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

Exact algebraic checks

Explicit cubic density and source audit › Exact algebraic checks · lines 1448–1521

Em01-F@3#eq:conformalcheck

The source formula gives

Q (h, G) = 0 for constant h, G,

and

Q (h, cη) = - 2 c G_{1} (h) for constant c .

For $h = e_{01}$ constant and $G = v η$ ,

Q (h, v η)_{μν} = h_{ν}^{ρ} \partial_{μ} \partial_{ρ} v + h_{μ}^{ρ} \partial_{ν} \partial_{ρ} v - η_{μν} h^{ρ σ} \partial_{ρ} \partial_{σ} v - h_{μν} □_{η} v .

For $v = x^{0} x^{2}$ , its only nonzero components are

Q (e_{01}, v η)_{12} = Q (e_{01}, v η)_{21} = 1.

For conformal directions,

Q (u η, v η)_{μν} = 2 u \partial_{μ} \partial_{ν} v + 2 v \partial_{μ} \partial_{ν} u + 3 (\partial_{μ} u \partial_{ν} v + \partial_{μ} v \partial_{ν} u) - 2 η_{μν} (u □_{η} v + v □_{η} u) - 3 η_{μν} \partial^{ρ} u \partial_{ρ} v .

For $u = v = x^{0}$ , all four diagonal components equal $3$ .

These evaluations follow by substitution into (eq:Qevaluated). An independent conformal check uses $g = e^{2 ω} η$ :

G_{μν} [g] = - 2 \partial_{μ} \partial_{ν} ω + 2 η_{μν} □_{η} ω + 2 \partial_{μ} ω \partial_{ν} ω + η_{μν} (\partial ω)^{2} .

Set $e^{2 ω} = 1 + s u + t v$ and extract its mixed coefficient. The determinant and inverse-metric factors cancel in this four-dimensional conformal Euler density, giving (eq:conformalcheck).

Finally, completeness of the basis implies for generic symmetric $h, G$

Q (h, G)_{μν} = L \sum (e_{L})_{μν} α, β, I, J \sum q_{α β}^{L, I, J} (\partial^{α} G_{J}) (\partial^{β} h_{I}) .

Together with Lemma lem:components, this checks the full-index and ten-component calculations without adding off-diagonal weights.

Displayed formulas in this passage

Em01-F@3#eq:conformalcheck · line 1477
$Q (u η, v η)_{μν} = 2 u \partial_{μ} \partial_{ν} v + 2 v \partial_{μ} \partial_{ν} u + 3 (\partial_{μ} u \partial_{ν} v + \partial_{μ} v \partial_{ν} u) - 2 η_{μν} (u □_{η} v + v □_{η} u) - 3 η_{μν} \partial^{ρ} u \partial_{ρ} v .$
unlabeled · line 1451
$Q (h, G) = 0 for constant h, G,$
unlabeled · line 1456
$Q (h, cη) = - 2 c G_{1} (h) for constant c .$
unlabeled · line 1461
$Q (h, v η)_{μν} = h_{ν}^{ρ} \partial_{μ} \partial_{ρ} v + h_{μ}^{ρ} \partial_{ν} \partial_{ρ} v - η_{μν} h^{ρ σ} \partial_{ρ} \partial_{σ} v - h_{μν} □_{η} v .$
unlabeled · line 1470
$Q (e_{01}, v η)_{12} = Q (e_{01}, v η)_{21} = 1.$
unlabeled · line 1494
$G_{μν} [g] = - 2 \partial_{μ} \partial_{ν} ω + 2 η_{μν} □_{η} ω + 2 \partial_{μ} ω \partial_{ν} ω + η_{μν} (\partial ω)^{2} .$
unlabeled · line 1509
$Q (h, G)_{μν} = L \sum (e_{L})_{μν} α, β, I, J \sum q_{α β}^{L, I, J} (\partial^{α} G_{J}) (\partial^{β} h_{I}) .$