Physical System

The line element is

$$ds^2 = -\alpha^2 dt^2 + \gamma_{ij} dx^i dx^j,$$ (part301)

where $\alpha$ is the lapse and $\gamma_{ij}$ the 3-metric. Defining $n$ to be the normal to the slice, we have the extrinsic curvature $K_{ij}$ given by

$$K_{ij} = \frac{1}{2}{\cal L}_n \gamma_{ij}$$ (part302)

where ${\cal L}$ is the Lie derivative.

The ADM equations then evolve the spatial three metric $\gamma_{ij}$ and the extrinsic curvature $K_{ij}$ using

$$\frac{d}{dt} \; \gamma_{ij} = -2\alpha K_{ij},$$ (part303)

$$\frac{d}{dt} \; K_{ij} = -D_{i}D_{j}\alpha + \alpha \left( \rule{0mm}{4mm} K K_{ij} - 2 K_{ik}K^{k}{}_{j} \right),$$ (part304)

with

$$\frac{d}{dt} = \partial_t .$$ (part305)

Here $D_{i}$ is the covariant derivative associated with the three-dimensional metric $\gamma_{ij}$.