Twisted geometries are area-metric geometries

TL;DR

This paper demonstrates that twisted geometries in Loop Quantum Gravity can be interpreted as arising from an area-metric, providing new insights into their structure and continuum limit behavior.

Contribution

It establishes a novel equivalence between twisted geometries and area-metrics, enabling new geometric notions and a microscopic understanding of spin foam geometries.

Findings

Twisted geometries can be understood as area-metrics in 4D.

New notions like signature and generalized triangle inequalities are defined.

Supports a microscopic interpretation of spin foam geometries.

Abstract

The quantum geometry arising in Loop Quantum Gravity has been known to semi-classically lead to generalizations of length-geometries. There have been several attempts to interpret these so called twisted geometries and understand their role and fate in the continuum limit of the spin foam approach to quantum gravity. In this paper we offer a new perspective on this issue by showing that the twisted geometry of a 4-simplex can be understood as arising from an area-metric (in contrast to the more particular length-metric). Such equivalence allows us to define notions like signature, generalized triangle inequalities and parallel transport for twisted geometries (now understood in a 4-dimensional setting), exemplifying how it provides a new handle to understand them. Furthermore, it offers a new microscopic understanding of spin foam geometries which is notably supported by recent studies of the continuum effective dynamics of spin foams.