Ngô Quốc Anh

December 19, 2012

How to decompose tensors into a purely spatial part and a timelike part?

Filed under: Riemannian geometry — Tags: — Ngô Quốc Anh @ 15:02

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

Let us recall that M (called spatial surface) is a hypersurface of V (called the spacetime) of the dimension n+1. At each point p\in M, the space of all spacetime vectors can be orthogonally decomposed as

\displaystyle T_p(V)=T_p(M) \oplus \text{span}(n),

where \text{span}(n) stands for the 1-dimensional subspace of T_p(V) generated by the unit normal vector n to the surface M.

To do so, we need two projection operators.

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

\displaystyle \begin{gathered} \gamma :{T_p}(V) \to {T_p}(M) \hfill \\ \qquad \quad\,\,\,v \mapsto v + (n \cdot v)n. \hfill \\ \end{gathered}

According to the above decomposition and thanks to

\displaystyle g_V = g_M + n \otimes n

with respect to any basis (e_i) of the space T_p(V), we have

\displaystyle \gamma_{ij}=g_{ij}+n_in_j,

which, by raising indices, gives

\displaystyle \gamma^i_{j}=\delta^i_{j}+n^in_j.


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