Autoparallel motion in a class of torsionful Schwarzschild spacetimes is completely integrable via four conserved quantities, and positive-energy autoparallels are repelled by the black hole.
Poincar\'e gauge gravity primer
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abstract
We give an introductory overview of the classical Poincar\'e gauge theory of gravity formulated on the spacetime manifold that carries the Riemann-Cartan geometry with nontrivial curvature and torsion. After discussing the basic mathematical structures at an elementary level in the framework of the standard tensor analysis, we formulate the general dynamical scheme of Poincar\'e gauge gravity for the class of Yang-Mills type models, and consider a selected number of physically interesting consequences of this theory.
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`Translation invariant' black hole: autoparallels and complete integrability
Autoparallel motion in a class of torsionful Schwarzschild spacetimes is completely integrable via four conserved quantities, and positive-energy autoparallels are repelled by the black hole.