Gauss-Codazzi equations (relativity)

The Gauss–Codazzi equations are the following collection of equations which relate the 4-dimensional Riemann tensor $R\_\{abcd\}$, Ricci tensor $R\_\{ab\}$ and Ricci scalar $R$ to their projection onto a 3-dimensional hypersurface embedded within 4-dimensional space-time, which will be denoted by $\{\}^\{(3)\}R\_\{abcd\}$, $\{\}^\{(3)\}R\_\{ab\}$ and $\{\}^\{(3)\}R$, respectively.

*$;^\{(3)\}R\_\{abcd\}\; =\; h^\{p\}\_\{a\}h^\{q\}\_\{b\}h^\{r\}\_\{c\}h^\{s\}\_\{d\}R\_\{pqrs\}pm\; K\_\{ac\}K\_\{bd\}\; mp\; K\_\{ad\}K\_\{bc\}$

*$;^\{(3)\}R\_\{bd\}\; =\; h^\{pr\}h^\{q\}\_\{b\}h^\{s\}\_\{d\}R\_\{pqrs\}pm\; KK\_\{bd\}\; mp\; K\_\{bc\}K^\{c\}\_\{;;d\}$

*$;^\{(3)\}R\; =\; R\; mp\; 2n^\{a\}n^\{b\}R\_\{ab\}\; pm\; K^\{2\}\; mp\; K\_\{ab\}K^\{ab\}$

*$;^\{(3)\}\; abla\_\{a\}K^\{a\}\_\{;;c\}\; -\; ;^\{(3)\}\; abla\_\{c\}K\; =\; h^\{a\}\_\{c\}R\_\{ab\}n^\{b\}$

The normal of a hypersurface $Sigma$ defined in space-time by $f(x)=0$ equals:$n\_\{a\}\; =\; frac\{partial\_\{a\}f\}\{sqrt\{pmpartial\_\{b\}f\; partial\_\{c\}f\; g^\{bc\},$ where the sign depends on whether $partial\_\{a\}f$ is time or space-like and choice of signature. The first fundamental form $h\_\{ab\}$ is the induced metric on the hypersurface related to the space-time metric as $g\_\{ab\}\; =\; h\_\{ab\}\; pm\; n\_\{a\}n\_\{b\}$.

The second fundamental form $K\_\{ab\}$ is the projection of $abla\; n$ into the hypersurface by $K\_\{ab\}\; =\; h\_\{a\}^\{c\}h\_\{b\}^\{d\}\; abla\_\{c\}n\_\{d\}$ with trace $K\; =\; K^\{a\}\{\}\_\{a\}$.