Can General Relativity and Curved Spaces Be Described in Complex Coordinates?

[Dmitry67]

I am not familiar with that stuff, so please don't laugh.

I know some facts about the geometry when coordinates are real. In pseudo-euclidean spaces (like in special relativity) T is also real, just the definitions of a distance is different.

R^2 = X^2+Y^2+Z^2 - T^2

But we can say that it is an euclidean space, but T is imaginary. So, if we assume that coordinates are complex, the only difference between euclidaen space and Minkowsky space is an orientation of a subset in a 4-dimensional complex space.

So far I hope it is correct.

My question is, what about General relativity and curved spaces? I read that for 3space+1time curved space can be put into 86 dimensional manifold with 3 timelike dimensions.

What if we work completely in the complex area, so there is no difference between space and time dimensions?

 

[tiny-tim]

I read that for 3space+1time curved space can be put into 86 dimensional manifold with 3 timelike dimensions.

uhh? 86?

did you get this from http://www.eng.uah.edu/~jacksoa/literature/MD_Int2.pdf? … 86 was just an academic example … it could have been 42 or 717 or …

 

[robphy]

I think it comes from the articles discussed in
https://www.physicsforums.com/showthread.php?p=1777820#post1777820

 

[Dmitry67]

it is from here: https://www.physicsforums.com/showthread.php?t=290098

If the metric for the higher dimensional pseudo-Riemannian manifold is required to restrict down to the metric for 4-dimensional spacetime, then it could take a lot of dimensions.

Chris Clarke* showed that every 4-dimensional spacetime can be embedded isometically in higher dimensional flat space, and that 90 dimensions suffices - 87 spacelike and 3 timelike. A particular spacetime may be embeddable in a flat space that has dimension less than 90, but 90 guarantees the result for all possible spacetimes.

* Clarke, C. J. S., "On the global isometric embedding of pseudo-Riemannian
manifolds," Proc. Roy. Soc. A314 (1970) 417-428