Note on complex metrics, complex time, and periodic universes

Motivated, on the one hand, by recent results on isochronous dynamical systems, and on the other, by quantum gravity applications of complex metrics, we show that, if such an enlarged class of metrics is considered, one can easily obtain periodic or bouncing complex solutions of Einstein's equations. It is found that, for any given solution $g_{\mu\nu}$ of the Einstein's equations, by means of a complex periodic change of time, one can construct infinitely many periodic or bouncing complex solutions $\hat g_{\mu\nu}$ that are physically indistinguishable from $g_{\mu\nu}$ over an arbitrarily long time interval. These results, which are based on the use of complex diffeomorphisms, point out an unacceptable arbitrariness in the theory. As we will show, a condition on the class of physically meaningful complex metrics proposed in [M. Kontsevich and G. B. Segal, Q. J. Math. 72, 673 (2021)] and discussed in [E. Witten, arXiv:2111.06514] solves this problem, restricting the family of admissible complex diffeomorphisms. We conclude by arguing that this condition can be viewed as a quantum-gravity generalization of the equivalence principle to complex spacetimes.