Manuscript I-B-VP@3 · I-B-VP-v3.tex · section · 93c04eb4

Signature ledger

Host, bundle, field normalization, and signature › Signature ledger · lines 459–492

On a developing chart,

X = r cos (α φ), Y = r sin (α φ) .

If a companion completion uses

η = d t^{2} - d X^{2} - d Y^{2} - d z^{2},

then

dev^{*} η = - g, - □_{g} = dev^{*} □_{η} .

The equation operator and action Hessian have opposite signs. This distinction matters when comparing inverse conventions.

Euclidean continuation uses

g_{E} = d τ^{2} + d r^{2} + α^{2} r^{2} d φ^{2} + d z^{2}, L_{E} = - \partial_{τ}^{2} + B .

With $t = - i τ$ , mixed stress components obey

T^{t}_{t} = T^{τ}_{τ}_{E}, T^{i}_{j} = T^{i}_{j}_{E} .

The static energy density is

ε = T_{tt} = - T^{t}_{t} .

Displayed formulas in this passage

unlabeled · line 462
$X = r cos (α φ), Y = r sin (α φ) .$
unlabeled · line 467
$η = d t^{2} - d X^{2} - d Y^{2} - d z^{2},$
unlabeled · line 471
$dev^{*} η = - g, - □_{g} = dev^{*} □_{η} .$
unlabeled · line 479
$g_{E} = d τ^{2} + d r^{2} + α^{2} r^{2} d φ^{2} + d z^{2}, L_{E} = - \partial_{τ}^{2} + B .$
unlabeled · line 484
$T^{t}_{t} = T^{τ}_{τ}_{E}, T^{i}_{j} = T^{i}_{j}_{E} .$
unlabeled · line 489
$ε = T_{tt} = - T^{t}_{t} .$