Holonomy in the Schwarzschild-Droste geometry

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-Nordström 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.

Loop variables in the geometry of a rotating black string

“…Recently, Rothman, Ellis and Murugan [5] studied some particular curves when a vector A is parallel transported in the geometry of Reissner-Nordströn and then observed very interesting properties of parallel transport that they denominated invariance band holonomy. In this section we analyze the band holonomy properties from the point of view of the parallel transport equation in the background of the rotating black string.…”

“…Using holonomy, Bollini, Giambiagi and Tiomno [4] investigated the Kerr black holes space-times and obtained many properties of this geometry. In a recent article, Rothman, Ellis and Murugan [5] investigated the holonomy in the Drost-Schwarzschild geometry obtaining very interesting results for a class orbits in this space-time. They found that holonomy in this space-time has a "quantization" property denominated band invariance holonomy.…”

Holonomy transformation in the FRW metric

“…This global property of the manifold can be used as a means of global classification of spacetimes, as pointed out by Rothman et al [6], who investigated the holonomy in the Schwarzschild-Droste geometry. The aim of present article is to investigate the loop variables in FRW spacetime with the objective of studying the global properties of this spacetime.…”

“…Using holonomy, Bollini, Giambiagi and Tiomno [5] investigated the Kerr black hole space-time and obtained many properties of this geometry. In a recent article, Rothman, Ellis and Murugan [6] investigated holonomy in the Schwarzschild-Droste geometry obtaining very interesting results for a class orbits in this space-time. They found that holonomy in this space-time has a "quantization" property denominated band invariance holonomy.…”

“…They found that holonomy in this space-time has a "quantization" property denominated band invariance holonomy. Other investigations on loop variables in the context of gravitational fields includes the computation of this mathematical object in the Schwarzschild-Droste geometry [6], in Taub-NUT space-time [7] and in the background spacetime generated by a rotating black string [8]. In a recent article the clock effects were investigated by Bini, Jantzen and Mashhoon [9], via holonomy transformation.…”