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
For any spacetime vector , since
we know that is purely spatial. This supports our construction of the orthogonal projection .
To project higher rank tensors into the spatial surface, each free index has to be contracted with a projection operator. It is sometimes convenient to denote this projection with a symbol , for example,
It can be checked that is nothing but the a spatial (induced) metric of .
The normal projector onto . According to the decomposition
one can easily see that the term gives us the normal part after the decomposition. In other words, the normal projector is nothing but
We can denote the normal projector by , for example, for any spacetime vector , we obtain
Once we have the above two projector, we can now use these two projection operators to decompose any tensor into its spatial and timelike parts. For example, given an arbitrary spacetime vector , we obtain
Now, given a -tensor , we get
Thus, after re-indexing, we obtain
Finally, for the Riemann curvature tensor , we obtain the following decomposition