Manuscript XIX-A@1 · XIX-A.tex · section · e30d8062

The regulated coordinate density

Third refusal: the density ramp is a named non-clock › The regulated coordinate density · lines 1023–1053

XIX-A@1#eq:ramp, XIX-A@1#eq:ramped-density

Choose a fixed $χ \in C^{\infty} (R)$ with

0 \leq χ \leq 1, χ (u) = 0 (u \leq 0), χ (u) = 1 (u \geq 1),

and a nonempty transition region. Put

σ_{ε} (t) = χ (\frac{t - t _{0}}{ε}), 0 < ε < t_{1} - t_{0} .

The X-A-type regulated density is

b_{ε, e}^{μν} = η^{μν} + σ_{ε} (t) (a_{e}^{μν} - η^{μν}) .

It defines a coordinate equation by replacing $a_{e}$ with $b_{ε, e}$ in (eq:harmonic), with the same zero coefficient Cauchy data.

This is the density-ramp placement described in the supplied X-A record CheungXA. It is not field-argument convolution, and it is not a scalar-clock equation.

Displayed formulas in this passage

XIX-A@1#eq:ramp · line 1032
$σ_{ε} (t) = χ (\frac{t - t _{0}}{ε}), 0 < ε < t_{1} - t_{0} .$
XIX-A@1#eq:ramped-density · line 1039
$b_{ε, e}^{μν} = η^{μν} + σ_{ε} (t) (a_{e}^{μν} - η^{μν}) .$
unlabeled · line 1026
$0 \leq χ \leq 1, χ (u) = 0 (u \leq 0), χ (u) = 1 (u \geq 1),$

The regulated coordinate density

Displayed formulas in this passage

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