Manuscript VII-E-ST@1 · VII-E-ST.tex · section · d28d9c5d

Transverse faces, edge laws, and affine charts

A four-face certificate and its supported 3+1 lift › Transverse faces, edge laws, and affine charts · lines 800–842

VII-E-ST@1#eq:charts

The rates

ℓ ˙_{-} (0^{-}) = - \frac{16}{25}, ℓ ˙_{+} (0^{+}) = \frac{16}{39}

show transversality. The four mixed adjacencies persist and satisfy $p ∣ x ∣ - q ∣ y ∣ = d s$ in their respective quadrants.

Use the following $2 + 1$ -dimensional Lorentz matrices, extended by the identity on $z$ :

C_{E} = 5/4 - 3/4 0 - 3/4 5/4 0 001, C_{W} = 5/4 3/4 0 3/4 5/4 0 001,

C_{N} = 13/12 0 - 5/12 010 - 5/12 0 13/12, C_{S} = 13/12 0 5/12 010 5/12 0 13/12 .

Direct multiplication verifies Lorentz orthogonality, determinant one, and future orientation. On each corresponding plane the transformed time coordinate is $s$ .

Set $Λ_{i \leftarrow j} = C_{i} C_{j}^{- 1}$ . Every closed ordered core word is the identity, since

(C_{i_{0}} C_{i_{1}}^{- 1}) (C_{i_{1}} C_{i_{2}}^{- 1}) \dots (C_{i_{m}} C_{i_{0}}^{- 1}) = I .

This proves the incoming triple, outgoing triple, and event word identities without interpreting the nontrivial frame changes as curvature.

Displayed formulas in this passage

unlabeled · line 803
$ℓ ˙_{-} (0^{-}) = - \frac{16}{25}, ℓ ˙_{+} (0^{+}) = \frac{16}{39}$
unlabeled · line 812
$C_{E} = 5/4 - 3/4 0 - 3/4 5/4 0 001, C_{W} = 5/4 3/4 0 3/4 5/4 0 001,$
VII-E-ST@1#eq:charts · line 819
$C_{N} = 13/12 0 - 5/12 010 - 5/12 0 13/12, C_{S} = 13/12 0 5/12 010 5/12 0 13/12 .$
unlabeled · line 834
$(C_{i_{0}} C_{i_{1}}^{- 1}) (C_{i_{1}} C_{i_{2}}^{- 1}) \dots (C_{i_{m}} C_{i_{0}}^{- 1}) = I .$

Transverse faces, edge laws, and affine charts

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