>>10758594

Yes! GR is all about how mass-energy affects geometry. More specifically, how it affects the spacetime metric, which is a generalization of a distance formula.

Generally speaking, you can start out choosing whatever coordinate system you like, as long as the coordinates vary smoothly and uniquely identify every spacetime point. However, without knowing how mass and energy are distributed throughout those coordinates, there is no way to translate coordinates into distances and time intervals. This is what's meant by the phrase "coordinates have no immediate metrical meaning."

Once you know the mass-energy distribution throughout your coordinates, you can calculate the metric, ie distance formula, for those coordinates. In the case of using cartesian coordinates in empty space, you end up with the Pythagorean theorem (more accurately, the Minkowski metric). In most real world scenarios, the metric will only deviate very very slightly from this, but around black holes and such, distances, areas, volumes, and time intervals will be different from what you naively calculate using the typical formulas based on coordinate radius, coordinate angle, and coordinate time.