# Does Contraction at High Speeds Happen in Any Dimension Besides Length?

I'm not sure if this is a duplicate.

Whenever physics buffs talk about Einstein's relativity (I forget which kind) at high speeds, they always talk about "length contraction", or shortening of the object due to the high speeds. This happens even at the relatively low speeds we travel at every day, but there is so little contraction we don't notice it. However, these physics buffs only talk about length contraction, implying a single dimension. Why do they never talk about contraction in more than 1 dimension? Is it because the contraction happens in the direction the object is going, which is usually only a single direction? If so, are there any cases which would involve 2- or 3-dimensional contraction?

• Maybe the question could be re-stated as this: if a front is propagating radially, e.g. a sphere inflating at high speed in radial direction, what happens to the distances at the surface of the sphere? Or is the question not relevant because we could not materialize a continuum that would spread radially?
– Joce
Commented Apr 23, 2014 at 8:19
• Sure. Or how about: What happens to length when the traveler is on a non-flat surface or geometry related to the observer, like a sphere or rectangular prism? What happens, according to the observer when the traveler goes around a "corner"? Or would that be a different question? Commented Apr 23, 2014 at 15:29
• Just asked a question like this here. Commented Apr 23, 2014 at 15:44

Is it because the contraction happens in the direction the object is going, which is usually only a single direction?

Yes, exactly. The dimensions perpendicular to the relative motion of the object and the observer are not affected. And as far as I know, there is no motion in which more than one (spatial) dimension is contracted.

• I figured. Just thought I'd ask. Commented Apr 22, 2014 at 17:48
• So if a front is propagating radially, e.g. a sphere inflating at high speed in radial direction, what happens to the distances at the surface of the sphere? Or is the question not relevant because we could not materialize a continuum that would spread radially?
– Joce
Commented Apr 23, 2014 at 12:44

A simple argument shows that contraction can only occur in the direction of motion: there is no way to uniquely specify any other axis in space along which an object could contract. If nature is to be consistent, then longitudinal contraction is the only choice.

" Is it because the contraction happens in the direction the object is going, which is usually only a single direction? If so, are there any cases which would involve 2- or 3-dimensional contraction?"

Yes. In order to illustrate the principle, uniform motion (constant velocity) is usually specified as occurring along a single spatial dimension, such as "the x-axis", with an equation of motion x = vt. Relativistic contraction occurs along the direction of motion, so in this case occurs in one dimension.

If, let's say, motion occurred along a 45 degree path (equation y = x = vt), then you would observe contraction in both the x and y directions. And a generalization to a diagonal in 3 dimensions will produce contraction in 3 dimensions.

It's just a matter of representation.

• You show how to get contraction along multiple coordinates, but still in one direction. Commented Apr 23, 2014 at 1:44
• I agree with @dmckee. One part of the question was if there is any way the motion could be in more than 1 direction at one time. Commented Apr 23, 2014 at 18:43