Ngô Quốc Anh

December 31, 2012

n+1 decomposition of the spacetime metric

Filed under: Uncategorized — Tags: — Ngô Quốc Anh @ 17:29

Let us introduce the components \gamma_{ij} of the n-metric \gamma of the hypersurface M with respect to the coordinates (x^i)

\displaystyle \gamma=\gamma_{ij}dx^i\otimes dx^j.

From the definition of the shift vector \beta, we can write

\displaystyle \beta_i=\gamma_{ij}\beta^j.

For the spacetime metric g of V, with respect to the coordinates (x^\alpha) we can also write

\displaystyle g=g_{\alpha\beta}dx^\alpha\otimes dx^\beta.

Keep in mind that \gamma=g|_M. Thanks to \partial_t = Nn+\beta, we obtain

\displaystyle \begin{array} {lcl}{g_{00}} &=&\displaystyle g({\partial _t},{\partial _t}) \hfill \\&=&\displaystyle g(Nn + \beta ,Nn + \beta ) \hfill \\ &=&\displaystyle {N^2}g(n,n) + g(\beta ,\beta ) \hfill \\ &=&\displaystyle - {N^2} + \gamma (\beta ,\beta ) \hfill \\ &=&\displaystyle - {N^2} + {\beta _i}{\beta ^i}. \hfill \\ \end{array}

Similarly, one can also obtain

\displaystyle\begin{array}{lcl} {g_{0i}} &=&\displaystyle g({\partial _t},{\partial _i}) \hfill \\&=&\displaystyle g(Nn + \beta ,{\partial _i}) \hfill \\&=&\displaystyle g({\beta _j}d{x^j},{\partial _i}) \hfill \\&=&\displaystyle {\beta _i}. \hfill \\ \end{array}

Finally, it is easy to see that

\displaystyle {g_{ij}} = g({\partial _i},{\partial _j}) = \gamma ({\partial _i},{\partial _j}) = {\gamma _{ij}}

for all i,j \geqslant 1.

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