Spinning geometry = Twisted geometry

TL;DR

This paper introduces spinning geometries as a continuous, piecewise-flat alternative to twisted geometries in loop gravity, interpreting fluxes as edge spins and ensuring geometric continuity across faces.

Contribution

It demonstrates that spinning geometries provide a continuous representation of loop gravity phase space, linking fluxes to edge spins and unifying twisted and continuous geometries.

Findings

Fluxes correspond to edge angular momenta.

Edges are necessarily helices with spin.

Spinning geometries form a continuous phase space representation.

Abstract

It is well known that the SU(2)-gauge invariant phase space of loop gravity can be represented in terms of twisted geometries. These are piecewise-linear-flat geometries obtained by gluing together polyhedra, but the resulting geometries are not continuous across the faces. Here we show that this phase space can also be represented by continuous, piecewise-flat three-geometries called spinning geometries. These are composed of metric-flat three-cells glued together consistently. The geometry of each cell and the manner in which they are glued is compatible with the choice of fluxes and holonomies. We first remark that the fluxes provide each edge with an angular momentum. By studying the piecewise-flat geometries which minimize edge lengths, we show that these angular momenta can be literally interpreted as the spin of the edges: the geometries of all edges are necessarily helices. We also show that the compatibility of the gluing maps with the holonomy data results in the same conclusion. This shows that a spinning geometry represents a way to glue together the three-cells of a twisted geometry to form a continuous geometry which represents a point in the loop gravity phase space.