Gauss-Codazzi equations

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The Gauss–Codazzi equations are the following collection of equations which relate the 4-dimensional Riemann tensor Rabcd, Ricci tensor Rab and Ricci scalar R to their projection onto a 3-dimensional hypersurface embedded within 4-dimensional space-time, which will be denoted by (3)Rabcd, (3)Rab 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)}\nabla_{a}K^{a}_{\;\;c} - \;^{(3)}\nabla_{c}K = h^{a}_{c}R_{ab}n^{b}

The normal of a hypersurface Σ defined in space-time by f(x) = 0 equals

n_{a} = \frac{\partial_{a}f}{\sqrt{\pm\partial_{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 hab 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 Kab is the projection of \nabla n into the hypersurface by K_{ab} = h_{a}^{c}h_{b}^{d}\nabla_{c}n_{d} with trace K = Kaa.

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