Manuscript Em01-D@2 · Em01-D-v2.tex · section · d3b49151

Coordinates and the exact measure

Exact nonphysical eight-variable diagnostic › Coordinates and the exact measure · lines 2003–2050

Use fixed-frame coordinates

b = a + B ℓ + u k + X, y = a + A ℓ + v k + Y, z = a + r,

with $B$ in a compact positive range on the patch. Then

(b - a)^{2} = 4 B u - ∣ X ∣^{2}, t = 2 u, s = 2 v .

Set

τ = (a, B, A, Y), d τ = d^{4} a d B d A d^{2} Y,

and

S_{κ} = {z = a, b = a + B ℓ, y = a + A ℓ + Y} \cap U .

This has codimension eight, but is not the physical aligned-null locus $N$ : $Y$ need not vanish.

Tests are extended by zero outside their compact support in $U$ . Define

F_{j} (τ; u, v, X) = (\partial_{b^{0}}^{j} Φ_{I_{*}}) (a, a + B ℓ + u k + X, a, a + A ℓ + v k + Y), j = 0, 1, 2.

The $b^{0}$ -derivatives are ambient derivatives taken before evaluation. Put

c_{0} = 1, c_{1} = - 2, c_{2} = 1,

P_{j} (S, T) = p (S) p^{(3 - j)} (S - T), T \geq 0.

Displayed formulas in this passage

unlabeled · line 2006
$b = a + B ℓ + u k + X, y = a + A ℓ + v k + Y, z = a + r,$
unlabeled · line 2012
$(b - a)^{2} = 4 B u - ∣ X ∣^{2}, t = 2 u, s = 2 v .$
unlabeled · line 2016
$τ = (a, B, A, Y), d τ = d^{4} a d B d A d^{2} Y,$
unlabeled · line 2021
$S_{κ} = {z = a, b = a + B ℓ, y = a + A ℓ + Y} \cap U .$
unlabeled · line 2033
$F_{j} (τ; u, v, X) = (\partial_{b^{0}}^{j} Φ_{I_{*}}) (a, a + B ℓ + u k + X, a, a + A ℓ + v k + Y), j = 0, 1, 2.$
unlabeled · line 2044
$c_{0} = 1, c_{1} = - 2, c_{2} = 1,$
unlabeled · line 2047
$P_{j} (S, T) = p (S) p^{(3 - j)} (S - T), T \geq 0.$