Ricci decomposition: Difference between revisions

Revision as of 19:32, 13 August 2005

Revision as of 19:43, 13 August 2005

Line 18:

:<math> S_{abcd} = \frac{R}{(n-1) \, (n-2)} \, H_{abcd}</math>

is built using the [[Ricci scalar]] <math>R = {R^m}_m</math>, where <math>R_{ab}</math> is the [[Ricci tensor]], and a tensor constructed algebraically from the [[metric tensor]] <math>g_{ab}</math>,

:<math>H_{abcd}=g_{a \left[ c \right.}g_{\left. d \right] b} = g_{ac} \, g_{bd} - g_{ad} \, g_{bc}</math>

Here, <math>n</math> is the dimension of the semi-Riemannian manifold under discussion.

The semi-traceless part

:<math>E_{abcd}=S_{a[c}g_{d]b}+S_{b[d}g_{c]a} = S_{ac} \, g_{bd} - S_{ad} \, g_{bc} + S_{bd} \, g_{ac} - S_{bc} \, g_{ad}</math>

is constructed algebraically using the metric tensor and the ''traceless part'' of the Ricci tensor

:<math> S_{ab} = R_{ab} - \frac{1}{n} \, g_{ab} \, R</math>