Caption: This image and discussion are incorrect, but I'm keeping image for future use maybe.

First, far from the black hole the geometry of spacetime is close to the gravity-free case: the nearly flat spacetime of pure special relativity.

A bit closer and test particles, planets, etc. would orbit here just as around a star. Spacetime is curved just enough to give what we think of as ordinary gravitational orbits. There is a small perturbation from gravity-free spacetime, but the curvature of space is almost??? below detectability in any direct sense.

As one gets close to the black hole in the region of strong gravity, space becomes noticeably curved and time runs slow as seen by a remote observer.

To gain some understanding of how the curvature of space manifests itself let us say we measure distances with measurements that are SIMULTANEOUSLY in frame of a remote observer at rest with the respect to the black holes.????

What one finds are that changes in RADIAL DISTANCE are larger than one would expect in flat space for observed changes in CIRCUMFERENCE.

This result is best illustrated by a figure that gives a common way of representing it that is suggestive of a gravitational well.

Credit/Permission: © David Jeffery, 2005 / Own work.
Image link: Itself.
Local file: local link: black_hole_schwarzschild_flamm_paraboloid.html.
File: Black hole file: black_hole_schwarzschild_flamm_paraboloid.html.