The Spin Holonomy Group In General Relativity

TL;DR

This paper investigates the time-independence of the spin holonomy group in general relativity, linking it to curvature self-duality, and classifies possible holonomy groups in various spacetime scenarios.

Contribution

It reveals the reasons for holonomy group invariance in vacuum and Einstein spaces, and classifies all potential holonomy groups in general relativity.

Findings

Holonomy group is preserved under time evolution in vacuum and Einstein spaces.

Presence of matter breaks the conservation of holonomy.

Holonomy groups can be classified and examples provided.

Abstract

It has recently been shown by Goldberg et al that the holonomy group of the chiral spin-connection is preserved under time evolution in vacuum general relativity. Here, the underlying reason for the time-independence of the holonomy group is traced to the self-duality of the curvature 2-form for an Einstein space. This observation reveals that the holonomy group is time-independent not only in vacuum, but also in the presence of a cosmological constant. It also shows that once matter is coupled to gravity, the "conservation of holonomy" is lost. When the fundamental group of space is non-trivial, the holonomy group need not be connected. For each homotopy class of loops, the holonomies comprise a coset of the full holonomy group modulo its connected component. These cosets are also time-independent. All possible holonomy groups that can arise are classified, and examples are given of connections with these holonomy groups. The classification of local and global solutions with given holonomy groups is discussed.