Manuscript En03-A@1 · En03-A.tex · section · 5398614e

Four affine faces

External conical geometry and certified local moves › Four affine faces · lines 1627–1684

En03-A@1#eq:boosts, En03-A@1#eq:core-transitions, En03-A@1#eq:opposite-transitions, En03-A@1#eq:mixedtransitions

The exchange is determined by the declared maximum slicing rule. It is not a uniqueness theorem over all foliations.

frames. For a slope $v_{i}$ , choose a Lorentz map with time coordinate

X_{i}^{0} = \frac{T - v _{i} \cdot ( x , y )}{1 - ∣ v _{i} ∣ ^{2}} .

On its face this equals $s$ . The concrete maps are

C_{E} C_{N} = 5/4 - 3/4 0 - 3/4 5/4 0 001, = 13/12 0 - 5/12 010 - 5/12 0 13/12, C_{W} C_{S} = 5/4 3/4 0 3/4 5/4 0 001, = 13/12 0 5/12 010 5/12 0 13/12 .

They are proper future-preserving Lorentz maps. Set

Λ_{i \leftarrow j} = C_{i} C_{j}^{- 1} .

The opposite-pair entries are

Λ_{E \leftarrow W} = 17/8 - 15/8 0 - 15/8 17/8 0 001, Λ_{N \leftarrow S} = 97/72 0 - 65/72 010 - 65/72 0 97/72 .

For $X_{+} = E, X_{-} = W, Y_{+} = N, Y_{-} = S$ ,

Λ_{X_{ϵ} \leftarrow Y_{ν}} = 65/48 - ϵ 13/16 ν 5/12 - ϵ 3/4 5/4 0 ν 25/48 - ϵ ν 5/16 13/12 .

Reverse maps are inverses; all core translations are zero.

Displayed formulas in this passage

En03-A@1#eq:boosts · line 1638
$C_{E} C_{N} = 5/4 - 3/4 0 - 3/4 5/4 0 001, = 13/12 0 - 5/12 010 - 5/12 0 13/12, C_{W} C_{S} = 5/4 3/4 0 3/4 5/4 0 001, = 13/12 0 5/12 010 5/12 0 13/12 .$
En03-A@1#eq:core-transitions · line 1658
$Λ_{i \leftarrow j} = C_{i} C_{j}^{- 1} .$
En03-A@1#eq:opposite-transitions · line 1663
$Λ_{E \leftarrow W} = 17/8 - 15/8 0 - 15/8 17/8 0 001, Λ_{N \leftarrow S} = 97/72 0 - 65/72 010 - 65/72 0 97/72 .$
En03-A@1#eq:mixedtransitions · line 1672
$Λ_{X_{ϵ} \leftarrow Y_{ν}} = 65/48 - ϵ 13/16 ν 5/12 - ϵ 3/4 5/4 0 ν 25/48 - ϵ ν 5/16 13/12 .$
unlabeled · line 1634
$X_{i}^{0} = \frac{T - v _{i} \cdot ( x , y )}{1 - ∣ v _{i} ∣ ^{2}} .$