At 02:24 PM 2/12/01 -0500, David Bowman wrote in effect
-- geodesics are "straight" ... as straight as they can be
-- geodesics (in curved spacetime) are "curved"
and basically I agree with what he said.
Here's my $.02 on the subject. We start with the question, "OK, if
geodesics are curved, in WHAT DIRECTION are they curved? Do they curve to
the left? Do they curve to the north?????"
I've seen some very famous cosmologists spout totally wrong answers to this.
The analogy to d=2 creatures crawling on a D=2 spherical or cylindrical
universe is instructive. We can see that their universe is embedded in our
D=3 universe, but they have no clue about that.
What we call great circles they call straight lines. We can see that they
curve through three dimensions, but this turns out to be the dimension that
they can't see at all.
A structure like that is tremendously rigid in two dimensions (in the plane
of the paper) but it would have very little rigidity in the third
dimension. If we wanted to make it rigid in the third dimension, we would
need to add additional girders, giving it a nontrivial size in the third
dimension.
Well, the d=2 creatures can build a d=2 lattice, but they can't build
anything that has any thickness in the third direction. They can't build
anything that gives them any "leverage" to control or even measure the
direction of curvature.
They can, however, infer SOME things about the curvature of their
world. In particular, they can measure something called the _intrinsic_
curvature. A sphere has intrinsic curvature; a cylinder does not. We d=3
creatures can use a flat piece of paper to make a map of a cylinder with no
distortion, but if we try to make a flat map of a spherical world,
something bad is going to happen. The d=2 creatures need to measure
curvature in other ways. One way is to look at the sum of the interior
angles of large triangles. On a sphere, you can erect a 90-90-90 triangle
as follows: one right angle at the equator, one right angle on the equator
90 degrees of longitude away, and the third 90 degree angle at the north
pole. That grossly violates the Euclidean-geometry rule that in flat space
the sum of the interior angles is 180 degrees. On a cylinder all the
triangles obey the Euclidean rule. The cylinder has zero intrinsic
curvature, and its residents do not know (or care) that it has extrinsic
curvature (assuming the cylinder is big enough that they can't
circumnavigate it). There are other ways to detect the presence or absence
of intrinsic curvature, but we need not go into them now.
Bottom line: If you are having trouble figuring out in which direction
spacetime is bending, don't feel too bad. There's nothing we can say about
the extrinsic curvature, and there's nothing we can do about it. All we
can do is measure the intrinsic curvature, e.g. by laying out big triangles
and measuring their corners.