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

Proposition: Radius is not determined by the declared data

The maps not supplied › A concrete metric nonidentifiability statement · lines 997–1032 · proof included or linked

XVII-A@1#prop:radius

Consider the family of models obtained by replacing $ℓ$ in (eq:metric) by any positive radius, while keeping the dimensionless coordinates, $K, d$ , the displayed mode normalization, $U$ , and the corresponding preparation fixed.

All boundary operator families, boundary states, and reconstructed records declared in this paper are identical throughout this family. Therefore no deterministic rule using only these data can return the correct radius for every member of the family. The corresponding metrics have different scalar curvature when their radii differ.

Proof. The finite matrices (eq:finite-oscillator), the bulk expansion (eq:bulk-field), the boundary family (eq:boundary-field), and the correspondence (eq:U) contain no $ℓ$ . Consequently the declared boundary states and the record channel are also independent of $ℓ$ . A deterministic rule applied to identical data must return the same answer.

For completeness, write

g_{ℓ} = e^{2 σ} (- d t^{2} + d x^{2}), σ = lo g ℓ - lo g cos x .

The two-dimensional conformal curvature formula gives

Scal (g_{ℓ}) = - 2 e^{- 2 σ} \partial_{x}^{2} σ = - \frac{2}{ℓ ^{2}} .

Distinct positive radii therefore give distinct scalar curvatures.

Displayed formulas in this passage

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$g_{ℓ} = e^{2 σ} (- d t^{2} + d x^{2}), σ = lo g ℓ - lo g cos x .$
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$Scal (g_{ℓ}) = - 2 e^{- 2 σ} \partial_{x}^{2} σ = - \frac{2}{ℓ ^{2}} .$

Proposition: Radius is not determined by the declared data

Displayed formulas in this passage

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