Manuscript XVII-A@1 · XVII-A.tex · proposition · 711d7ee4

Proposition: Fixed-background field equation

Objects fixed before the reconstruction map › Declared bulk field and named observable · lines 301–321 · proof included or linked

XVII-A@1#prop:wave

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The matrix field (eq:bulk-field) satisfies

□_{g_{ℓ}} Φ = 0

and has zero leading boundary value at both ends of the strip.

Proof. In two dimensions,

□_{g_{ℓ}} = \frac{cos ^{2} x}{ℓ ^{2}} (- \partial_{t}^{2} + \partial_{x}^{2}) .

Each term of (eq:bulk-field) is annihilated by the operator in parentheses because its temporal and spatial frequencies are both $n$ . The sine spatial profile vanishes at $x = - π /2$ and $x = π /2$ . Finite summation preserves both assertions.

Displayed formulas in this passage

unlabeled · line 304
$□_{g_{ℓ}} Φ = 0$
unlabeled · line 312
$□_{g_{ℓ}} = \frac{cos ^{2} x}{ℓ ^{2}} (- \partial_{t}^{2} + \partial_{x}^{2}) .$

Nearby in this section

section: Declared bulk field and named observable
remark: Remark: The observable is not an unproved continuum compression
section: Declared bulk field and named observable