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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 ?
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.
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