Manuscript XIV-E@1 · XIV-E.tex · lemma · e89fd2a5

Lemma: Local zero locus

The elementary product phase is not a common symbol calculus › The common zero set of the cone equations · lines 926–963 · proof included or linked

XIV-E@1#lem:zerolocus

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On these neighborhoods,

q_{R} = q_{1} = q_{2} = 0 ⟺ (b, a, y) \in N^{\circ} .

Consequently the auxiliary critical set of (eq:phase) is

C_{Φ} = N_{att} \times (0, \infty)^{3} \times R^{4} .

Proof. Set $ξ = b - a$ and $η^{'} = y - a$ . The equations imply

ξ^{2} = (η^{'})^{2} = (ξ - η^{'})^{2} = 0, ξ \cdot η^{'} = 0.

The time-order inequalities make $ξ$ and $η^{'}$ nonzero future null vectors. Write

ξ = β (1, n), η^{'} = α (1, m), α, β > 0.

Their scalar product is $α β (1 - n \cdot m)$ , so it vanishes exactly when $n = m$ . The second time inequality gives $α < β$ . This is the copied parameterization of $N^{\circ}$ .

The auxiliary derivatives of $Φ$ are

- q_{R}, q_{1}, q_{2}, a - z .

Their simultaneous vanishing gives the stated critical set.

Displayed formulas in this passage

unlabeled · line 929
$q_{R} = q_{1} = q_{2} = 0 ⟺ (b, a, y) \in N^{\circ} .$
unlabeled · line 935
$C_{Φ} = N_{att} \times (0, \infty)^{3} \times R^{4} .$
unlabeled · line 942
$ξ^{2} = (η^{'})^{2} = (ξ - η^{'})^{2} = 0, ξ \cdot η^{'} = 0.$
unlabeled · line 948
$ξ = β (1, n), η^{'} = α (1, m), α, β > 0.$
unlabeled · line 959
$- q_{R}, q_{1}, q_{2}, a - z .$

Lemma: Local zero locus

Displayed formulas in this passage

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