On 1/5/2019 1:28 PM, [email protected] wrote:
The relation is provided by the metric. If you choose different
coordinate systems (e.g. cylindrical or spherical or whatever)
then there is different metric tensor. So the integral along the
path of g_ab dx^a dx^b is the same.
Brent
*I assume you're showing why the proper time along a given path is the
same for all observers, and this has nothing to do with coordinate
time being unrelated to proper time. AG *
Coordinate time between events A and B is just delta(x^0) = x^0(B) -
x^0(A). Just like the longitudinal distance between LA and NY is
Long(LA)-Long(NY). But the driving distance between LA and NY depends
on the path you take and is an integral along that path which includes
changes in latitude:
S^2 = INT_path g_ab dx^a dx^b = INT_path [ dlong*dlong*cos^2(lat) +
dlat*dlat]
Notice the cos^2 factor because the space isn't flat.
So in GR coordinate time is related to proper time; it contributes a
term in accordance with the metric that describes the curvature of the
spacetime. But there are other terms from the spatial coordinates and
even cross terms and the terms are weighted by the metric factors that
describe the shape of the space.
Brent
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