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

A quantitative bump diagnostic

The empty fixed-input Einstein jet › A quantitative bump diagnostic · lines 1152–1177

XI-A@1#eq:bump

Because $g$ rises from zero to one somewhere, there is an interior open bump region on which $\partial_{t} g \geq c > 0$ . For nonnegative $f$ supported there with $\int f d^{4} x = 1$ ,

\int f (\partial_{μ} g) T_{0}^{μ 0} [ϕ_{m}] d^{4} x \geq \frac{c}{2} > 0.

This is an evaluated conservation obstruction, not an Einstein solution.

In the massless diagnostic $ϕ_{0} = t$ , take also $γ = ψ η$ , $ψ$ smooth. Then

T_{0}^{00} = \frac{1}{2}, (\partial_{μ} g) T_{0}^{μ 0} = \frac{1}{2} \partial_{t} g, G_{lin} [ψ η]_{μν} = - \partial_{μ} \partial_{ν} ψ + η_{μν} □ ψ .

Choosing a conformal test tensor does not alter the nonzero number (eq:bump) and does not solve the coupled metric equation. This is a check of the compatibility gate only.

Displayed formulas in this passage

XI-A@1#eq:bump · line 1157
$\int f (\partial_{μ} g) T_{0}^{μ 0} [ϕ_{m}] d^{4} x \geq \frac{c}{2} > 0.$
unlabeled · line 1167
$T_{0}^{00} = \frac{1}{2}, (\partial_{μ} g) T_{0}^{μ 0} = \frac{1}{2} \partial_{t} g, G_{lin} [ψ η]_{μν} = - \partial_{μ} \partial_{ν} ψ + η_{μν} □ ψ .$