# Metric defined with a non-coordinate basis

There is no such thing. The geometry of spacetime does not change. Spacetime is a 4-dimensional geometry that already includes all the information about what we normally think of as "change" over the entire history of the universe. What we call "change" is not the geometry of spacetime changing; it is just looking at different segments of our worldlines that lie in different parts of the unchanging spacetime geometry.

Also, as I have already said, there is no way to identify "corresponding points" in different geometries, so the conceptual picture you are implicitly using when you think of "geometry changing" is not well-defined.
Whatever you use to describe the geometry of the spacetime, I really find difficulties to comprehend they can contain all the infomation of the past of the universe.

PeterDonis
Mentor
2019 Award
I think the way to solve differential equations similar to parallel transport equations is the method of integral production, and in my opinion, this is the the only way to solve such problem, and the calculation is too complicated to get an exact solution, because the solution can be expressed as the Dyson expansion.and only in a book I find someone to solve the problem in such a way.
I don't know what books you're looking at, but they don't seem to be GR textbooks. Try looking at some GR textbooks.

if the curvature of the space is not zero(the parallel of a vector is path dependent), we can not get an exact solution the parallel transport of a vector along a path.
Certainly you can. Parallel transport along a path is uniquely defined in a curved space. What is not well-defined is parallel transport between two points without specifying a path. But if you specify a path you specify a unique result for parallel transport.

PeterDonis
Mentor
2019 Award
Whatever you use to describe the geometry of the spacetime, I really find difficulties to comprehend they can contain all the infomation of the past of the universe.
Then you need to spend some time learning GR from GR textbooks.

PeterDonis
Mentor
2019 Award