and scientific progress:
"Once you start applying common-sense (i.e. consistent logical reasoning) to Holy Scriptures and theological dogmas, there's no telling what sort of trouble you can get yourself into."
Common sense has been defined as "the sum total of all the prejudices you formed before you were 16."

When you define it as "consistent logical reasoning", you leave out the unspoken subsequent clause ; "- consistent with the assumptions and conclusions derived from previous experience". In this case, that previous experience consists of non-relativistic phenomena.

Chief among these invalid assumptions is that of the linear addition of velocities, usually exemplified by the "sports car on a train" thought experiment. It's certainly in line with our everyday experience, but it simply isn't true. And it is obviously impossible for light to have a velocity which is independent of the relative velocities of different observers. Any such suggestion is contrary to all common sense.

And, as I said, that's where it all starts to go wrong for you.

"Eppur se muove."
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Old 2nd October 2015, 03:46 PM   #9
Darwin123
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Originally Posted by wogoga View Post
Here the question arises:
How can a ruler, contracted from a nominal length of 1 light year to 0.01 LY, be on the other hand expanded to 100 LY with respect to the same rest-frame length-units?
The answer is astonishingly simple:
In the noninertial reference frame of rocket while the thrust is nonzero, the ruler expanded more than a thousand fold. It can be shown using the FULL Lorentz transform that when the 'proper acceleration' of the rocket is 'a',

x' = x (1+ax/c^2)/sqrt(1-[v/c]^2)
where,
a=F/m
where F is t he thrust and m is the mass of the rocket.

The man in the rocket see the earth fall toward him. However, the distance the earth has to fall increases with 'a'. This is a 'length dilation' caused by the nonzero proper acceleration.


You could see the gravitational potential in this equation. The gravitational potential during turn around is equal to 'ax'.

According to my last count, you had six concrete errors before I posted this reply. You correct about three of these concrete errors leaving you with only three concrete errors. However, I just pointed out another concrete error.

4) The distances measured by an observer vary with the proper acceleration of the observer.

Hence, you are back to four concrete errors.

I recommend that you read that Wikipedia article on 'proper acceleration' very carefully. You keep on making concrete errors that confirm your first concrete error where 'acceleration is negligible.' Acceleration is not negligible if it is 'proper acceleration.'

Now would you please write down the total Lorentz transformation so that I know you are actually interested?
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Old 2nd October 2015, 04:51 PM   #10
Darwin123
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Originally Posted by Darwin123 View Post
In the noninertial reference frame of rocket while the thrust is nonzero, the ruler expanded more than a thousand fold. It can be shown using the FULL Lorentz transform that when the 'proper acceleration' of the rocket is 'a',

x' = x (1+ax/c^2)/sqrt(1-[v/c]^2)
where,
a=F/m
where F is t he thrust and m is the mass of the rocket.
Correction. These are increments of distance, not absolute distance.

x_2'-x_1'=[x_2-x_1] (1+ax_1/c^2)/sqrt(1-[v/c]^2)
where
a=F/m
and
2|x_2-x_1|<<|x_2+x_1|.

Again, this is easily derived using calculus.

However, in the case of a 'sudden' turn around.

Before,
x_A'=[X+vt]/sqrt(1-[v/c]^2)
After
x_B' = [X-vt]/sqrt(1-[v/c]^2)

x_A'-x_B' = [2vt]sqrt(1-[v/c]^2)

That is the approximately the extra distance that the 'earth' has to move in the reference frame of the twin in the rocket before the two twins meet. As 'v' increases, the distance increases.

The distances also change due to the proper acceleration. Einstein showed this in the case of orbit around the star, where the ratio of circumference to radius was not pi.

Proper acceleration of an observer affects both measurements of length and measurements of time in the noninertial frame of the observer. The proper acceleration is basically the force on the observer divided by the mass of the observer. Therefore, the measurements of space an time are effected by the FORCE on the observer.

You could have guessed this if you had paid attention to units when you read Einsteins paper. He talked about a 'fictitious gravitational field'. If you had checked the units, you would have seen that the units of this 'fictitious gravitational field' were [m/s^2]. In other words, they were units of acceleration.

The 'fictional gravitational field' is the negative of the 'proper acceleration'. Since Einstein does use 'the fictional gravitational field' in his theory, you can't say that he ignores 'proper acceleration'.

This is why I insist on some discussion of force when someone presents an 'SR paradox'. A reference frame is inertial ONLY if there is no force on the observer.
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