<論文>Parallel Propagatorの数学的考察

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Freely falling particle moves along the geodesic, the time-like curve, which defines and probes the structure of curved space-time. The tangential vector of the geodesic then forms the parallel vector field along that curve. The parallel propagator was investigated by J.L.Synge in detail, which is the linear operator to yield the parallel vector field or parallel tensor one along a geodesic. As is well known, the spin vector of the gyroscope or the polarization vector of electromagnetic waves is approximately parallel transported along the geodesic. Firstly, this paper shows that the parallel propagator forms a 1-parameter group of transformations of the tangent bundle along a curve generated from a 1-parameter group of transformation of a manifold. Secondly, it is shown that the equation of the parallel transport of a vector in an arbitrary space-time reduces to the Riccati's differential equation. The power series expression of the parallel propagator is also given through the transformation of the orthonormal frame. Finally, the method for obtaining the exact expression of the parallel propagator is presented, restricted to any time-like geodesic lying in the equatorial plane in the stationary axisymmetric space-time.
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