Comments

This thorn evolves the standard ADM equations, see [7]. The line element is

$$ds^2 = -(\alpha^2 - \beta^i \beta_i) dt^2 + \beta_i dt dx^i + \gamma_{ij} dx^i dx^j,$$ (part211)

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

$$K_{ij} = \frac{1}{2}{\cal L}_n \gamma_{ij}$$ (part212)

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

$$\frac{d}{dt} \; \gamma_{ij} = -2\alpha K_{ij},$$ (part213)

$$\frac{d}{dt} \; K_{ij} = -D_{i}D_{j}\alpha + \alpha \left( \rule{0mm}{4mm} R_{ij} + K K_{ij} \right.$$

$$\left. - 2 K_{ik}K^{k}{}_{j} - {}^{(4)}R_{ij} \right),$$ (part214)

with

$$\frac{d}{dt} = \partial_t - \cal{L}_{\beta}$$ (part215)

and where $\cal{L}_{\beta}$ is the Lie derivative with respect to the shift vector $\beta^i$. Here $R_{ij}$ is the Ricci tensor and $D_{i}$ the covariant derivative associated with the three-dimensional metric $\gamma_{ij}$. The 4-dimensional Ricci tensor ${}^{(4)}R_{ij}$ is usually written in terms of the energy density $\rho$ and stress tensor $S_{ij}$ of the matter as seen by the normal (Eulerian) observers:

$${}^{(4)}R_{ij} = 8 \pi \left[ S_{ij} - \frac{1}{2} \left( S - \rho \right) \right] .$$ (part216)