Has the time-space curvature a complex form ; due to the complex form of time-space ?

Question

It is necessary for time-space curvature to be considered :

1-Rellationship between time factor and space factor .

2-Existence of complex material (mass) to make complex curvature (due to complex form of space-time).

Why haven't considered ?

Comments

If your spacetime manifold admits a complex structure, then you can use complex coordinates to express curvature forms. You can find it from the book "Geometry, Topology and Physics" by Nakahara.

Answer 1

The notion of complex curvature does exist: see https://hal.archives-ouvertes.fr/hal-02948386/

For instance, a complex circle with (complex) radius R has a (complex) radius of curvature equal to R (see page 17).

The author says the paper is to appear in the Journal of Symplectic Geometry.

Answer 2

You are taking space time curvature way to literally. Think of it as as mass increases in a confined space -> harder for any single unit of such mass in that space to move -> unit of mass therefore must move slower because no room to move -> therefore it appears that time is slowing down for such mass (this is called both gravity and gravitational time dilation - one and the same thing). When things move in space time they are impacted by various differences in levels of mass in the "mass in a confined space" so things in space time slow and go faster accordingly - > this gives appearance of curves in space time.

Answer 3

As I said before, the notion of complex curvature does exist.

One could imagine that there exists a complex metric such that the operator annihilation verifies Dirac or Klein Gordon equations associated with this metric while the operator creation would verify these same equations but for the conjugate metric.

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