Today, we aim to talk about how to decompose tensors into a purely spatial part which lies in the hypersurfaces and a timelike part which is normal to the spatial surface ?

Let us recall that (called spatial surface) is a hypersurface of (called the spacetime) of the dimension . At each point , the space of all spacetime vectors can be orthogonally decomposed as

where stands for the 1-dimensional subspace of generated by the unit normal vector to the surface .

To do so, we need two projection operators.

**The orthogonal projector onto **. In the literature, there exists such an operator, denoted by the symbol , given by

According to the above decomposition and thanks to

with respect to any basis of the space , we have

which, by raising indices, gives

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