On Wed, 21 Jan 2004 17:42:03 +0000 (UTC),
(Mahdiarnt) wrote:

This question is about the meaning of the length measurment of events,
from a particular observer, in terms of the metric tensor of the
4-dimensional space-time.

Here is the way I view the situation. The fundamental physical
quantity that describes the geometry of space-time is the Lorentz
interval, which is invariant to all observers.

Think of the geometry of a ball. At any point on the ball, (a 2d
surface), there will be a "flat" plane tangent to the ball.
Similarly, at any point in space-time, there will be a "flat" 4
dimensional "tangent space" that's tangent to the manifold. This
tangent space will have the familiar metric from special relativity

ds^2 = dx^2 + dy^2 + dz^2 - cdt^2 (1)

Now, if we are given the Lorentz interval in special relativity, how
do we separate it into space and time components? Well, we have to
pick some specific observer, moving with some specific velocity. The
choice of this observer defines how the "time" arrow points. The
directions orthogonal to the direction chosen to be "time" are
"space".

In general relativity, the issue is much the same, but we have to
choose an entire coordinate system. The choice of coordinates is
arbitrary and can be made in any manner that's convenient. In special
relativity we can think of the coordinate axes as straight lines - in
general relativity, space-time itself is curved, so the co-ordinate
axes themselves are in general also allowed to be curved.

In special relativity we could make one choice for the direction of
time, and because of flatness we could translate this choice
everywhere. In general relativity, because of the curved nature of
the manifold, parallel transport of a vector representing time will
depend on the path along which the vector is transported. (Given the
metric, we can view a vector transported over an infinitesimal
distance as being "parallel transported" if, and only if, the vector
and the transported vector are the two opposite sides of a
parallelogram.) Thus we have to make a separate choice of the
direction that is timelike at every point in space-time to remove this
ambiguity. We are free, in general, to do this in whatever manner we
like, but usually some ways will be more convenient than others. One
measure of convenience might be the elimination of unnecessary
torsion.

Once we have made the choice of coordinate systems, the issues of
separating the lorentz interval into space and time components becomes
easy. We can imagine the "grid" structure of the coordinate system
drawn by lines of a constant coordinate. We require any specific set
of coordinates to map to only one point in space-time as a requirement
for a well-behaved coordinate system.