Asymptotic analysis of the EPRL model with timelike tetrahedra

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.

Einstein equation from covariant loop quantum gravity in semiclassical continuum limit

“…But the discussion and results are valid or any other spinfoam models which have both the correct large spin asymptotics, and the flatness (e.g. the model with timelike tetrahedra [43] and its recent asymptotical analysis [10]). …”

“…[6][7][8][9][10]). It has been shown that if one doesn't consider the spin-sum, but consider the spinfoam (partial) amplitude with fixed spins, the large spin asymptotics of the amplitude give the Regge action of gravity, being a discretization of the Einstein-Hilbert action on the triangulation.…”

The Hessian in Spin Foam Models

“…For the given normal N 0 i (see [8]) with the norm c i = |N 0 i | 2 ∈ {−1, 1} we can decompose the bivector B as follows…”

“…The bivectors at this stationary point are denoted B 0 ij (in the simplex frame) and B 0 ′ ij = (g 0 i ) −1 B 0 ij (see the beginning of section 3). The geometric bivectors B ∆ ij are described in [8] and appear in section 6.…”

“…the hessian at that point has no zero eigenvectors, after gauge fixing) or not. These issues were summarized in our previous paper [8].…”