On 23/08/2023 19:57, root wrote:
For the derivation of the equations of general relativity
is spacetime represented as a four dimensional manifold
in 5 Cartesian dimensions? In other words, is the curvature
of spacetime represented as curvature with respect to
a Cartesian coordinate system?
Not necessary to do that. The curvature of a GR metric tensor can be
encoded as a symmetric matrix in a 4D manifold.
This isn't a bad introduction in Wiki:
https://en.wikipedia.org/wiki/Metric_tensor_(general_relativity)
A flat spacetime is the simplest possible with coordinates (t,x,y,z) and
-c^2, 1, 1, 1 down the diagonal.
Schwarzchild is the next simplest GR metric for a mass M, followed by
Kerr for a rotating object which describes most astrophysical objects.
https://en.wikipedia.org/wiki/Kerr_metric
(maths starting to get a lot more difficult here)
This paper "The Kerr spacetime: A brief introduction" from 2007 might
answer some of the OP's question at least as it applies to astrophysics.
https://arxiv.org/abs/0706.0622
--
Martin Brown
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