Manuscript XI-A@1 · XI-A.tex · theorem · b8211ecc

Theorem: Empty fixed-input Einstein jet

The empty fixed-input Einstein jet › A fixed input for which the jet is empty · lines 1106–1141 · proof included or linked

XI-A@1#thm:empty, XI-A@1#eq:empty

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For every nonzero compact switch of the frozen class,

J_{g} (ϕ_{m}) = \emptyset .

The obstruction occurs in the first sourced metric coefficient, before the second-order scalar or metric coefficients can repair it.

Proof. The input has no spatial derivative. A direct computation gives

T_{0}^{00} [ϕ_{m}] = \frac{1}{2}, T_{0}^{0 i} [ϕ_{m}] = 0,

and

T_{0}^{ij} [ϕ_{m}] = {\frac{1}{2} cos (2 m t) δ^{ij}, \frac{1}{2} δ^{ij}, m > 0, m = 0.

Consequently the $ν = 0$ component of (eq:compatibility) is

\frac{1}{2} \partial_{t} g = 0.

A nonzero function compactly supported in the slab interior cannot be independent of time: for each fixed spatial point it vanishes near both time boundaries. Since the switch has a nonempty plateau, $\partial_{t} g$ is not identically zero. The necessary compatibility condition fails.

Displayed formulas in this passage

XI-A@1#eq:empty · line 1109
$J_{g} (ϕ_{m}) = \emptyset .$
unlabeled · line 1119
$T_{0}^{00} [ϕ_{m}] = \frac{1}{2}, T_{0}^{0 i} [ϕ_{m}] = 0,$
unlabeled · line 1124
$T_{0}^{ij} [ϕ_{m}] = {\frac{1}{2} cos (2 m t) δ^{ij}, \frac{1}{2} δ^{ij}, m > 0, m = 0.$
unlabeled · line 1133
$\frac{1}{2} \partial_{t} g = 0.$

Theorem: Empty fixed-input Einstein jet

Displayed formulas in this passage

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