# Asymptotic analysis of the EPRL model with timelike tetrahedra

## TL;DR

This paper performs a stationary phase analysis of the EPRL spin foam model with timelike tetrahedra, revealing that the amplitude's phase aligns with the Regge action and includes Lorentzian, Euclidean, and split signature sectors.

## Contribution

It extends the analysis of the EPRL model to include timelike tetrahedra and unified treatment of different signatures, revealing new split signature solutions.

## Key findings

- Stationary points correspond to 4-simplices with phases matching Regge action.
- The model includes Lorentzian, Euclidean, and split signature 4-simplices.
- The analysis unifies different signature sectors in the EPRL model.

## Abstract

We perform the stationary phase analysis of the vertex amplitude for the EPRL spin foam model extended to include timelike tetrahedra. We analyse both, tetrahedra of signature $---$ (standard EPRL), as well as of signature $+--$ (Hnybida-Conrady extension), in a unified fashion. However, we assume all faces to be of signature $--$. The stationary points of the extended model are described again by $4$-simplices and the phase of the amplitude is equal to the Regge action. Interestingly, in addition to the Lorentzian and Euclidean sectors there appear also split signature $4$-simplices.