Why does length contraction only occur parallel to the direction of motion?

[harrylin]

In my scenario 'assuming no time dilation' is not an option! My scenario is one where we are making observations of a moving clock, therefore time dilation is an intrinsic part.

Then I may have misunderstood your scenario, as I think that you describe two systems in which clocks are observed that are in rest in each system. Please check.

Please can you show me your mathematical workings step by step using my initial values for l, c and v. Also an explanation of how 'assuming zero time dilation' is possible in a situation where we are observing a moving clock. [..]

Yes, I will, as I promised, after you reply my repeated question to you so as to be sure that we are "tuned to the same frequency". So, for the third time: please clarify if you agree with the part that you did not cite, here it is again:

Your question was why length contraction only occurs parallel to the direction of motion.
I interpreted your question as "why are the Lorentz transformations the only feasible solution", while you seem to have meant "how do the Lorentz transformation work". Correct?

 

[peterspencers]

I believe that my initial 'why' question has been answered through my calculations, this has been confirmed to me by someone else in another post. I am now attempting to vindicate that explanation. My question may have been indirectly about the Lorentz transformation, I am still only aware of what it is and how it works so much as my scenario and calculations betray.

I am fascinated at the thought of there being other ways to mathematically and intuitively describe and explain the reasons for length contraction. Before I begin along that line of questioning however I must first clarify if my current understanding (based upon my scenario) is sufficient. As you have stated that it is not...

Then I may have misunderstood your scenario, as I think that you describe two systems in which clocks are observed that are in rest in each system. Please check.

No my scenario does not describe 2 clocks that are at rest, why would I have included v (velocity) in my equations if that were the case?

I hope I have answered you questions sufficiently, I must ask again, in light of my attempt at clarification... if my scenario and calculations are sufficient to explain length contraction and time dilation?

 

[harrylin]

[..] I am fascinated at the thought of there being other ways to mathematically and intuitively describe and explain the reasons for length contraction. Before I begin along that line of questioning however I must first clarify if my current understanding (based upon my scenario) is sufficient. As you have stated that it is not... [...]

It's sufficient to understand how the Lorentz transformations work. It's not sufficient to determine that for relativity the Lorentz transformations are the only solution. And apart of the relativity principle there is also a physical reason for length contraction (yes we could discuss that later!).

No my scenario does not describe 2 clocks that are at rest, why would I have included v (velocity) in my equations if that were the case?

I did not think that you describe 2 clocks that are in rest, so probably I did understand your scenario which looks to me the "standard" one.

I hope I have answered you questions sufficiently, I must ask again, in light of my attempt at clarification... if my scenario and calculations are sufficient to explain length contraction and time dilation?

Yes that's sufficient to get a basic understanding of how length contraction and time dilation work. And here's your calculation redone for an alternative solution* (I simply copy-pasted from you with a few modifications) :

A light clock onboard a spacecraft moves past the Earth parallel to an observer. The lightclock measures the time it takes for light waves to bounce between two mirrors. There are two sets of mirrors, one on the vertical axis perpendicular to the direction of travel and also a horizontal set, in line with the direction of travel.

c = 10 m/s (speed of light)
v = 6 m/s (velocity)
l = 4 m ( length between two mirrors)

A time measurement onboard the craft should yield according to the craft's reference frame:

ts = 2l/c = 0.8 s

Then from Earth's reference frame we calculate the time (uncorrected for relativity) by using the following two equations, for the vertical clock we use:

te = \frac{ts}{√1-v^2/c^2}

te = 1 second

Then for the horizontal clock:

te = \frac{2lc}{c^2-v^2}

te = 1.25 seconds

...clearly there is something wrong here, both clocks should agree on the time in both directions.

We could for example propose a length transformation by a factor γ² in the horizontal clock:

Lo = the proper length (the length between the mirrors in their rest frame)

L_h = Lo(1-v²/c²)

and end up with a value of 2.56 for L_h in the horizontal clock...

so we go back to te = \frac{2lc}{c^2-v^2}

this time with 2.56 as our value for L_h, and then we get:

te = 1 second

Similarly for the vertical clock we can propose a length transformation by a factor γ:

L_v = Lo(√1-v²/c²)

now with 3.2 as our value for L_v, and then we get:

te = 1 second

This means that the 'light distance' in both directions as measured with a clock is equal and the same in the rest frame and the moving frame. And this solution works with a time dilation factor of 1 (no time dilation).*based on that scenario there's in fact an infinite number of solutions, indicated with the multiplication factor l; see equation 1 of http://en.wikisource.org/wiki/On_the_Dynamics_of_the_Electron_%28June%29

 

[vin300]

Then for the horizontal clock:

te = \frac{2lc}{c^2-v^2}

te = 1.25 seconds

There's been a mistake here. The formula I found for te is te = \frac{2l}{\sqrt{c^2-v^2}}. It gives te = 1 second for this case.

 

[harrylin]

There's been a mistake here. The formula I found for te is te = \frac{2l}{\sqrt{c^2-v^2}}. It gives te = 1 second for this case.

That can hardly be right, even when correcting for a typo in your equation: MMX would not have been performed with equal round times for equal lengths. Peterspencer's equation (which I copied from his post #24) is easy to derive, see https://en.wikisource.org/wiki/On_the_Relative_Motion_of_the_Earth_and_the_Luminiferous_Ether, page 336:
T + T_/ = 2D \frac{V}{V^2-v^2}

 

[vin300]

My derivation was similar to that except that I used the contracted distance instead of D. Wouldn't the horizontal light clock be length-contracted? l=4m is its proper length, right?

 

[harrylin]

My derivation was similar to that except that I used the contracted distance instead of D. Wouldn't the horizontal light clock be length-contracted? l=4m is its proper length, right?

Sure - please read my post again! I had even added "(uncorrected for relativity)" to my copy of peterspencer in order to clarify that he calculated for the un-contracted distance. And even then, it may be difficult to understand my post without reading the discussion starting from post #24.

 

[Histspec]

The general relation for length changes, by which horizontal (longitudinal) and vertical (transverse) travel times become the same in this light-clock experiment, is
\frac{L_{l}}{\gamma\phi}\ :\ \frac{L_{t}}{\phi}
with \gamma=1/\sqrt{1-v^{2}/c^{2}}.

Though \phi can be chosen at will. Some examples:

\phi=1 gives the standard expression for length contraction (peterspencers case).
\phi=\gamma. No time dilation, but length contraction is squared (Harrylin's case).
\phi=1/\gamma. No length contraction, but the vertical length and thus time dilation is elongated (equivalent to the "Voigt transformation").

 

[harrylin]

The general relation for length changes, by which horizontal (longitudinal) and vertical (transverse) travel times become the same in this light-clock experiment, is
\frac{L_{l}}{\gamma\phi}\ :\ \frac{L_{t}}{\phi}
with \gamma=1/\sqrt{1-v^{2}/c^{2}}.

Though \phi can be chosen at will. Some examples: [..]

Yes, your ∅ is the same as Poincare's l (see bottom of post #33). Thanks for that nice elaboration.

 

[Histspec]

Yes, your ∅ is the same as Poincare's l (see bottom of post #33). Thanks for that nice elaboration.

Or the same as Einstein's \phi(v) in §3 of his 1905 paper.
http://www.fourmilab.ch/etexts/einstein/specrel/www/

See also Lorentz's 1904 paper, on page 822: \left(\frac{1}{kl},\ \frac{1}{l},\ \frac{1}{l}\right)
http://en.wikisource.org/wiki/Electromagnetic_phenomena

Or Wiki:
http://en.wikipedia.org/wiki/Michel...Length_contraction_and_Lorentz_transformation

 

[peterspencers]

ok... many thanks harrylin for taking the time to explain that too me :) I think I finally understand what you mean.

Also thanks to Histspec, I think you may have just answered my next question! I may need to ask for an intuitive explanation of exactly what you have done here.

I'd just quickly like to clear up something first in regards to harrylin's explanation. Firstly am I right in concluding that what you have just shown me tells us that my scenario is not a complete 'proof' of length contraction, time dilation? This being the case because within my example's set parameters, we are able to satisfy all of lights prerequisites with an infinite number of different length contractions available in both axis and in the presence of zero time dilation.

If my conclusion is correct then I would like to offer the following idea that I feel vindicates my example as an adequate proof of TD, LC...

I believe there are a number of unstated precepts, or prerequisites of nature, ontop of the ones already stated in regards to light, that your counter argument does not satisfy, but my original calculations do satisfy. These are...

Firstly, if we were to contract length on the vertical axis, then the example I have been given earlier, with the train and the tunnel; where in one frame of reference there is a crash, and in another no crash. Then we would be violating the continuity of reality. This seems to be a fairly solid precept of nature, ones equations must ensure that events in all frames of reference agree. (even if they don't agree on the order or timing of events, as shown in another train thought experiment where einstein shows how relativity can break the simultaneity of events) if something happens in one reference frame, then it must at some point also happen in another.

Secondly, If we are to contract length, then we must also dilate time and in a related fashion. Space, time and speed (as given in length, velocity and time in our example) are the way in which we understand movement. Without movement there is no time and space; and vise versa. If we imagined a universe where there was no movement whatsoever, there would be no time or space. As soon as something moves, time and space are born, so much as to say that they are all parts of the same whole. Speed = distance/time, Time = distance x speed. These basic high school equations follow this precept, if we alter the value of one, then the rest must reflect this; how can it be that other equations describing nature shouldn't have to follow this principle?

I appreciate that this is perhaps a little philosophical, although I do feel that the examples I have given are backed up as being 'necessary precepts to accurately describe nature' in other proven areas of physics.

With this in mind, I feel that within the set scenario and parameters of my example, the equations I have presented of LC, TD are indeed the only solution to satisfy all of natures (currently known) mandatory precepts.

 

[harrylin]

ok... many thanks harrylin for taking the time to explain that too me :) I think I finally understand what you mean.

Also thanks to Histspec, I think you may have just answered my next question! I may need to ask for an intuitive explanation of exactly what you have done here.

I'd just quickly like to clear up something first in regards to harrylin's explanation. Firstly am I right in concluding that what you have just shown me tells us that my scenario is not a complete 'proof' of length contraction, time dilation? [..]

Exactly - that was the point that I made in post #20.

If my conclusion is correct then I would like to offer the following idea that I feel vindicates my example as an adequate proof of TD, LC...

I believe there are a number of unstated precepts, or prerequisites of nature, ontop of the ones already stated in regards to light, that your counter argument does not satisfy, but my original calculations do satisfy. These are...

Firstly, if we were to contract length on the vertical axis, then the example I have been given earlier, with the train and the tunnel; where in one frame of reference there is a crash, and in another no crash. [..]

Yes, that's what we discussed in posts #4 #5 #20; so now you understand what we meant.

Secondly, If we are to contract length, then we must also dilate time and in a related fashion. Space, time and speed (as given in length, velocity and time in our example) are the way in which we understand movement. [...] if we alter the value of one, then the rest must reflect this; how can it be that other equations describing nature shouldn't have to follow this principle?
I appreciate that this is perhaps a little philosophical [..]

That one is indeed too philosophical for me.

With this in mind, I feel that within the set scenario and parameters of my example, the equations I have presented of LC, TD are indeed the only solution to satisfy all of natures (currently known) mandatory precepts.

All of our perceived "nature's mandatory precepts" are simply based on observation, and however unlikely, there still remain a few insufficiently tested hypotheses - future surprises are not ruled out.

 

[phyti]

ok... many thanks harrylin for taking the time to explain that too me :) I think I finally understand what you mean.

Firstly am I right in concluding that what you have just shown me tells us that my scenario is not a complete 'proof' of length contraction, time dilation? This being the case because within my example's set parameters, we are able to satisfy all of lights prerequisites with an infinite number of different length contractions available in both axis and in the presence of zero time dilation.

If my conclusion is correct then I would like to offer the following idea that I feel vindicates my example as an adequate proof of TD, LC...

Your looking through the wrong end of the telescope!

1. Your scenario is not a proof, but a theoretical demonstration of how/why these phenomona occur.
2. Light does not have prerequisites. Time dilation (td) and length contraction (lc) result from motion of mass and a constant & independent speed of light. If light speed added to object speeds in the manner of vectors, there would be no td and lc.
3. The 'infinite' ways of lc is just a gimmick called scaling, useless and redundant.
If you examine the equations, you should see lc & td are functions of v/c, i.e. the ratio of object speed to light speed. The range of v/c is (0 to 1). The corresponding range of gamma is (1 to *unbounded). (*I avoid the useless term infinity). The important relation is, a one to one correspondence for every gamma value to every v/c value. There is no necessity for higher orders of gamma. There is also the em field responsible for lc which only requires one application of gamma.

If we imagined a universe where there was no movement whatsoever, there would be no time or space.

...or anyone to think about it!

As soon as something moves, time and space are born, so much as to say that they are all parts of the same whole. Speed = distance/time, Time = distance x speed. These basic high school equations follow this precept, if we alter the value of one, then the rest must reflect this; how can it be that other equations describing nature shouldn't have to follow this principle?

We are aware of many things that aren't moving and for variable time intervals.

I appreciate that this is perhaps a little philosophical, although I do feel that the examples I have given are backed up as being 'necessary precepts to accurately describe nature' in other proven areas of physics.

With this in mind, I feel that within the set scenario and parameters of my example, the equations I have presented of LC, TD are indeed the only solution to satisfy all of natures (currently known) mandatory precepts.

What about quantum physics?

your post 30:

In answer to your questions, I make the claim of TD being unavoidable where the clock is moving based on the fact that the motion of the overall clock increases the path the photon travels to complete one bounce between both mirrors. As the speed of light is constant this action theifore takes longer than it does in the rest frame, hence the time dilation.

If you extend this idea to atoms (nuclei embedded in elctron clouds), the em interactions are less frequent, thus weaker. This allows atoms closer separations, but they need relative motion to close the ranks, and it's the same motion that causes the em field deformation. You will have answered your original question.

 

[harrylin]

[..] I would like to offer the following idea that I feel vindicates my example as an adequate proof of TD, LC...[..]

[..] Your scenario is not a proof, but a theoretical demonstration of how/why these phenomona occur.

Thanks for catching that one - I overlooked that that logical error still continued.

 

[ghwellsjr]

Firstly, if we were to contract length on the vertical axis, then the example I have been given earlier, with the train and the tunnel; where in one frame of reference there is a crash, and in another no crash. Then we would be violating the continuity of reality. This seems to be a fairly solid precept of nature, ones equations must ensure that events in all frames of reference agree. (even if they don't agree on the order or timing of events, as shown in another train thought experiment where einstein shows how relativity can break the simultaneity of events) if something happens in one reference frame, then it must at some point also happen in another.

A frame is a man-made concept. It doesn't exist in nature. It allows us to put meaning into nature that otherwise wouldn't exist. That is the assignment of coordinates to events. The relationship between the coordinates in one frame and another frame is purely a mathematical exercise. Things happen in nature. How we describe things is based on our definitions. Einstein's Special Relativity provides us with a simple and meaningful way to do this. You should not think of different frames as causing different things to happen.

Secondly, If we are to contract length, then we must also dilate time and in a related fashion. Space, time and speed (as given in length, velocity and time in our example) are the way in which we understand movement. Without movement there is no time and space; and vise versa. If we imagined a universe where there was no movement whatsoever, there would be no time or space. As soon as something moves, time and space are born, so much as to say that they are all parts of the same whole. Speed = distance/time, Time = distance x speed. These basic high school equations follow this precept, if we alter the value of one, then the rest must reflect this; how can it be that other equations describing nature shouldn't have to follow this principle?

Not in the high school I went to. If Speed = distance/time then Time = distance/speed not distance x speed. Again, I get the impression that you think nature is forced to follow our mathematical description of it but it's the other way around. We are forced to find a mathematical description that follows what we can measure and observe of nature. That's what the enterprise of science is all about and it's very hard work. Of course, once Einstein did the very hard work of discovering a way to define and thus to describe nature mathematically, the rest of us can jump right into his simple theory of Special Relativity. It might be hard to find the needle in the haystack but once its location is known, it's easy for others to locate it.

 

[peterspencers]

A frame is a man-made concept. It doesn't exist in nature. It allows us to put meaning into nature that otherwise wouldn't exist. That is the assignment of coordinates to events. The relationship between the coordinates in one frame and another frame is purely a mathematical exercise. Things happen in nature. How we describe things is based on our definitions. Einstein's Special Relativity provides us with a simple and meaningful way to do this.

I understand that the frame is our viewpoint and isn't a 'thing' in itself.

You should not think of different frames as causing different things to happen.

I dont, why do you think I did?

Not in the high school I went to. If Speed = distance/time then Time = distance/speed not distance x speed.

Yes, I feel stupid.

Again, I get the impression that you think nature is forced to follow our mathematical description of it but it's the other way around. We are forced to find a mathematical description that follows what we can measure and observe of nature.

Yes I can see how I've given this impression, I do see what you mean.
In my last post I was simply trying to say that the equations I posted to explain the light clock example seem to not contradict any of our current explinations of nature. I was attempting to show how, the counter argument posted by harrylin about 'scaling' seemed irrelevant at rendering the light clock example insufficient to explain lc, td.

The 'infinite' ways of lc is just a gimmick called scaling, useless and redundant.

(posted by 'phiti')

As far as I can tell the light clock example is completely sufficient to explain lc, td ( based on what we currently think we know about nature, and yes I know that still dosent make it a complete 'proof'!). unless anybody has a different reason why it isnt?

 

[ghwellsjr]

Peterspencers, do you realize that if you follow Einstein's rules for making a coordinate system in one frame, then it is simply math that gets you the corresponding coordinates in any other frame using the Lorentz Transformation process. Your sentence, "This seems to be a fairly solid precept of nature, ones equations must ensure that events in all frames of reference agree" imlies that it's something in nature that causes the consistency between frames rather than mere math.

 

[harrylin]

[..] In my last post I was simply trying to say that the equations I posted to explain the light clock example seem to not contradict any of our current explinations of nature. I was attempting to show how, the counter argument posted by harrylin about 'scaling' seemed irrelevant at rendering the light clock example insufficient to explain lc, td. [..]

Before that post of yours, I already elaborated on my agreement with that (post #26).

My counter argument was merely to show that your light clock example is insufficient to "conclude that the length must contract around the moving pulse of light to preserve its consistency for all frames of reference". (emphasis mine)

 

[peterspencers]

imlies that it's something in nature that causes the consistency between frames rather than mere math.

I was trying to say that a train cannot, crash in one reference frame and not crash in another, the 'thing' we are labeling as a crash is still 'thing' regardless of how we measure it through mathematics. Is not the case as you see it?

 

[peterspencers]

My counter argument was merely to show that your light clock example is insufficient to "conclude that the length must contract around the moving pulse of light to preserve its consistency for all frames of reference". (emphasis mine)

I can't see how the scaling argument has any value. Dosent the 'train crashing and not crashing' argument rule out scaling as a valid counter argument, leaving only the Lorentz transformations as the 'must' contraction?

 

[zonde]

I can't see how the scaling argument has any value. Dosent the 'train crashing and not crashing' argument rule out scaling as a valid counter argument, leaving only the Lorentz transformations as the 'must' contraction?

You seem to miss important point about this scaling function - it is not frame independent. If you require that transformation is symmetric (you invoke principle of relativity) then scaling function is ruled out.
Speaking about 'train crashing and not crashing' argument, it can be solved without contradictions if you scale down when you perform transformation in one direction and scale up when you perform reverse transformation.
So the point is that you have to invoke physical principle i.e. principle of relativity to get to the Lorentz transformation.

 

[ghwellsjr]

I was trying to say that a train cannot, crash in one reference frame and not crash in another, the 'thing' we are labeling as a crash is still 'thing' regardless of how we measure it through mathematics. Is not the case as you see it?

If you are saying that an event (a crash) exists in all frames, then yes, that is correct. But then all events, even ones where nothing "happens" beyond a particular time at a particular place, exist in all frames.

Look, it's like saying that every temperature on the Fahrenheit scale exists on the Centigrade scale or like saying that every weight on the Imperial scale exists on the Metric scale. Frames are just coordinates that we use to pinpoint events, they can't change or have any influence on what is actually happening. Would you say that if you measure a temperature in Fahrenheit and then you calculate the temperature in Centigrade that you have performed two measurements?

 

[peterspencers]

Rightyho so your saying reality is the way it is, and we can superimpose an infinite number of names, measurements and interpretations onto it but this dosent change it. So in answer, we would be measuring once? Also would you say reality has one true set of principles that we may one day discover? Or would you say reality is so infinite, in every sence, that understanding will always be an illusion?

 

[harrylin]

I can't see how the scaling argument has any value. Dosent the 'train crashing and not crashing' argument rule out scaling as a valid counter argument, leaving only the Lorentz transformations as the 'must' contraction?

Yes, in post #20 I explained that on order to conclude "must", you need to add such an argument; your light clock calculation is not sufficient for that. Once more (I copy-paste):

"you could assume, for example, that the vertical contraction is gamma and the horizontal contraction gamma square. Then with zero time dilation your calculation will also work.

However (in addition to other examples already given), imagine that two identical high objects collide; from SR symmetry they should have identical damage. Or alternatively, imagine a very fast bullet going through a narrow tube; it must not be possible to know which one "moves absolutely faster", and neither can it be that the bullet is smaller than the tube and also bigger than the tube when it passes through, so that a collision happens and also doesn't happen."

Therefore, in order for the two postulates to hold, the Lorentz transformations appear to be the only solution.

But as I and others have explained this in so many posts, and you seem to understand this, I won't try to explain it again!

 

[ghwellsjr]

Rightyho so your saying reality is the way it is, and we can superimpose an infinite number of names, measurements and interpretations onto it but this dosent change it. So in answer, we would be measuring once?

The basic problem is that once light has left you, you cannot tell when it arrives at any particular location remote from you because you have nothing faster than light with which to track its progress. The best we can do is have the light reflect off of a distant object and see when it gets there but we are not really seeing when it gets there, we are seeing the total round-trip time that it takes for the light to get to the distant object and for the reflection to get back to us.

Whenever we do this, we get the same value of time for the round-trip for any given measured distance. The time will always be twice the distance divided by c. This is a measurement. But we cannot know if the light spent the same amount of time getting to the remote object as it did to get back. For all we know, it could have spent 1/4 of the total time getting there and 3/4 of the total time getting back. Or any other pair of ratios that add up to one. Nature will not reveal to us the answer to this question.

Now you might think that the issue could be resolved by taking a second clock identical to the one that we are using to make our measurement and after making sure they read exactly the same time, move it to the remote location. The problem is that if we then bring it back, the two clocks will have a different time on them and the difference is larger the faster we move the clock.

But then you might think that if we move it slow enough, we can minimize the difference between the two clocks (after we bring the second clock back), which is true but we still don't know if when we get the clock to the remote location, which will take a certain amount of time as measured on the moving clock, that it takes the same amount of time as measured on the moving clock, to bring it back.

Einstein showed us the way out of this dilemma. We simply define time on the remote clock such that the light takes the same amount to time to get to the remote object as it does to get back. So after adjusting our remote clock so that this is true, if we then measure when the light gets to the remote object, are we making a measurement, or simply reading back the time we previously put there? This process is the basis of defining an inertial Reference Frame in Special Relativity by expanding the process to an infinite number of imaginary clocks located at every position throughout space.

Now if someone else, moving inertially with respect to us follows the same process with his own set of rulers and clocks and we each "measure" the time and location of an event remote from both of us, we can get different answers but do you consider this difference to be somehow a difference in reality or even a difference in measurement, or rather merely a difference in definition?

Also would you say reality has one true set of principles that we may one day discover?

No, we've already discovered the one true set of principles.

Or would you say reality is so infinite, in every sence, that understanding will always be an illusion?

No, we have an excellent and concrete way of understanding it.

 

[peterspencers]

Yes, in post #20 I explained that on order to conclude "must", you need to add such an argument; your light clock calculation is not sufficient for that. Once more (I copy-paste):

Thankyou harrylin I finally see why this argument needs to be added :) The example on its own dosent encompass the equivalence principle and you need this to show why the Lorentz transformations are the only ones that fit the bill.

Thankyou for your patience in explaining that too me, I am incredibly grateful.

 

[harrylin]

Thankyou harrylin I finally see why this argument needs to be added :) The example on its own dosent encompass the equivalence principle and you need this to show why the Lorentz transformations are the only ones that fit the bill.

Thankyou for your patience in explaining that too me, I am incredibly grateful.

You're welcome!

Note that "equivalence principle" is something else (related to general relativity); here we discussed the relativity principle of special relativity.