Manuscript II-B@1 · II-B-v1.tex · lemma · bfd2eb3c

Lemma: Restricted Einstein cubic

Complete contacts and the exact three-row certificate › Derivation of the reference value · lines 913–970 · proof included or linked

II-B@1#lem:cubic, II-B@1#eq:ABB

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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}) .

Consequently, for incoming $p + q + r = 0$ ,

v_{3} (A_{p}, B_{q}, B_{r}) = - 2 (p^{0} q^{0} + p^{0} r^{0} + q^{0} r^{0}) = (p^{0})^{2} + (q^{0})^{2} + (r^{0})^{2} .

No on-shell hypothesis on $p, q, r$ is required.

Proof. Set $q_{i} = 1 + κ H_{i}$ , $q_{0} = 1$ , and $ϕ_{i} = \frac{1}{2} lo g q_{i}$ , $ϕ_{0} = 0$ . For a diagonal metric the nonzero 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) .

Substitution into (eq:GG) gives

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 a prefactor is

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

For spatial $a$ this is zero. For $a = 0$ it is the remaining $H_{k}$ . Since $η^{00} = - 1$ , the asserted cubic follows.

Polarize it with one $A$ wave and two separately labelled $B$ waves. For each choice of undifferentiated wave the summed tensor coefficient is $- 2$ ; for example,

A_{11} (B_{22} B_{33} + B_{33} B_{22}) = - 2.

Each time derivative contributes $- i k^{0}$ , giving the first expression in (eq:ABB). The second uses energy conservation.

Displayed formulas in this passage

II-B@1#eq:ABB · line 929
$v_{3} (A_{p}, B_{q}, B_{r}) = - 2 (p^{0} q^{0} + p^{0} r^{0} + q^{0} r^{0}) = (p^{0})^{2} + (q^{0})^{2} + (r^{0})^{2} .$
unlabeled · line 916
$h_{μν} = diag (0, H_{1}, H_{2}, H_{3}), H_{1} + H_{2} + H_{3} = 0,$
unlabeled · line 920
$E_{3} = - \int d^{4} x (H_{1} H ˙_{2} H ˙_{3} + H_{2} H ˙_{1} H ˙_{3} + H_{3} H ˙_{1} H ˙_{2}) .$
unlabeled · line 942
$Γ^{a}_{aa} = \partial_{a} ϕ_{a}, Γ^{a}_{ab} = \partial_{b} ϕ_{a}, Γ^{a}_{bb} = - \frac{g _{bb}}{g _{aa}} \partial_{a} ϕ_{b} (a \neq = b) .$
unlabeled · line 949
$q_{1} q_{2} q_{3} a = 0 \sum^{3} \frac{η ^{aa}}{q _{a}} 1 \leq i < j \leq 3 i, j \neq = a \sum \frac{\partial _{a} H _{i} \partial _{a} H _{j}}{q _{i} q _{j}} .$
unlabeled · line 956
$\frac{1}{2} (H_{1} + H_{2} + H_{3}) - H_{a} - H_{i} - H_{j}, H_{0} = 0.$
unlabeled · line 965
$A_{11} (B_{22} B_{33} + B_{33} B_{22}) = - 2.$

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section: Derivation of the reference value