Been reading Gravitation, by Thorne, Wheeler, and Misner. On page 48:
"Geometric Objects are objects in Spacetime that exist independently of coordinate systems or reference frames."
This made me wonder... is Distance, then, a Geometric Object?
For you can have two coordinate systems, one the X-Y Cartesian system, another the Polar system with Angle and Radius. In the first system, we can have the point 1,1.
In the second system, this same point cannot be described with rationals. It needs an irrational value that is an infinite sequence of digits. Namely, sqrt2.
So you see, the geometric object, the point, doesn't exist in exactly the same way in both coordinate systems. In one, it's coordinates are rationas. In another, the Radius becomes irrational.
This makes me wonder if... if "Distance" is not actually a Geometric Object? What is a GO then?