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arXiv · 1906.11805

On the discrete Christoffel symbols

V. M. Khatsymovsky
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Abstract

The piecewise flat spacetime is equipped with a set of edge lengths and vertex coordinates. This defines a piecewise affine coordinate system and a piecewise affine metric in it, the discrete analogue of the unique torsion-free metric-compatible affine connection or of the Levi-Civita connection (or of the standard expression of the Christoffel symbols in terms of metric) mentioned in the literature, and, substituting this into the affine-connection form of the Regge action of our previous work, we get a second order form of the action. This can be expanded over metric variations from simplex to simplex. For a particular periodic simplicial structure and coordinates of the vertices, the leading order over metric variations is found to coincide with a certain finite difference form of the Hilbert-Einstein action.

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V. M. Khatsymovsky. 2019-06-27. On the discrete Christoffel symbols. https://doi.org/10.1142/s0217751x19501860

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