Manuscript I-B-VP-1@1 · I-B-VP-1-v1.tex · assumption · 6d700be9

Assumption: Smooth completion and density

The completion, regular data, and Ward prescription › Geometry and the additional smooth structure · lines 231–255

I-B-VP-1@1#ass:completion, I-B-VP-1@1#eq:density

Identify the topological completion with

M = R_{y}^{1 + 1} \times R_{x}^{2}, y = (t, z), x = (x^{1}, x^{2}) = (r cos φ, r sin φ) .

Equip it with this Cartesian smooth structure and the auxiliary smooth metric

g ˉ = - d t^{2} + d z^{2} + (d x^{1})^{2} + (d x^{2})^{2} .

The pairing density is

d Ω = α d t d z d^{2} x .

It equals the host volume density on $M$ .

All tensor distributions below are sections of the ordinary smooth tensor bundles of $M$ . Stress is extended as a mixed tensor $T^{μ}_{ν}$ ; the current is extended first as a contravariant vector $J^{μ}$ .

Displayed formulas in this passage

I-B-VP-1@1#eq:density · line 245
$d Ω = α d t d z d^{2} x .$
unlabeled · line 234
$M = R_{y}^{1 + 1} \times R_{x}^{2}, y = (t, z), x = (x^{1}, x^{2}) = (r cos φ, r sin φ) .$
unlabeled · line 241
$g ˉ = - d t^{2} + d z^{2} + (d x^{1})^{2} + (d x^{2})^{2} .$

Assumption: Smooth completion and density

Displayed formulas in this passage

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