Manuscript XXII-E@1 · XXII-E.tex · proposition · 51c99d2f

Proposition: A derivative-amplification witness

Error control, coverage, and derivative limits › Transport convergence does not control second derivatives · lines 1272–1311 · proof included or linked

XXII-E@1#prop:derivative, XXII-E@1#eq:derivative-witness

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On a fixed bounded comparison region, let

f_{n} (x) = n^{- 2} sin (n x^{1}), h^{(n)}_{μν} = 2 f_{n} η_{μν} .

Then $h^{(n)} \to 0$ uniformly together with its first derivatives, but

E_{η} [h^{(n)}]_{00} = (m - 2) sin (n x^{1})

does not converge uniformly to zero on any region whose $x^{1}$ projection contains an interval.

Moreover $e^{2 f_{n}} η$ has exactly the same pointwise cones as $η$ for every $n$ .

Proof. The zeroth and first derivative bounds are respectively $O (n^{- 2})$ and $O (n^{- 1})$ . Since

\partial_{1}^{2} f_{n} = - sin (n x^{1}), □ f_{n} = sin (n x^{1}), \partial_{0}^{2} f_{n} = 0,

equation (eq:conformal-einstein) gives (eq:derivative-witness). The supremum of its absolute value reaches $m - 2$ on such a region for all sufficiently large $n$ . Positive conformal multiplication preserves the cones.

Displayed formulas in this passage

unlabeled · line 1275
$f_{n} (x) = n^{- 2} sin (n x^{1}), h^{(n)}_{μν} = 2 f_{n} η_{μν} .$
XXII-E@1#eq:derivative-witness · line 1282
$E_{η} [h^{(n)}]_{00} = (m - 2) sin (n x^{1})$
unlabeled · line 1299
$\partial_{1}^{2} f_{n} = - sin (n x^{1}), □ f_{n} = sin (n x^{1}), \partial_{0}^{2} f_{n} = 0,$

Nearby in this section

section: Transport convergence does not control second derivatives