Abstract
This is a technical chapter to prepare the following ones. We motivate and perform a conformal decomposition of the 3-metric on each hypersurface of a 3+1 slicing. To avoid dealing with tensor densities, we introduce a background flat 3-metric. The link between the connections of the physical 3-metric and the conformal one is exhibited, leading to the computation of the Ricci tensor of the conformal 3-metric. Two associated decompositions of the extrinsic curvature are presented, with two different conformal rescalings of the traceless part. The 3+1 Einstein equations are then rewritten in terms of the conformal quantities. Finally, we discuss the Isenberg-Wilson-Mathews approximation to general relativity, which amounts to assuming that the conformal 3-metric is flat and that the slicing is maximal.
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Notes
- 1.
See also Ref. [2] which is freely accessible on the web.
- 2.
The \(C^k_{\;\,ij}\) are not to be confused with the components of the Cotton tensor discussed in Sect. 7.1 Since we shall no longer make use of the latter, no confusion may arise.
- 3.
Notice that we are using a hat, instead of a tilde, to distinguish this quantity from that defined by (7.63).
- 4.
To be discussed in Sect. 10.2.2.
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Gourgoulhon, É. (2012). Conformal Decomposition. In: 3+1 Formalism in General Relativity. Lecture Notes in Physics, vol 846. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-24525-1_7
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