Difference between proper time and coordinate time

[cianfa72]

All of the things you discuss are theoretical items; they're not the same as having actual, physical observers that can exchange information in order to physically assign coordinates.

Yes, nevertheless there is in principle a physical procedure to assign them (at least on the exterior region). Take for example the Schwarzschild $r$ coordinate: it is basically the length of maximal circumference divided by $2 \pi$.

See also Exploring Black Holes, 2nd edition - Taylor, Wheeler -- ch 2 section 4

 

[PeterDonis]

nevertheless there is in principle a physical procedure to assign them (at least on the exterior region).

Yes, in the exterior region. But not in the interior region. And the interior region is part of the spacetime, so you do not have a physical procedure to assign coordinates on the entire spacetime, only on a portion of it. Which was my point.