Manuscript VII-F@1 · VII-F.tex · section · 2d81c03f

A finite rational chart table

Adapted affine charts and exact holonomy › A finite rational chart table · lines 1007–1074

Let

Q_{i} = diag (1, Rot (iπ /2)), i = 0, 1, 2.

In sector coordinates define

B_{h} = 3 - 2 - 2 - 2 21 - 2 12,

and

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 .

Direct multiplication gives

B_{h}^{T} η B_{h} = B_{x}^{T} η B_{x} = B_{y}^{T} η B_{y} = η .

They have determinant one and positive time orientation.

For a face in sector $i$ , use the following affine coordinate changes from the cut-space reference coordinates:

face V_{i} U_{i}^{-} U_{i}^{+} Q_{i}, Z_{i} A_{f} (z) Q_{i} B_{h} Q_{i}^{- 1} z - 3 s e_{0} Q_{i} B_{y} Q_{i}^{- 1} z - \frac{5}{2} s e_{0} Q_{i} B_{x} Q_{i}^{- 1} z - \frac{5}{2} s e_{0} z equation in the adapted chart T^{'} = 0 T^{'} = 0 T^{'} = 0 T^{'} = s + 2.

Here $e_{0} = (1, 0, 0)^{T}$ . The same symbol $Q_{i}$ for a sector rotation and a collar face is distinguished by context: $Q_{i} B_{h} Q_{i}^{- 1}$ is a matrix expression, whereas $Q_{i} = [A_{i}, D_{i}, D_{i + 1}, A_{i + 1}]$ denotes a cell.

These charts are used on the regular part of the spacetime. Their formulas extend to face closures, but no claim is made that the cone apex has a regular affine neighbourhood.

Displayed formulas in this passage

unlabeled · line 1010
$Q_{i} = diag (1, Rot (iπ /2)), i = 0, 1, 2.$
unlabeled · line 1015
$B_{h} = 3 - 2 - 2 - 2 21 - 2 12,$
unlabeled · line 1024
$B_{x} = 5/4 - 3/4 0 - 3/4 5/4 0 001, B_{y} = 5/4 0 - 3/4 010 - 3/4 0 5/4 .$
unlabeled · line 1040
$B_{h}^{T} η B_{h} = B_{x}^{T} η B_{x} = B_{y}^{T} η B_{y} = η .$
unlabeled · line 1053
$face V_{i} U_{i}^{-} U_{i}^{+} Q_{i}, Z_{i} A_{f} (z) Q_{i} B_{h} Q_{i}^{- 1} z - 3 s e_{0} Q_{i} B_{y} Q_{i}^{- 1} z - \frac{5}{2} s e_{0} Q_{i} B_{x} Q_{i}^{- 1} z - \frac{5}{2} s e_{0} z equation in the adapted chart T^{'} = 0 T^{'} = 0 T^{'} = 0 T^{'} = s + 2.$

Nearby in this section

remark: Remark: Exact arithmetic