Complex-valued spacetime curvature

Question

I've just been reading about tachyons and tachyonic fields, and although they probably don't exist/are wildly unstable, I'm curious: What does imaginary mass do to spacetime curvature? Does ‘complex valued curvature’ even have a single plausible meaning the way real-valued curvature does?

Clement · Accepted Answer

The notion of complex curvature does exist: see Y. Martinez-Maure, Real and complex hedgehogs, their symplectic area,curvature and evolutes. 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.

Complex-valued spacetime curvature

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