Note on complex metrics, complex time and periodic universes

TL;DR

The paper explores how complex metrics in Einstein's equations can produce periodic or bouncing solutions, revealing arbitrariness in the theory that can be addressed by a proposed condition related to quantum gravity.

Contribution

It introduces a condition on complex metrics that restricts complex diffeomorphisms, reducing arbitrariness and extending the equivalence principle to complex space-times.

Findings

Infinite periodic and bouncing solutions can be generated via complex time transformations.

A proposed condition restricts complex diffeomorphisms, addressing arbitrariness.

The condition can be viewed as a quantum-gravity extension of the equivalence principle.

Abstract

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 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 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, that 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 arguing that this condition can be viewed as a quantum-gravity generalization of the equivalence principle to complex space-times.