Kruskal-Szekeres Radius: Explained for Beginners in GR

[smoothoperator]

I'm a beginner in GR (as you may conclude from some of my previous posts) so any help is greatly appreciated.

I was recently studying alternative metrics for the Schwarzschild metric and one of them was the Kruskal Szekeres metric.

In Schwarzschild, the radius r is defined which is the radius from the centre of the object with mass we are sutdying. In K-S metric the Tortoise coordinate was mentioned with a value greater than the classical r in Schwarzschild.

So my question is, if we use a different metric, in this case the K-S metric, does the radial spatial distance between two points change? And can anybody give me a concrete example, for instance a distance of 1m near Earth would be what distance according to Kruskal and Szekeres coordinates?

Thanks in advance.

 

[Orodruin]

Kruskal-Szekeres is not an alternative metric, it is a different coordinate system of the same manifold with the same metric. In the overlapping regions, the KS coordinates describe the very same physical situation. Just like you can use polar coordinates in $\mathbb R^2$ without changing the fact that you are describing $\mathbb R^2$.

 

[Matterwave]

Proper distances, being proper, do not depend on the particular coordinate system you choose. If you chose 1 path, parametrized by $\lambda$, between two points $1,2$, the proper distance between them is given by $$s=\int_1^2 \sqrt{g_{\mu\nu}\frac{dx^\mu}{d\lambda}\frac{dx^\nu}{d\lambda}}d\lambda$$ You should verify for yourself that a change of coordinates changes $g_{\mu\nu}\rightarrow g_{\mu'\nu'}$ and $x^\mu(\lambda)\rightarrow x^{\mu'}(\lambda)$ but does not change $s$.