Spin networks and cosmic strings in 3+1 dimensions

Abstract

Spin networks, the quantum states of discrete geometry in loop quantum gravity, are directed graphs whose links are labeled by irreducible representations of SU(2), or spins. Cosmic strings are 1-dimensional topological defects carrying distributional curvature in an otherwise flat spacetime. In this paper we prove that the classical phase space of spin networks coupled to cosmic strings may obtained as a straightforward discretization of general relativity in 3+1 spacetime dimensions. We decompose the continuous spatial geometry into 3-dimensional cells, which are dual to a spin network graph in a unique and well-defined way. Assuming that the geometry may only be probed by holonomies (or Wilson loops) located on the spin network, we truncate the geometry such that the cells become flat and the curvature is concentrated at the edges of the cells, which we then interpret as a network of cosmic strings. The discrete phase space thus describes a spin network coupled to cosmic strings. This work proves that the relation between gravity and spin networks exists not only at the quantum level, but already at the classical level. Two appendices provide detailed derivations of the Ashtekar formulation of gravity as a Yang-Mills theory and the distributional geometry of cosmic strings in this formulation.

Dual2+1Dloop quantum gravity on the edge

“…We expect to find both 3+1D general relativity and 3+1D teleparallel gravity as different polarizations of the discrete phase space. The discretization in 3+1 dimensions will be presented in an upcoming paper [35].…”

A Short Survey of Time Machines

“…The fractional Dirac delta distribution 11 discussed here in this Subsection is based on the Fourier transform method, and is derived using Mandlebrot scaling-under variance constraint/κ ∈ (0, 1] (FYI, the Cantor set has dimension less than "1") [57]. Videlicet, consider the line element given here by…”

“…However, given all that was already said, the TM we intend to modify is founded on spinning topological solitons or sometimes called cosmic strings. Anyway, it was conjectured that cosmic strings emerged ex nihilo, via phase transitions in the early history of the universe [10,11,12].…”