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

Assumed geometry and independently declared boundary region

Objects fixed before the reconstruction map › Assumed geometry and independently declared boundary region · lines 196–226

XVII-A@1#eq:metric, XVII-A@1#eq:region-point

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Fix $ℓ > 0$ . The background is the universal-cover global $AdS_{2}$ strip

M = R \times (- π /2, π /2), g_{ℓ} = \frac{ℓ ^{2}}{cos ^{2} x} (- d t^{2} + d x^{2}) .

The coordinate $t$ is dimensionless. The conformally rescaled metric extends to $x = \pm π /2$ . The right conformal boundary is

\partial_{R} M = {(t, π /2) : t \in R} .

Before introducing the field or a reconstruction map, declare

I_{B} = [- π /2, π /2], B = {(t, π /2) : t \in I_{B}}, p_{*} = (0, 0) \in M .

Thus $B$ is a boundary time interval. It is not a spatial tensor-factor subregion at one time. Its definition uses the assumed conformal completion of (eq:metric); it is not obtained by renaming any region from the supplied manuscripts.

The metric, its conformal completion, its coordinates, and the location $p_{*}$ are inputs.

Displayed formulas in this passage

XVII-A@1#eq:metric · line 200
$M = R \times (- π /2, π /2), g_{ℓ} = \frac{ℓ ^{2}}{cos ^{2} x} (- d t^{2} + d x^{2}) .$
XVII-A@1#eq:region-point · line 212
$I_{B} = [- π /2, π /2], B = {(t, π /2) : t \in I_{B}}, p_{*} = (0, 0) \in M .$
unlabeled · line 208
$\partial_{R} M = {(t, π /2) : t \in R} .$