Just wanted to add to this: When has <math>\Lambda</math> failed? It's a perfectly legitimate term in the EFE's, as csmcmillion has stated. The righthand side of it is (in <math>G=c=1</math> units) is <math>8 \pi T^{\alpha \beta }</math>. You need something on the left which is a measure of curvature, i.e., the Einstein tensor <math>G^{\alpha \beta}</math>. Why the Einstein tensor? Why not the Ricci tensor? Why not the scalar curvature times the metric? The reason is that the stress-energy tensor (<b>T</b>) has to have zero divergence: <math>\nabla_{\beta} T^{\alpha \beta} = 0 </math>. The Einstein tensor has zero divergence as well. But, we also have <math> \nabla_{\gamma} g^{\alpha \beta} = 0</math>. So we can safely add in <math>\Lambda g^{\alpha \beta}</math> to the left hand side of the EFE without violating local conservation of energy-momentum. Using this gives <math>G^{\alpha \beta}+\Lambda g^{\alpha \beta} = 8 \pi T^{\alpha \beta}. </math> [[User:AndyFrankinson|AndyFrankinson]] 19:46, 15 November 2011 (EST) | Just wanted to add to this: When has <math>\Lambda</math> failed? It's a perfectly legitimate term in the EFE's, as csmcmillion has stated. The righthand side of it is (in <math>G=c=1</math> units) is <math>8 \pi T^{\alpha \beta }</math>. You need something on the left which is a measure of curvature, i.e., the Einstein tensor <math>G^{\alpha \beta}</math>. Why the Einstein tensor? Why not the Ricci tensor? Why not the scalar curvature times the metric? The reason is that the stress-energy tensor (<b>T</b>) has to have zero divergence: <math>\nabla_{\beta} T^{\alpha \beta} = 0 </math>. The Einstein tensor has zero divergence as well. But, we also have <math> \nabla_{\gamma} g^{\alpha \beta} = 0</math>. So we can safely add in <math>\Lambda g^{\alpha \beta}</math> to the left hand side of the EFE without violating local conservation of energy-momentum. Using this gives <math>G^{\alpha \beta}+\Lambda g^{\alpha \beta} = 8 \pi T^{\alpha \beta}. </math> [[User:AndyFrankinson|AndyFrankinson]] 19:46, 15 November 2011 (EST) |