The general covariant approach to general relativity is not adapted to our experience from everyday life. The most intuitive concept is not that of spacetime geometry, but rather that of a time succession of space geometries. This ‘flowing geometries’ picture could be easily put into the computer, by discretizing the time coordinate, in the same way that the continuous time flow of the real life is coded in terms of a discrete set of photograms in a movie.
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Bona, C., Bona-Casas, C., Palenzuela-Luque, C. (2009). The Evolution Formalism. In: Elements of Numerical Relativity and Relativistic Hydrodynamics. Lecture Notes in Physics, vol 783. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-01164-1_2
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