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

1. Aug 4, 2012

### peterspencers

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

2. Aug 4, 2012

### HallsofIvy

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.

3. Aug 4, 2012

### 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.

4. Aug 4, 2012

### cepheid

Staff Emeritus
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.

Last edited: Aug 4, 2012
5. Aug 4, 2012

### bcrowell

Staff Emeritus
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.

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.

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]

Last edited by a moderator: May 6, 2017
6. Aug 4, 2012

### cepheid

Staff Emeritus
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.

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 think that might be the same idea, yeah.

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).

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.

Last edited by a moderator: May 6, 2017
7. Aug 4, 2012

### Muphrid

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.

8. Aug 5, 2012

### TrickyDicky

You mean then the EM field is not really a vector. Not the electric field.

9. Aug 5, 2012

### 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.

10. Aug 5, 2012

### bcrowell

Staff Emeritus
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.

11. Aug 6, 2012

### zonde

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

12. Aug 6, 2012

### 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.

13. Aug 6, 2012

### bcrowell

Staff Emeritus
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.

14. Aug 6, 2012

### cepheid

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

15. Aug 7, 2012

### phyti

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.

16. Aug 8, 2012

### 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".

17. Aug 11, 2012

### valentin mano

It is very simple.just read the Lorenz transformations.

18. Aug 13, 2012

### 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.

19. Aug 16, 2012

### 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?

20. Aug 16, 2012

### 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.

Last edited: Aug 16, 2012