Manuscript VII-F@1 · VII-F.tex · section · 2d81c03f

Rest data and face normals

The particle packet and the geometric rule › Rest data and face normals · lines 1275–1312

The geometric particle packet is

Π_{⋆} = (P_{⋆}, μ, W, u_{⋆} = e_{0}, cell label P, reference framing) .

It is the same packet on both germs.

The future unit normal of a $V_{i}$ face, in sector coordinates, is

n_{h} = (3, 2, 2) .

The future unit normals of $U_{i}^{+}$ and $U_{i}^{-}$ are respectively

n_{x} = (\frac{5}{4}, \frac{3}{4}, 0), n_{y} = (\frac{5}{4}, 0, \frac{3}{4}) .

They are rotated into the developed sector by $Q_{i}$ .

Consequently the particle's face-relative rapidities satisfy

cosh χ_{h} = 3, cosh χ_{k} = \frac{5}{4},

χ_{h} = arcosh 3, χ_{k} = lo g 2.

The reference rest vector remains $e_{0}$ . A stationary worldline therefore does not imply zero rapidity relative to every incident face.

Displayed formulas in this passage

unlabeled · line 1278
$Π_{⋆} = (P_{⋆}, μ, W, u_{⋆} = e_{0}, cell label P, reference framing) .$
unlabeled · line 1286
$n_{h} = (3, 2, 2) .$
unlabeled · line 1290
$n_{x} = (\frac{5}{4}, \frac{3}{4}, 0), n_{y} = (\frac{5}{4}, 0, \frac{3}{4}) .$
unlabeled · line 1298
$cosh χ_{h} = 3, cosh χ_{k} = \frac{5}{4},$
unlabeled · line 1304
$χ_{h} = arcosh 3, χ_{k} = lo g 2.$