Discrete Geodesics and Cellular Automata
This paper proposes a dynamical notion of discrete geodesics, understood as straightest trajectories in discretized curved spacetime. The proposed notion is generic, as it is formulated in terms of a general deviation function, but readily specializes to metric spaces such as discretized pseudo-riemannian manifolds. It is effective: an algorithm for computing these geodesics naturally follows, which allows numerical validation—as shown by computing the perihelion shift of a Mercury-like planet. It is consistent, in the continuum limit, with the standard notion of timelike geodesics in a pseudo-riemannian manifold. Whether the algorithm fits within the framework of cellular automata is discussed at length.
KeywordsDiscrete connection Parallel transport General relativity Regge calculus
This work has been funded by the ANR-12-BS02-007-01 TARMAC grant, the ANR-10-JCJC-0208 CausaQ grant, and the John Templeton Foundation, grant ID 15619. Pablo Arrighi benefited from a visitor status at the IXXI institute of Lyon.
- 2.Arrighi, P., Dowek, G.: Discrete geodesics. arXiv Pre-print, with program available when downloading source (2015)Google Scholar
- 3.Arrighi, P., Facchini, S., Forets, M.: Quantum walks in curved spacetime (2015). Pre-print arXiv:1505.07023
- 8.d’Inverno, R.: Introducing Einstein’s Relatvity. Oxford University Press, USA (1899)Google Scholar
- 9.Gandy, R.: Church’s thesis and principles for mechanisms. In: Barwise, J., Keisler, H.J., Kunen, K. (eds.) The Kleene Symposium. North-Holland Publishing Company, Amsterdam (1980)Google Scholar
- 15.Polthier, K., Schmies, M.: Straightest geodesics on polyhedral surfaces. In: Discrete Differential Geometry: An Applied Introduction, SIGGRAPH 2006, p. 30 (2006)Google Scholar