Manuscript VIII-A@3 · VIII-A-v3.tex · section · 2398a747

Affine conventions

Spinless affine geometry and realization › Affine conventions · lines 580–624

VIII-A@3#A:eq:affine-product, VIII-A@3#A:eq:rotation, VIII-A@3#A:eq:packet-holonomy

Use coordinates $z = (T, X, Y)$ with

η = diag (- 1, 1, 1) .

Affine isometries are written $(L, b) \in ISO^{+} (2, 1)$ , with $L \in SO^{+} (2, 1)$ , and act by

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

Thus

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

For $0 < μ < 2 π$ , define

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

Let $e_{0} = (1, 0, 0)^{T}$ and $W_{0} = R e_{0}$ .

For a reference affine frame $A = (C, a)$ , put

u H_{Π} = C e_{0}, W = a + R u, = A J (μ) A^{- 1} = (C R (μ) C^{- 1}, a - C R (μ) C^{- 1} a) .

The vector $u$ is future unit timelike.

Displayed formulas in this passage

VIII-A@3#A:eq:affine-product · line 588
$(L, b) z = L z + b, (L, b) (L^{'}, b^{'}) = (L L^{'}, L b^{'} + b) .$
VIII-A@3#A:eq:rotation · line 599
$R (μ) = 100 0 cos μ sin μ 0 - sin μ cos μ, J (μ) = (R (μ), 0) .$
VIII-A@3#A:eq:packet-holonomy · line 614
$u H_{Π} = C e_{0}, W = a + R u, = A J (μ) A^{- 1} = (C R (μ) C^{- 1}, a - C R (μ) C^{- 1} a) .$
unlabeled · line 583
$η = diag (- 1, 1, 1) .$
unlabeled · line 594
$(L, b)^{- 1} = (L^{- 1}, - L^{- 1} b) .$