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

Saw

Maybe other answers say the same in a more technical manner, maybe not, I have not checked. In any case, I will share the explanation I usually give myself.

If observers measure different values for a given property, it is because they do it from different perspectives or, more technically, reference frames, that is to say, situations/circumstances which have an impact on the measurement process of the relevant property. This definition entails that observers should not disagree, however, if they come to measure another property with regard to which their circumstances are identical, that is to say, with respect to which their perspective or reference frame is the same.

Take the easy example of two persons standing on the ground and looking at each other from a distance. A sees B smaller and vice versa. That is because A looks at B from a distance and vice versa. They have different perspectives on each other, in this respect. But they do see the same distance between the the two of them, for they have the same perspective on this issue.

The same happens with length contraction. A and B have different states of motion, but only in one direction. Hence they have a different perspective in this respect, i.e. in terms of measuring this property, length in the direction of their relative motion, say the X axis. However, they do not have different states of motion in the Y axis, for example. In this respect they share the same perspective; for this purpose, they occupy the same reference frame; hence they measure the same values.

peterspencers

Ok so thankyou all for the help, I think I'm nearly there. I think I may have a clear understanding, is this a correct explination.....

If I have a light clock on a fast moving spacecraft being observed from earth, with a vertical set of mirrors (perpendicular to the motion of the overall clock) and a horizontal set of mirrors. The mirrors time the pulses of light, eminating from the same source. I then apply the equations for calculating the time it takes for the light pulses to bounce between each set of mirrors using the values of c = 10, v = 6 and l = 4.
I find initially that the horizontal bounces take 1.25 seconds and the vertical bounces take only 1 second. It's only when I apply the lorenz transformation to the length in the horizontal clock to give me new decreased value for l of 3.2, that I discover my time calculations for both mirrors agree on 1 second. I then conclude that the length must contract around the moving pulse of light to preserve its consistancy for all frames of reference.
Is this correct?

harrylin

Yes but it's insufficient: the Lorentz transformations state that there is no vertical contraction, so that is what you want to prove (or make plausible). You demonstrated that based on the starting assumptions (equal speed of light etc), the vertical contraction factor must differ from the horizontal one. However, 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.

peterspencers

How will my calculation show length contraction with, zero time dillation? My calculation shows that time on the ship (ts) would be 0.8 and time from say earth (te) would be 1. Also why would I want to make vertical contraction possible? The vertical clock dosent contract, the path the photon takes is extended, as per a little pythagoras, hence the time dilation.

ghwellsjr

I can't make sense of your setup. You haven't said what c, v and l are and you haven't said what their units are. Usually, we reserve the letter "c" to be the speed of light and "v" is a velocity. It's a little unusual for someone to set the speed of light to be 10, we usually make c = 1 to make the equations simpler. And you haven't said what equations you are using nor what l applies to.

Specifically, I can't figure out how you got 1 second for the vertical bounce.

harrylin

1. Your question is "Why does length contraction only occur parallel to the direction of motion?". Although you now suggest the contrary, I thought that you did not intend to discover what the Lorentz transformations state (contrary to valentin). As a matter of fact, they state that y=y' and z=z', so if that was your question and you did not want to prove what you assumed, then that was the answer and the end of this topic.

2. With zero time dilation instead of time dilation by a factor gamma, you can search if it is possible to obtain the same return times with your setup. Then you will find that this is possible if the length is decreased by a factor gamma square and the width and height by a factor gamma. If you don't get that, then you made a calculation error.

3. In my post I explained that that solution is nevertheless not an option if we want the PoR to hold, so that we assume that length contraction only occurs parallel to the direction of motion. Lorentz and Einstein gave other examples with the same conclusion.

peterspencers

I apologise for not showing my workings, here they are:

I have a light clock onboard a spacecraft moving past the earth parallel to an observer. The lightclock measures the time it takes for light waves to bounce between two mirriors. I have two sets of mirriors, 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)

Firstly I take a time measurement onboard the craft:

[itex]ts[/itex] = 2l/c = 0.8

Then from earth's reference frame I calculate the time by using the following two equations, for the vertical clock I use:


[itex]te[/itex] = [itex]\frac{ts}{√1-vv/cc}[/itex] (please excuse my writing vv/cc, I mean v²/c² however when I enter [sup[/sup] inside the fraction text it donsent seem to work :P im totally new to all this, trying to work it out as I go along, any help would be most kind)

[itex]te[/itex] = 1 second

Then for the horizontal clock:

[itex]te[/itex] = [itex]\frac{2lc}{cc-vv}[/itex] (apologies again the bottom half should read c²-v²)

[itex]te[/itex] = 1.25 seconds

.....clearly there is something wrong here, both clocks shold agree on the time.

So I apply the lorentz transformation to l in the horizontal clock:

[itex]Lo[/itex] = the proper length (the length between the mirrors in their rest frame)

[itex]L[/itex] = [itex]Lo[/itex]√1-v²/c²

and end up with a value of 3.2 for l in the horizontal clock....

so I go back to [itex]te[/itex] = [itex]\frac{2lc}{cc-vv}[/itex] (apologies again the bottom half should read c²-v²)

this time with 3.2 as my value for l, and then I get:

[itex]te[/itex] = 1 second

This means that the amount of distance the light can cover between each set of mirrors is equal, so even though the length is contracted in one clock, the 'light distance' is equal, as it is in both the rest frame and the moving frame.

Now both clocks agree and i'm very happy :) .... I hope!

Is this correct ??

peterspencers

Why is my above calculation insufficient to explain length contraction? I dont follow the reasoning here, please could someone explain (to a lamen) if the above quote really does apply to my above calculation.

harrylin

The Lorentz transformations tell you directly that there is no length contraction parallel to the direction of motion; to find that out you don't need to make such a calculation. However, 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".

- Your calculation is fine to show how length contraction is a necessary element of the Lorentz transformations, and how that works.
- Your calculation doesn't show why that is the only feasible possibility based on the postulates.

Once more: if you assume that there is no time dilation but all distances contract by an additional factor γ (so that the length contracts by γ² and the width by a factor γ), then your scenario will also work (the 'light distance' in both directions as measured with a clock is equal and the same in the rest frame and the moving frame).

Histspec

One can use the Robertson-Mansouri-Sexl test theory, which shows that three experiments are necessary to derive the following parameter of the Lorentz transformation:
[itex]\alpha[/itex] for time changes
[itex]\beta[/itex] for longitudinal length changes
[itex]\delta[/itex] for transverse length changes

The Michelson-Morley experiment measures the combination of [itex]\beta[/itex] and [itex]\delta[/itex].
The Kennedy-Thorndike experiment measures the combination of [itex]\alpha[/itex] and [itex]\beta[/itex].

In order to obtain the individual values, you have to measure one of those quantities directly.

For instance, the Ives-Stilwell experiment measures [itex]\alpha[/itex] in accordance with time dilation. Using this value of [itex]\alpha[/itex] with Kennedy-Thorndike shows that [itex]\beta[/itex] must be in accordance with relativistic length contraction in longitudinal direction; combining this value of [itex]\beta[/itex] with Michelson-Morley shows that [itex]\delta[/itex] must be zero.

Therefore, length contraction in transverse direction is experimentally excluded - only relativistic length contraction in longitudinal direction is allowed.

peterspencers

In my scenario I have a rest observation and a moving one. Where the clock is being observed in motion, velocity is an obvious consideration. As soon as velocity is acting upon the clock the equations for calculating the time change and time dilation becomes an inevitable and unavoidable factor. This in turn leads to the nesicarry appliance of length contraction along the horizontal axis to then preserve equal light time between both sets of mirriors in both frames of reference.

I can only assume no time dilation in the clocks rest frame, in this instance all the prerequisites are met without the need to contract any lengths.

It sounds like you are describing a totally different situation, and have misinterpreted my scenario, in which case I apologise for my poor initial explanation. If I am still mistaken then please could you describe step by step using mathematics and my initial values, the situation you are describing. Many thanks

harrylin

I agree with that; however, based on what fact or assumption do you make that claim?

I clarified how you can get the same result from your scenario, based on the assumption of no time dilation (if you ask me to show how exactly, I'll gladly do that later; it seems rather obvious to me). The related transformations are of course different from the Lorentz transformations, according to which y'=y. Thus, please clarify if you agree with the part that you did not cite. What exactly did you intend with your "why" question?

peterspencers

Your explanation still dose not explain how 'assuming zero time dilation' is possible when observing my moving light clock.

My scenario is one where we are making observations of a moving clock, therefore time dilation is an intrinsic part.

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

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.

And, my initial 'why' question, I believe, has been answered through my calculations. This has also been verified in another thread (which you have picked up) so I am now attempting to understand your claim that my scenario is insufficient to completely describe the answer to my initial 'why' question.

harrylin

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.
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 explination. 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....

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

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!).
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.
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:

[itex]ts[/itex] = 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:

[itex]te[/itex] = [itex]\frac{ts}{√1-v^2/c^2}[/itex]

[itex]te[/itex] = 1 second

Then for the horizontal clock:

[itex]te[/itex] = [itex]\frac{2lc}{c^2-v^2}[/itex]

[itex]te[/itex] = 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:

[itex]Lo[/itex] = the proper length (the length between the mirrors in their rest frame)

[itex]L_h[/itex] = [itex]Lo[/itex](1-v²/c²)

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

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

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

[itex]te[/itex] = 1 second

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

[itex]L_v[/itex] = [itex]Lo[/itex](√1-v²/c²)

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

[itex]te[/itex] = 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...ron_%28June%29

vin300

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