# Eikonal Blog

## 2010.07.22

### Library of metrics

Filed under: General relativity, geometry — Tags: , , — sandokan65 @ 09:42

In $D=1+3$ dimensions the ADM (Arnowitt-Deser-Misner) split of the metric is given by ($i,j\in\overline{1,3}$, $\mu,\nu\in\overline{0,3}$):

$ds^2 = -N^2 dt^2 + g_{ij}^{(3)}(dx^i+N^i dt)(dx^j+N^j dt)$
where $N$ is the lapse variable, $N^i$ is the shift 3-vector.

Here:

• $R = K_{ij}K^{ij} - K^2 + R^{(3)} + 2\nabla_\mu (n^\mu \nabla_\nu n^\nu - n^\nu \nabla_\nu n^\mu)$ is the Ricci scalar,
• $K_{ij} :\equiv \frac1{2N} (\dot{g}_{ij}^{(3)} - \nabla_i^{(3)} N_j - \nabla_j^{(3)} N_i)$ is the extrinsic curvature, $K :\equiv g^{ij}K_{ij}$.

Source: “Unifying inflation with dark energy in modified F(R) Horava-Lifshitz gravity” by E. Elizalde, S. Nojiri, S. D. Odintsov, D. Saez-Gomez; arXiv.org > hep-th > arXiv:1006.3387 (http://arxiv4.library.cornell.edu/abs/1006.3387).

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