It's been about two decades since I took anything vaguely resembling a
science course. So, as I hang my head in shame, can someone explain to
me the difference between spatial geometry and the geometry I agonized
over in high school?

<[Only registered users see links. ]> wrote in message
news:1133917133.881483.117670@g47g2000cwa.googlegr oups.com...

I think Euclid defined the basics of earthly geometry. The idea is that the
shortest time (distance) to get from one point to another is to follow a
straight line drawn between them. However, on a larger scale this is not
necessarily true. In the vicinity of high gravitational forces the shortest
distance is not to follow the path light takes from one point to another.
Gravity refracts light.

<[Only registered users see links. ]> wrote in message
news:1133917133.881483.117670@g47g2000cwa.googlegr oups.com...

Your high school geometry was built in a universe 'populated' by points,
lines, and planes. It is an excellent geometry for its intended purposes -
measuring small tracts of land, designing roads and buildings, and in
general dealing with the things men build and use. A 'straight' line between
two points is defined as the path traced by a point through the empty space
with no change of direction from one end to the other.

The geometry of space and time useful to astronomers, physicists,
cosmologists, and the like is the geometry of a universe populated by mass,
energy, and fields. In this geometry, a 'straight' line between two points
is defined as the path taken by a mass through the field with no change of
*energy* from one point to the other.

When masses and energies are more important considerations than directions
and distances the latter is preferred.