Manuscript Em01-F@3 · Em01-F-v3.tex · section · 225e59af

Density and scalar vertex

Density and scalar vertex · lines 1046–1068

For example,

D_{dens} (e_{00}, e_{00}) = diag (\frac{3}{4}, \frac{1}{4}, \frac{1}{4}, \frac{1}{4}),

D_{dens} (e_{01}, e_{01}) = diag (- 1, 1, - 1, - 1) .

These follow both from (eq:Ddensity) and from directly inverting the corresponding one-parameter metric matrices. For arbitrary $h, G$ , bilinearity and basis completeness give

D_{dens}^{μν} (G, h) = I, J \sum D^{μν, I, J} h_{I} G_{J} .

Thus the full-index and ten-component density calculations agree.

Displayed formulas in this passage

unlabeled · line 1049
$D_{dens} (e_{00}, e_{00}) = diag (\frac{3}{4}, \frac{1}{4}, \frac{1}{4}, \frac{1}{4}),$
unlabeled · line 1054
$D_{dens} (e_{01}, e_{01}) = diag (- 1, 1, - 1, - 1) .$
unlabeled · line 1062
$D_{dens}^{μν} (G, h) = I, J \sum D^{μν, I, J} h_{I} G_{J} .$

Nearby in this section

section: Density and scalar vertex
proposition: Proposition: Identified density derivative
corollary: Corollary: Derived matter vertex