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

• peterspencers
In summary, the reason space time does not contract uniformly in every direction around a fast moving object is because velocity is a vector quantity and any contraction due to velocity cannot be perpendicular to the velocity. This is because lengths only need to contract along the direction of motion to preserve the speed of light. Additionally, length contraction is necessary to maintain a constant speed of light in all inertial frames of reference. This is supported by a thought experiment involving a train and a wall, where the only way to resolve a paradox is if there is no contraction in the perpendicular directions. This is also consistent with the axiomatization of special relativity as a theory of space, time, and causality.
peterspencers
why does space time not contract uniformly in every direction around a fast moving object?

I guess the real question is "why would you think it should contract in all directions?" The fact that there is contraction at all is an unexpected (by classical physics) result that is forced on us by experimental results. And those experimental results show a contraction only in the direction of motion.

The simplest answer to your question is that velocity is a vector quantity. Any contraction due to velocity couldn't very well be perpendicular to the velocity because there is no velocity in that direction.

The reason length contraction occurs in the first place is to preserve a constant speed of light for all inertial frames of reference. If an observer says that you're moving in only the x direction, then lengths only need to contract along that direction for you to preserve the speed of light. Since you have no motion in the y or z direction, no length contraction is needed in the perpendicular directions.

There's also a very simple thought experiment that can be used to show that length contraction can't happen in directions perpendicular to the direction of motion.

Suppose a train is moving alongside a vertical wall that has a blue horizontal line painted on it. The blue line is painted to be at the (rest) height of the centres of the train's windows, as measured the in wall frame. (Let's say we have an observer at rest in the wall's frame who drew the blue line in advance based on what the train's blueprints said the height of the windows was).

An observer on the train has a paint brush and a can of red paint. He sticks his hand out the centre of a window and touches the brush to the wall, so that a red horizontal line is drawn as the train moves forward.

Suppose length contraction in the vertical direction *did* happen? Then the train observer claims that the wall is in motion, and would see vertical distances on the wall as being shorter than they appear to the wall observer. As a result, the blue line, which according to the wall observer, is as high as the centreline of the windows, would appear lower than this height to the train observer. So, the prediction is that the red line drawn by the train observer would be parallel to, but above the blue line that was drawn by an observer stationary w.r.t. to the wall.

But if we repeat this reasoning using the logic of the wall observer, we get a different answer. The wall observer claims that the train is in motion, and therefore vertical distances on the train appear shortened to him. In particular, the vertical height of the centreline of the moving train's windows appears shorter to him than what was claimed in the train's blueprints. So, this observer predicts that the red line drawn by the train observer will be parallel to, but below the blue line.

So, we have a logical contradiction, one that no amount of juggling of reference frames can resolve. At the end of the day, either the red line has to be above the blue line, or the blue line has to be above the red line. The only way to resolve this paradox is if the amount of vertical length contraction is 0, and therefore the red line lies directly on top of the blue line, according to both observers.

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weezy
I like cepheid's argument, because it's rigorous and also conceptually simple. My only minor criticism is that it depends on a symmetry principle that wasn't invoked explicitly in #4. A's velocity relative to B and B's velocity relative to A point in opposite directions. It's possible that one of these directions produces contraction, and one expansion. To rule this out, we need to assume that space is isotropic.

I would have to look more carefully, but cepheid's argument may be the same as the "nails on rulers" argument given here: https://www.physicsforums.com/showthread.php?p=2108296#post2108296

Another argument is the following. In 1+1 dimensions, one can prove straightforwardly that Lorentz transformations must preserve area. (For a proof, see http://www.lightandmatter.com/html_books/0sn/ch07/ch07.html#Section7.2 , caption to figure j.) By a similar argument, Lorentz transformations in 2+1 dimensions must preserve volume. The only way that both of these can be true is if lengths in the transverse direction are preserved.

HallsofIvy said:
The simplest answer to your question is that velocity is a vector quantity. Any contraction due to velocity couldn't very well be perpendicular to the velocity because there is no velocity in that direction.

I don't buy this at all. The electric field is a vector, but under a Lorentz boost, its component perpendicular to the boost can certainly change.

Mark M said:
The reason length contraction occurs in the first place is to preserve a constant speed of light for all inertial frames of reference. If an observer says that you're moving in only the x direction, then lengths only need to contract along that direction for you to preserve the speed of light. Since you have no motion in the y or z direction, no length contraction is needed in the perpendicular directions.

IMO this is logically backwards, since Einstein's 1905 axiomatization of SR is clearly a mistake, with the benefit of 107 years of historical hindsight. We see SR now as a theory of space, time, and causality, in which light plays no central role. More appropriate axiomatizations have been known since 1911; see our FAQ: https://www.physicsforums.com/showthread.php?t=534862

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bcrowell said:
I like cepheid's argument, because it's rigorous and also conceptually simple.

Thanks! I wish I could say that I came up with it myself. I related it from memory (i.e. I understand the argument, so I can recount it myself), but it was something I read in Introduction to Electrodynamics by David J. Griffiths (and he explains it far less verbosely). He, in turn, says in the book that he adapted it from Spacetime Physics by Taylor and Wheeler.

bcrowell said:
My only minor criticism is that it depends on a symmetry principle that wasn't invoked explicitly in #4. A's velocity relative to B and B's velocity relative to A point in opposite directions. It's possible that one of these directions produces contraction, and one expansion. To rule this out, we need to assume that space is isotropic.

Interesting, I hadn't thought about that. What do you mean by "space is isotropic?" In this case it sounds like you are saying that, "the laws of physics are the same regardless of what direction you're moving in." Is that basically it?

bcrowell said:
I would have to look more carefully, but cepheid's argument may be the same as the "nails on rulers" argument given here: https://www.physicsforums.com/showthread.php?p=2108296#post2108296

I think that might be the same idea, yeah.

bcrowell said:
Another argument is the following. In 1+1 dimensions, one can prove straightforwardly that Lorentz transformations must preserve area. (For a proof, see http://www.lightandmatter.com/html_books/0sn/ch07/ch07.html#Section7.2 , caption to figure j.) By a similar argument, Lorentz transformations in 2+1 dimensions must preserve volume. The only way that both of these can be true is if lengths in the transverse direction are preserved.

That's a cool link! I read the section that you were referring to, and I like the way they just used reasoning from the five postulates to arrive at the necessary geometric properties of the transformation. (EDIT: "They" being you, I gather).
bcrowell said:
IMO this is logically backwards, since Einstein's 1905 axiomatization of SR is clearly a mistake, with the benefit of 107 years of historical hindsight. We see SR now as a theory of space, time, and causality, in which light plays no central role. More appropriate axiomatizations have been known since 1911; see our FAQ: https://www.physicsforums.com/showthread.php?t=534862
Hmmm, interesting. I don't think I have any problem with Einstein's original axiomatization, but even with his original axiomatization, it seems that MarkM's argument is backwards as you suggest (no offence intended), because the constancy of the speed of light is assumed, and then length contraction is derived as a consequence of it, not the other way around.

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bcrowell said:
I don't buy this at all. The electric field is a vector, but under a Lorentz boost, its component perpendicular to the boost can certainly change.

Eh, misconception. The electric field is not really a vector. Its transformation follows from the full transformation of $F_{\mu \nu}$--i.e., from the transformations of the tx, ty, and tz planes. The electric field is a field of planes, and those planes' transformations are entirely in agreement with what you expect by boosting the individual vectors that span them.

Muphrid said:
Eh, misconception. The electric field is not really a vector. Its transformation follows from the full transformation of $F_{\mu \nu}$--i.e., from the transformations of the tx, ty, and tz planes. The electric field is a field of planes, and those planes' transformations are entirely in agreement with what you expect by boosting the individual vectors that span them.
You mean then the EM field is not really a vector. Not the electric field.

I mean, yeah, you can keep calling the electric field a vector, but then you keep having to remember, well, the ways that it isn't one. Just calling a rabbit a rabbit in the first place is cleaner than calling it a duck first.

Muphrid said:
Eh, misconception. The electric field is not really a vector. Its transformation follows from the full transformation of $F_{\mu \nu}$--i.e., from the transformations of the tx, ty, and tz planes. The electric field is a field of planes, and those planes' transformations are entirely in agreement with what you expect by boosting the individual vectors that span them.

Everything you say is true, depending on one's notion of a vector. There are really two definitions of a vector that are commonly used: (A) the definition of a 3-vector from freshman mechanics, and (B) the definition of a 4-vector from SR. The electric field fits definition A but not definition B. However, the argument given in #2 doesn't make use of any specific properties of the B definition as opposed to the A definition, and the electric field is a counterexample under the A definition, so the argument can't be correct.

Another point to make about #3 is that most people these days are introduced to SR through the pedagogical device of the light clock. In the light clock argument, a necessary assumption is that there is no transverse length contraction. If we admit the possibility of transverse length contraction, then the result of the light-clock argument is underdetermined. You really need some other argument, such as #4, to make the light clock derivation logically complete.

cepheid said:
Interesting, I hadn't thought about that. What do you mean by "space is isotropic?" In this case it sounds like you are saying that, "the laws of physics are the same regardless of what direction you're moving in." Is that basically it?
I asked about this symmetry assumption some time ago and at the end I was convinced that it is principle of relativity and nothing else.
You can look here - https://www.physicsforums.com/showthread.php?t=544580

Isotropy means all directions in spacetime are equivalent. It captures both the principle of relativity and the usual idea that in space alone there is no preferred direction. (In spacetime, a "direction" can also mean a timelike direction, so the two concepts are unified into one.) Isotropy means that (in SR) we live in a vector space in which we are free to choose a basis, and any choice of basis should give equivalent results to all others.

Homogeneity is a related concept, but it means that there is no preferred origin or center of spacetime. This tells us that only intervals between spacetime locations (or quantities derived from such intervals) are physically meaningful, so while we can choose a coordinate origin, this choice should not have physical consequences. It can only be a matter of convenience.

Isotropy can give some insight into boosts. Isotropy is what gives us the ability to rotate and boost coordinate systems without changing physically significant quantities. Pick any plane in spacetime, and we are free thanks to isotropy to choose any two basis vectors in that plane and any two basis vectors out of that plane. Isotropy gives us the freedom to reselect the basis vectors in that plane without affecting the ones out of the plane. When that plane is, say, the tx-plane, this means we change the time and x-coordinates of events without changing the y- and z-coordinates, for instance.

So, we see that any change of basis whose effects can be confined to a plane can only change components of vectors (i.e. 4-vectors) in that plane and not components out of the plane. I don't presume to say isotropy is the natural or best starting point, but the relationship between isotropy and the ability to freely choose a basis is one I find compelling. It is a symmetry, and with every symmetry comes freedom. This is something worth reiterating throughout physics, regardless of the exact topic at hand.

cepheid said:
Interesting, I hadn't thought about that. What do you mean by "space is isotropic?" In this case it sounds like you are saying that, "the laws of physics are the same regardless of what direction you're moving in." Is that basically it?

I would say that it's the principle that the laws of physics don't distinguish any direction in space from any other. As a special case, you can apply it to a velocity vector.

bcrowell said:
I would say that it's the principle that the laws of physics don't distinguish any direction in space from any other. As a special case, you can apply it to a velocity vector.

Makes sense to me. Thanks (also to Muphrid and zonde) for the clarification.

peterspencers said:
why does space time not contract uniformly in every direction around a fast moving object?

1. Motion extends the distance photons move between em interactions because light speed is constant.
2. The em fields are weaker by 1/λ in the direction of motion and any transverse direction due to lower frequency of interactions, i.e. time dilation.
3. This allows mass particles to compress in the direction of motion during acceleration.

If light speed added vectorially to object speeds, step 1 and the remaining sequence would not occur.

This is the same as asking why the universe does not have 4 spatial dimensions.
Because it does not.
When you ask a question like "why is x this way", you're asking for a decomposition of the fact into other facts that you can readily accept.
For example, why do we fall? Because the Earth exerts a force on us.
Here, the fact that masses exert forces on other masses is the "other fact".

It is very simple.just read the Lorenz transformations.

Battlemage!
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.

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?

peterspencers said:
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?
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.

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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.

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peterspencers said:
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?
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.

peterspencers said:
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.
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.

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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:

$ts$ = 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:$te$ = $\frac{ts}{√1-vv/cc}$ (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 I am totally new to all this, trying to work it out as I go along, any help would be most kind)

$te$ = 1 second

Then for the horizontal clock:

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

$te$ = 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:

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

$L$ = $Lo$√1-v2/c2

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

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

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

$te$ = 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 ??

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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.

Why is my above calculation insufficient to explain length contraction? I don't follow the reasoning here, please could someone explain (to a lamen) if the above quote really does apply to my above calculation.

peterspencers said:
Why is my above calculation insufficient to explain length contraction? I don't follow the reasoning here, please could someone explain (to a lamen) if the above quote really does apply to my above calculation.
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).

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peterspencers said:
why does space time not contract uniformly in every direction around a fast moving object?

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:
$\alpha$ for time changes
$\beta$ for longitudinal length changes
$\delta$ for transverse length changes

The Michelson-Morley experiment measures the combination of $\beta$ and $\delta$.
The Kennedy-Thorndike experiment measures the combination of $\alpha$ and $\beta$.

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

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

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

harrylin said:
- 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).

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

peterspencers said:
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. [..]
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?

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.

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peterspencers said:
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 explination 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?

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...

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?

peterspencers said:
[..] 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 Lh 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 Lh, 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 Lv, 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

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harrylin said:
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.

vin300 said:
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:
$T + T_/$ = $2D \frac{V}{V^2-v^2}$

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