Manuscript VIII-C@3 · VIII-C-v3.tex · section · 3b070436

Packets and affine frames

Complete affine matching at every command boundary › Packets and affine frames · lines 228–277

VIII-C@3#eq:rotation, VIII-C@3#eq:packet

Use developed coordinates

z = (T, X, Y)^{T}, η = diag (- 1, 1, 1), e_{0} = (1, 0, 0)^{T} .

Affine isometries act and compose by

(L, b) z = L z + b, (L, b) (L^{'}, b^{'}) = (L L^{'}, L b^{'} + b),

and

(L, b)^{- 1} = (L^{- 1}, - L^{- 1} b) .

Define

R (μ) = 100 0 cos μ sin μ 0 - sin μ cos μ, J (μ) = (R (μ), 0) .

The reference packet is

Π_{μ} = (p, μ, W_{0}, e_{0}, id, J (μ)), W_{0} = R e_{0} .

Here $p$ is a geometric particle identifier.

For a declared affine frame $A = (L_{A}, b_{A})$ , the coherently transported packet has

W_{Π} = A W_{0}, u_{Π} = L_{A} e_{0}, H_{Π} = A J (μ) A^{- 1} .

In particular,

H_{Π} (A (t e_{0})) = A (t e_{0}) for every t \in R .

The frame, worldline, rest vector, deficit, and holonomy are transported together.

Displayed formulas in this passage

VIII-C@3#eq:rotation · line 245
$R (μ) = 100 0 cos μ sin μ 0 - sin μ cos μ, J (μ) = (R (μ), 0) .$
VIII-C@3#eq:packet · line 256
$Π_{μ} = (p, μ, W_{0}, e_{0}, id, J (μ)), W_{0} = R e_{0} .$
unlabeled · line 231
$z = (T, X, Y)^{T}, η = diag (- 1, 1, 1), e_{0} = (1, 0, 0)^{T} .$
unlabeled · line 236
$(L, b) z = L z + b, (L, b) (L^{'}, b^{'}) = (L L^{'}, L b^{'} + b),$
unlabeled · line 241
$(L, b)^{- 1} = (L^{- 1}, - L^{- 1} b) .$
unlabeled · line 265
$W_{Π} = A W_{0}, u_{Π} = L_{A} e_{0}, H_{Π} = A J (μ) A^{- 1} .$
unlabeled · line 271
$H_{Π} (A (t e_{0})) = A (t e_{0}) for every t \in R .$