A note on the admissibility of complex BTZ metrics

TL;DR

This paper tests Witten's admissibility criterion for complex metrics using quasi-Euclidean BTZ solutions, confirming that the criterion excludes overspinning metrics that would otherwise cause inconsistencies in the gravitational path integral.

Contribution

It provides a detailed analysis of the admissibility of complex BTZ metrics, validating Witten's criterion and clarifying the role of overspinning solutions in quantum gravity.

Findings

Admissibility criterion successfully excludes overspinning metrics.

Smoothness and admissibility are confirmed for all quasi-Euclidean BTZ metrics.

The results support the consistency of the gravitational path integral with Witten's criterion.

Abstract

We perform a nontrivial check of Witten's recently proposed admissibility criterion for complex metrics. We consider the `quasi-Euclidean' metrics obtained from continuing the BTZ class of metrics to imaginary time. Of special interest are the overspinning metrics, which are smooth in this three-dimensional context. Their inclusion as saddle points in the gravitational path integral would lead to puzzling results in conflict with those obtained using other methods. It is therefore encouraging that the admissibility criterion discards them. For completeness, we perform an analysis of smoothness and admissibility for the family of quasi-Euclidean BTZ metrics at all values of the mass and angular momentum.