Why do we always talk only about length contraction in the longitudinal direction and not contraction in other transversal dimensions?

Question

why do we always talk only about length contraction and not contraction in other dimensions? if spacetime is uniform in all directions then change in one dimension must lead to a change in other dimensions right?

Answer 1

I shall simply reproduce the relevant section from Introduction to Electrodynamics, 4th ed. by Griffiths.

A moving object is shortened only along the direction of its motion:

Dimensions perpendicular to the velocity are not contracted.

Indeed, in deriving the time dilation formula I took it for granted that the height of the train is the same for both observers. I’ll now justify this, using a lovely gedanken experiment suggested by Taylor and Wheeler.⁸

Imagine that we build a wall beside the railroad tracks, and $1~\mathrm{m}$ above the rails (as measured on the ground), we paint a horizontal blue line. When the train goes by, a passenger leans out the window holding a wet paintbrush $1~\mathrm{m}$ above the rails, as measured on the train, leaving a horizontal red line on the wall. Question: Does the passenger’s red line lie above or below our blue one? If the rule were that perpendicular directions contract, then the person on the ground would predict that the red line is lower, while the person on the train would say it’s the blue one (to the latter, of course, the ground is moving). The principle of relativity says that both observers are equally justified, but they cannot both be right. No subtleties of simultaneity or synchronization can rationalize this contradiction; either the blue line is higher or the red one is—unless they exactly coincide, which is the inescapable conclusion. There cannot be a law of contraction (or expansion) of perpendicular dimensions, for it would lead to irreconcilably inconsistent predictions.

_{⁸E. F. Taylor and J. A. Wheeler, Spacetime Physics 2nd ed. (San Francisco: W. H. Freeman, 1992). A somewhat different version of the same argument is given in J. H. Smith, Introduction to special relativity (Champaign, IL: Stipes, 1965).}

Answer 2

The mistake you made is that whilst spacetime is isotropic, relative velocity is not. By its very presence, it breaks the symmetry of spacetime. Hence length contraction can be asymmetric, and it is, it is in the direction of motion.

Answer 3

Nope, In which direction length contraction going to occur depends on the direction of the velocity of the object's frame relative to the rest frame (or observers frame).

In the simple instances, when the body moves along the positive $x$-axis, We get a length contraction given by $$\Delta x'=\frac{\Delta x}{\gamma}$$ The anisotropy of the space is due to this direction of velocity.

If the velocity of the object would be in some other direction then of course the length contraction will occur differently.

Answer 4

Length contraction is an effect caused by the relativity of simultaneity. It is a property of spacetime, not of objects as such.

Suppose you have two points in space that are at rest relative to each other a distance d apart. If you measure the distance in any other frame in which the points are moving along the line that joins them you get a shorter value for d. The reason is that in the other frame simultaneous measurements of the positions of the two points are actually measurements at two separate moments in the rest frame . The position of the leading point is measured before the trailing point, which gives the trailing point a chance to move ahead before it is measured, so the recorded gap between the two points is shortened.

If you consider two additional points so that the four in total form a square, the two leading points (ie the leading corners of the square) are measured simultaneously in the moving frame, so the distance between them (ie at right angles to the direction of travel) is properly measured and doesn't become shortened.

So you see that the effect applies to the space itself, not just to objects.

As to how 'real' the effect is, you can find lots of writing by heavyweight physicists over the years that shows they clearly believe some kind of force acts on moving objects causing them to compress. I just can't buy into that. A thought experiment that makes the idea of 'real' compression hard to credit is simply to consider ten observers moving relative to you at ten different speeds. According to each of them you are currently compressed by ten different amounts all at once, which is clearly impossible.