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

Adapted affine charts

The complete fixed-cone construction › Adapted affine charts · lines 1739–1785

VIII-A@3#A:eq:general-boost, VIII-A@3#A:eq:chart-table

For $∣ g ∣ < 1$ , define

B (g) = (γ - γ g - γ g^{T} I_{2} + (γ - 1) g g^{T} / ∣ g ∣^{2}), γ = (1 - ∣ g ∣^{2})^{- 1/2},

with $B (0) = I_{3}$ . It sends $T = c + g \cdot x$ to a plane with first coordinate $γ c$ .

Let $O_{i} = diag (1, Rot (iπ /2))$ , and use

B_{h} = 3 - 2 - 2 - 2 21 - 2 12, B_{x} = 5/4 - 3/4 0 - 3/4 5/4 0 001, B_{y} = 5/4 0 - 3/4 010 - 3/4 0 5/4 .

They are, respectively, $B ((2/3, 2/3))$ , $B ((3/5, 0))$ , and $B ((0, 3/5))$ . Direct multiplication verifies $B^{T} η B = η$ , determinant one, and future orientation.

The adapted charts are

face V_{i} U_{i}^{-} U_{i}^{+} Q_{i}, resting exterior A_{f} (z) O_{i} B_{h} O_{i}^{- 1} z - 3 s e_{0} O_{i} B_{y} O_{i}^{- 1} z - \frac{5}{2} s e_{0} O_{i} B_{x} O_{i}^{- 1} z - \frac{5}{2} s e_{0} z

Displayed formulas in this passage

VIII-A@3#A:eq:general-boost · line 1742
$B (g) = (γ - γ g - γ g^{T} I_{2} + (γ - 1) g g^{T} / ∣ g ∣^{2}), γ = (1 - ∣ g ∣^{2})^{- 1/2},$
VIII-A@3#A:eq:chart-table · line 1774
$face V_{i} U_{i}^{-} U_{i}^{+} Q_{i}, resting exterior A_{f} (z) O_{i} B_{h} O_{i}^{- 1} z - 3 s e_{0} O_{i} B_{y} O_{i}^{- 1} z - \frac{5}{2} s e_{0} O_{i} B_{x} O_{i}^{- 1} z - \frac{5}{2} s e_{0} z$
unlabeled · line 1757
$B_{h} = 3 - 2 - 2 - 2 21 - 2 12, B_{x} = 5/4 - 3/4 0 - 3/4 5/4 0 001, B_{y} = 5/4 0 - 3/4 010 - 3/4 0 5/4 .$

Adapted affine charts

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