Metric Affine Manifold: Dynamics in General Relativity by Aleks Kleyn (PDF)

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I tell about different mathematical tool that is important in general relativity. The text of the book includes definition of geometric object, concept of reference frame, geometry of metric affinne manifold. Using this concept I learn dynamics in general relativity.We call a manifold with torsion and nonmetricity the metric affine manifold. The nonmetricity leads to a difference between the auto parallel line and the extreme line, and to a change in the expression of the Frenet transport. The torsion leads to a change in the Killing equation. We also need to add a similar equation for the connection.The dynamics of a particle follows to the Frenet transport. The analysis of the Frenet transport leads to the concept of the Cartan connection which is compatible with the metric tensor. We need additional physical constraints to make a nonmetricity observable.

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