Gregg,
Yes, I heard the Penrose's talk with Weinstein. Non-Euclidean
geometry (Riemannian sphere) indeed has been a great knife to cut
many an apple. Relativity led to creation of various
transformations because as the frames of references & medium
of communication (space) change calculations.
Figure one and two are trajectories of submarines travelling from
same point A to B in the ocean.
Question - Which of the two figures is more likely to
take lesser time to travel would the submarine be travelling at
constant speed in both cases.
Those who answered Figure 1 are wrong. Because for someone
floating on the ocean Figure 1 seems straight. But for someone far
in space or sky, Figure two is straight, quite like a straight
line joining two points on a circle is always shorter than the
circumferential arc.
Similarly, London, Paris & Moscow actually form a right
angled triangle on earth but Pythagoras theorem does not hold true
for them. While the earth's surface is Riemannian, our local space
is Euclidean. Therefore, we need to make adjustments. Quite
similarly, the curvature of space changes with its average
density. Light moves in a straight line, but straight for an
observer inside curved space is curved for an observer far away,
where the curvature does not exist. So that which is straight for
one inside, is curved for one outside the impact of mass.
The time for a laser signal propagating in a laid straight cable
will be more than that needed for a laser signal propagating
straight in air. So for our frame of reference we need to make
transformations to calculate the correct time taken by light when
it passes in proximity to large stars or blackholes.
In context of my writings - it meant that representations allow for simplification and resolution of problems. The curved space is real, its transformation on a complex plane, helps us calculate time taken to travel through it from point of view of our non-curved space. So the curvature is real, its transformation is a mathematical representation for precision of calculation.
There are numerous non-spherical representational geometries. As
an example two parallel lines on a hyperbolic planes (Thermal
Plant condensers if your have seen them) never meet (like parallel
lines) but the distance between them is not constant.
TY
DL
