Gdot-Kdot

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Notation:

g_{ab} is the spatial metric, \alpha is the scalar lapse function, \beta^a is the shift vector, D_a is the spatial covariant derivative, and K_{ab} is the extrinsic curvature.

Equations of motion:

In terms of the time derivative operator \partial_\perp \equiv \partial_t - {\mathcal L}_\beta, the equations of motion are:


\begin{array}{rcl}
\partial_\perp g_{ab} & = &  -2\alpha K_{ab}  \\
\partial_\perp K_{ab} & = & \alpha K K_{ab} - 2\alpha K_{ac}K^c_b + \alpha R_{ab} - D_a D_b\alpha  \\
\end{array}

where R_{ab} is the spatial Ricci tensor. Constraints and constraint evolution system:


\begin{array}{rcl}
{\mathcal H} & \equiv & K^2 - K_{ab}K^{ab} + R = 0 \\
{\mathcal M}_a & \equiv & D_b K^b_a - D_a K  = 0 \\
\partial_\perp{\mathcal H} & = &  2\alpha K {\mathcal H} -4{\mathcal M}^a D_a\alpha - 2\alpha D_a {\mathcal M}^a  \\
\partial_\perp{\mathcal M}_a & = & -\frac{1}{2}\alpha D_a{\mathcal H} - {\mathcal H}D_a\alpha + \alpha K {\mathcal M}_a \\
\end{array}

Other relations:


\partial_\perp(D_a {\mathcal M}^a) = -\frac{1}{2} \alpha D^2 {\mathcal H} - \frac{3}{2} D_a\alpha D^a{\mathcal H} - {\mathcal H} D^2\alpha 
    + \alpha K D_a{\mathcal M}^a + 2 D_a(\alpha K^a_b{\mathcal M}_b)
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