GR geometry question

[root]

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?

Thanks.

[Martin Brown]

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