Manuscript II@2 · II.tex · lemma · 50d64def

Lemma: Trace-free diagonal restriction

The three labels and the reference value › Restricted Einstein cubic · lines 1139–1216 · proof included or linked

II@2#lem:diagonal, II@2#eq:diagonal-cubic, II@2#eq:exact-diagonal

On the linear subspace

h_{μν} = diag (0, H_{1}, H_{2}, H_{3}), H_{1} + H_{2} + H_{3} = 0,

the cubic coefficient is, modulo a divergence,

E_{3} = - \int d^{4} x (H_{1} H ˙_{2} H ˙_{3} + H_{2} H ˙_{1} H ˙_{3} + H_{3} H ˙_{1} H ˙_{2}) .

It may be polarized using any three fields in this subspace, including an off-shell insertion.

Proof. Set

q_{i} = 1 + κ H_{i}, q_{0} = 1, ϕ_{i} = \frac{1}{2} lo g q_{i}, ϕ_{0} = 0.

The nonzero diagonal-metric Christoffel patterns are

Γ^{a}_{aa} = \partial_{a} ϕ_{a}, Γ^{a}_{ab} = \partial_{b} ϕ_{a}, Γ^{a}_{bb} = - \frac{g _{bb}}{g _{aa}} \partial_{a} ϕ_{b} (a \neq = b) .

Three-distinct-index entries vanish. Substituting these patterns and collecting each derivative direction gives

g^{μν} (Γ^{α}_{μ β} Γ^{β}_{ν α} - Γ^{α}_{μν} Γ^{β}_{α β}) = 2 a = 0 \sum^{3} g^{aa} i < j i, j \neq = a \sum \partial_{a} ϕ_{i} \partial_{a} ϕ_{j} .

Since $\partial_{a} ϕ_{i} = κ \partial_{a} H_{i} / (2 q_{i})$ , the density (eq:GammaGamma) becomes

q_{1} q_{2} q_{3} a = 0 \sum^{3} \frac{η ^{aa}}{q _{a}} 1 \leq i < j \leq 3 i, j \neq = a \sum \frac{\partial _{a} H _{i} \partial _{a} H _{j}}{q _{i} q _{j}} .

The coefficient of $κ$ in the prefactor of the term indexed by $a, i, j$ is

\frac{1}{2} (H_{1} + H_{2} + H_{3}) - H_{a} - H_{i} - H_{j}, H_{0} = 0.

For spatial $a$ , the three indices exhaust $1, 2, 3$ , so this coefficient vanishes. For $a = 0$ , it equals the remaining $H_{k}$ . The factor $η^{00} = - 1$ proves (eq:diagonal-cubic).

As a normalization check, the time-derivative quadratic part is

- i < j \sum H ˙_{i} H ˙_{j} = \frac{1}{2} i \sum H_{i} ˙^{2}

on the trace-free subspace, in agreement with (eq:PF). Restriction to a linear subspace commutes with polarization. No field equation for the inserted fields was used.

Displayed formulas in this passage

II@2#eq:diagonal-cubic · line 1146
$E_{3} = - \int d^{4} x (H_{1} H ˙_{2} H ˙_{3} + H_{2} H ˙_{1} H ˙_{3} + H_{3} H ˙_{1} H ˙_{2}) .$
II@2#eq:exact-diagonal · line 1190
$q_{1} q_{2} q_{3} a = 0 \sum^{3} \frac{η ^{aa}}{q _{a}} 1 \leq i < j \leq 3 i, j \neq = a \sum \frac{\partial _{a} H _{i} \partial _{a} H _{j}}{q _{i} q _{j}} .$
unlabeled · line 1142
$h_{μν} = diag (0, H_{1}, H_{2}, H_{3}), H_{1} + H_{2} + H_{3} = 0,$
unlabeled · line 1162
$q_{i} = 1 + κ H_{i}, q_{0} = 1, ϕ_{i} = \frac{1}{2} lo g q_{i}, ϕ_{0} = 0.$
unlabeled · line 1167
$Γ^{a}_{aa} = \partial_{a} ϕ_{a}, Γ^{a}_{ab} = \partial_{b} ϕ_{a}, Γ^{a}_{bb} = - \frac{g _{bb}}{g _{aa}} \partial_{a} ϕ_{b} (a \neq = b) .$
unlabeled · line 1176
$g^{μν} (Γ^{α}_{μ β} Γ^{β}_{ν α} - Γ^{α}_{μν} Γ^{β}_{α β}) = 2 a = 0 \sum^{3} g^{aa} i < j i, j \neq = a \sum \partial_{a} ϕ_{i} \partial_{a} ϕ_{j} .$
unlabeled · line 1200
$\frac{1}{2} (H_{1} + H_{2} + H_{3}) - H_{a} - H_{i} - H_{j}, H_{0} = 0.$
unlabeled · line 1209
$- i < j \sum H ˙_{i} H ˙_{j} = \frac{1}{2} i \sum H_{i} ˙^{2}$

Lemma: Trace-free diagonal restriction

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