2.3 The 3+1 decomposition of the spacetime

Although the spacetime description of general relativity is very elegant, the covariant form of Einstein equations is not suitable to describe how an initial configuration evolves towards the future. It is, therefore, more intuitive to instead consider a succession of spacetime geometries, where the evolution of a given slice is given by the Einstein equations (for more detailed treatments see [4, 24, 34, 96]). In order to convert the four-dimensional, covariant Einstein equations to a more intuitive “space+time” or 3+1 decomposition, the following steps are taken:
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