Holonomy in the Schwarzschild-Droste Geometry

TL;DR

This paper investigates the properties of holonomy in Schwarzschild-Droste and Reissner-Nordström geometries, revealing quantized invariance and differences in behavior for extremal cases, with implications for quantum transport in curved spacetimes.

Contribution

It provides a detailed analysis of holonomy effects in specific curved spacetimes, including quantization phenomena and extensions to spinors and extremal cases.

Findings

Holonomy around circular orbits shows quantized band structure.

Radial holonomy exhibits distinct features for different metrics.

Extremal Reissner-Nordström case behaves qualitatively differently.

Abstract

Parallel transport of vectors in curved spacetimes generally results in a deficit angle between the directions of the initial and final vectors. We examine such holonomy in the Schwarzschild-Droste geometry and find a number of interesting features that are not widely known. For example, parallel transport around circular orbits results in a quantized band structure of holonomy invariance. We also examine radial holonomy and extend the analysis to spinors and to the Reissner-Nordstr\"om metric, where we find qualitatively different behavior for the extremal ($Q = M$) case. Our calculations provide a toolbox that will hopefully be useful in the investigation of quantum parallel transport in Hilbert-fibered spacetimes.