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

Mixed curvature calculation

Explicit cubic density and source audit › Mixed curvature calculation · lines 1306–1366

Em01-F@3#eq:mixedconnection, Em01-F@3#eq:mixedRicci, Em01-F@3#eq:mixedEinstein

The insertion uses

Q (h, G) = D J_{2} [G] h .

The dynamical field $h$ occupies the $a$ -slot; the marked field $G$ occupies the $z$ -slot.

Define

D (h, G)_{μν}^{ρ} = - \frac{1}{2} (h^{ρ σ} S (G)_{σ μν} + G^{ρ σ} S (h)_{σ μν}),

R_{1} (h)_{μν} = \partial_{ρ} C (h)_{μν}^{ρ} - \partial_{ν} C (h)_{μ ρ}^{ρ}, r_{1} (h) = η^{μν} R_{1} (h)_{μν},

G_{1} (h)_{μν} = R_{1} (h)_{μν} - \frac{1}{2} η_{μν} r_{1} (h) .

Set

U (h, G)_{μν} = \partial_{ρ} D (h, G)_{μν}^{ρ} - \partial_{ν} D (h, G)_{μ ρ}^{ρ} + C (h)_{ρ α}^{ρ} C (G)_{μν}^{α} + C (G)_{ρ α}^{ρ} C (h)_{μν}^{α} - C (h)_{ν α}^{ρ} C (G)_{μ ρ}^{α} - C (G)_{ν α}^{ρ} C (h)_{μ ρ}^{α},

s (h, G) = η^{μν} U (h, G)_{μν} - h^{μν} R_{1} (G)_{μν} - G^{μν} R_{1} (h)_{μν},

E (h, G)_{μν} = U (h, G)_{μν} - \frac{1}{2} η_{μν} s (h, G) - \frac{1}{2} (h_{μν} r_{1} (G) + G_{μν} r_{1} (h)) .

Displayed formulas in this passage

Em01-F@3#eq:mixedconnection · line 1316
$D (h, G)_{μν}^{ρ} = - \frac{1}{2} (h^{ρ σ} S (G)_{σ μν} + G^{ρ σ} S (h)_{σ μν}),$
Em01-F@3#eq:mixedRicci · line 1339
$U (h, G)_{μν} = \partial_{ρ} D (h, G)_{μν}^{ρ} - \partial_{ν} D (h, G)_{μ ρ}^{ρ} + C (h)_{ρ α}^{ρ} C (G)_{μν}^{α} + C (G)_{ρ α}^{ρ} C (h)_{μν}^{α} - C (h)_{ν α}^{ρ} C (G)_{μ ρ}^{α} - C (G)_{ν α}^{ρ} C (h)_{μ ρ}^{α},$
Em01-F@3#eq:mixedEinstein · line 1358
$E (h, G)_{μν} = U (h, G)_{μν} - \frac{1}{2} η_{μν} s (h, G) - \frac{1}{2} (h_{μν} r_{1} (G) + G_{μν} r_{1} (h)) .$
unlabeled · line 1309
$Q (h, G) = D J_{2} [G] h .$
unlabeled · line 1325
$R_{1} (h)_{μν} = \partial_{ρ} C (h)_{μν}^{ρ} - \partial_{ν} C (h)_{μ ρ}^{ρ}, r_{1} (h) = η^{μν} R_{1} (h)_{μν},$
unlabeled · line 1333
$G_{1} (h)_{μν} = R_{1} (h)_{μν} - \frac{1}{2} η_{μν} r_{1} (h) .$
unlabeled · line 1351
$s (h, G) = η^{μν} U (h, G)_{μν} - h^{μν} R_{1} (G)_{μν} - G^{μν} R_{1} (h)_{μν},$