# Special Relativity – Understanding Euclidean Geometry in a Non-Inertial Frame

classical-electrodynamicsclassical-field-theoryfield-theoryspecial-relativity

Refer, "The classical theory of Fields" by Landau&Lifshitz (Chap 3).
Consider a disk of radius R, then circumference is $2 \pi R$. Now, make this disk rotate at velocity of the order of c(speed of light). Since velocity is perpendicular to radius vector, Radius does not change according to the observer at rest. But the length vector at boundary of disk, parallel to velocity vector will experience length contraction . Thus, $\dfrac{\text{radius}}{\text{circumference}}>\dfrac{1}{2\pi}$ , when disc is rotating. But this violates rules of Euclidean geometry.

What is wrong here?

However, the non-existence of such a material may be shown even microscopically. It is not possible to "order" any solid object to keep the proper distances at every moment because the distance between two atoms (or points on the solid object) may only be measured with a delay $\Delta t = \Delta x / c$ simply because no information may move faster than light. That's why it's always possible to squeeze any rod on one end and the opposite end of the rod won't move at least for this $\Delta t = \Delta x / c$. This relationship between the "limited speed of signals by $c$" and "non-existence of rigid objects in relativity" was already pointed out by Max von Laue in 1911.