Constructing allowed complex metrics from black holes

TL;DR

The work presents a constructive method to obtain Kontsevich-Segal (KS) admissible complex metrics by applying diffeomorphisms that swap time and radius and by performing complex coordinate mappings on black-hole solutions. This yields time-dependent backgrounds, including KS-allowed static metrics from Schwarzschild interiors and dynamical metrics for de Sitter, Kantowski-Sachs, and Kerr Gowdy-type cosmologies, with explicit KS validity intervals governed by $\sum_{\mu} |\text{Arg}(\lambda_\mu)|<\pi$. The authors connect KS viability to horizon structure and classical break-time notions, showing, for example, a finite KS-valid window $\tau < \pi/(6H)$ in dynamical de Sitter and region-dependent admissibility in Kerr interiors. Overall, the paper provides a practical framework to realize KS-permissible curved backgrounds for quantum field theory in curved spacetime, linking black-hole interior geometry to cosmological-like metrics.

Abstract

We use diffeomorphic mappings to connect black hole metrics with complex solutions allowed by the Kontsevich-Segal criterion. By swapping radial and time-like coordinates and applying complex mappings, we derive dynamic metrics suitable for a Quantum Field Theory. This is shown for static and rotating black holes, mapping their interiors into the Kantowski-Sachs and a specific Gowdy-type cosmological model. We offer interpretations of the period during which the Kontsevich-Segal criterion holds.

Figures (2)

• Figure 1: This figure shows a density plot of the function in Eq. (4.13) for the particular values $M=1000$, $a=100$ and ratio $\lambda =0.1$. In this case the location of horizons in non-reduced units is; $t_{-} = 5.012562893$, $t_{+}=1994.9874371066$ and $t_0=1066.09$. In the region $t_0 < t < t_{+}$ the criterion is met for all values of the angle $\delta$.
• Figure 2: This figure shows the raise of the percentage of validity of the criterion within the horizons as the ratio $\lambda$ decreases, namely in the limit $a\rightarrow 0$. The particular value $t_0$ at which the criterion ceases to be valid approaches the singularity.