On some global problems in the tetrad approach to quasi-local quantities

TL;DR

This paper investigates the topological obstructions in the tetrad approach to quasi-local quantities, showing conditions for trivializability of frame bundles and classifying homotopy classes of spin frames on 2-surfaces.

Contribution

It demonstrates the global trivializability of frame bundles near spacelike 2-surfaces under certain conditions and classifies homotopy classes of spin frames based on surface genus.

Findings

Lorentz frame bundle is always trivializable near orientable surfaces.

All spin frames in the same spinor structure share the same homotopy class.

Number of homotopy classes of Lorentz frames depends on surface genus g.

Abstract

The potential global topological obstructions to the tetrad approach to finding the quasi-local conserved quantities, associated with closed, orientable spacelike 2-surfaces S, are investigated. First we show that the Lorentz frame bundle is always globally trivializable over an open neighbourhood U of any such S if an open neighbourhood of S is space and time orientable, and hence a globally trivializable SL(2,C) spin frame bundle can also be introduced over U. Then it is shown that all the spin frames belonging to the same spinor structure on S have always the same homotopy class. On the other hand, on a 2-surface with genus g, there are $2^{2g}$ homotopically different Lorentz frame fields, and there is a natural one-to-one correspondence between these homotopy classes and the different SL(2,C) spinor structures.