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

#### peterspencers

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

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#### HallsofIvy

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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.

#### Mark M

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.

#### cepheid

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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

#### bcrowell

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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 [Broken] , 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.

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.

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 [Broken]

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#### cepheid

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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.

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?

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.

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 [Broken] , 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).

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 [Broken]

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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#### Muphrid

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.

#### TrickyDicky

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.

#### Muphrid

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.

#### bcrowell

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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.

#### zonde

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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

#### Muphrid

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.

#### bcrowell

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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.

#### cepheid

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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.

#### phyti

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.

#### GarageDweller

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".

#### valentin mano

It is very simple.just read the Lorenz transformations.

• Battlemage!

#### Saw

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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

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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#### 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.

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#### ghwellsjr

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Gold Member
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.

#### harrylin

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. :tongue2:

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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#### 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:

$ts$ = 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 im 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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#### peterspencers

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 dont follow the reasoning here, please could someone explain (to a lamen) if the above quote really does apply to my above calculation.

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