# Explanation for “if all accelerated systems are equivalent, then Euclidean geometry cannot hold in all of them”

I'm doing an EPQ (mini college research paper) on gravity, and I found a site that explained things in simple terms. I am having trouble understanding how Einstein came to his revelation space-time was curved.

Einstein also realised that the gravitational field equations were bound to be non-linear and the equivalence principle appeared to only hold locally.

and Einstein said

If all accelerated systems are equivalent, then Euclidean geometry cannot hold in all of them.

Can anybody help?

Consider flat space (i.e. Minkowski), viewed in a rotating frame (in e.g. cylindrical coordinates one just replaces $\phi$ by $\phi'=\phi+\omega t$). One can calculate (without too much trouble) that, in these coordinates, a spatial line element can be expressed in terms of the canonical cylindrical coordinates as $$d\ell^2=dr^2+dz^2+\frac{r^2d\phi^2}{1-\frac{\omega^2r^2}{c^2}}$$ Now, note that if we consider a unit disc in the $z=\text{constant}$ plane, we find $$d\ell=\frac{2\pi}{\sqrt{1-\frac{\omega^2}{c^2}}}>2\pi\hspace{1cm}\iff \omega>0$$
The startling conclusion is that this observer will measure the circumference of a disc of radius $r$ to be $C>2\pi r$ for any $\omega>0$. Hence, Euclidean geometry does not hold universally, even in flat space, if we relax the assumption that 'inertial frames' are somehow privileged, i.e. if we take this calculation seriously. Realizing that there is a need to consider (relatively) accelerating frames as equivalent was one of the major breakthroughs that needed to be made in order to arrive at the theory of general relativity.