On the discrete Christoffel symbols

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.

21 pages, explanations and refs added

References in corpus (3)

• Quantum Gravity on the Lattice
• Gravitational Wilson Loop and Large Scale Curvature
• Wilson loops in CDT quantum gravity

Cited by in corpus (6)

• On the discrete version of the Schwarzschild problem
• On the discrete version of the black hole solution
• On the gravitational diagram technique in the discrete setup
• On the discrete version of the Kerr-Newman solution
• On the discrete version of the Reissner-Nordström solution
• On the discrete version of the Kerr geometry