Manuscript VIII-B@3 · VIII-B-v3.tex · section · 3aa6baa1

Intrinsic slice defects

Intrinsic slice defects · lines 2277–2325

VIII-B@3#eq:incoming-defect, VIII-B@3#eq:outgoing-defect

The intrinsic particle defect of a slice need not equal the spacetime deficit in the packet.

On an incoming core face, the induced oblique metric is

G_{h} = G_{α} - (a a) (a a) .

The boundary ray tangents have squared norm $1 - a^{2}$ and inner product $z_{α} - a^{2}$ . Thus

θ_{h} (μ) = arccos (\frac{z _{α} - a ^{2}}{1 - a ^{2}}), κ_{P}^{-} (μ) = 2 π - 3 θ_{h} (μ) .

On $U_{i}^{-}$ , the ray and bisector tangents have oblique coordinates $(1, 0)$ and $(1, 1)$ . Their angle satisfies

cos θ_{k} = \frac{1 + z _{α}}{2 ( 1 + z _{α} ) - b ^{2}}, tan θ_{k} = \frac{S _{α} ^{2} - b ^{2}}{1 + z _{α}} = \frac{4}{5} tan \frac{α}{2} .

Consequently,

θ_{k} (μ) = arctan (\frac{4}{5} tan \frac{α}{2}), κ_{P}^{+} (μ) = 2 π - 6 θ_{k} (μ) .

Positive-definite face metrics give $0 < θ_{h} < π$ and $0 < θ_{k} < π /2$ .

At $μ_{0}$ ,

κ_{P}^{-} = 2 π - 3 arccos (- 4/5), κ_{P}^{+} = 2 π - 6 arctan (4/5) .

Neither replaces $μ$ in (E2).

Displayed formulas in this passage

VIII-B@3#eq:incoming-defect · line 2290
$θ_{h} (μ) = arccos (\frac{z _{α} - a ^{2}}{1 - a ^{2}}), κ_{P}^{-} (μ) = 2 π - 3 θ_{h} (μ) .$
VIII-B@3#eq:outgoing-defect · line 2309
$θ_{k} (μ) = arctan (\frac{4}{5} tan \frac{α}{2}), κ_{P}^{+} (μ) = 2 π - 6 θ_{k} (μ) .$
unlabeled · line 2283
$G_{h} = G_{α} - (a a) (a a) .$
unlabeled · line 2300
$cos θ_{k} = \frac{1 + z _{α}}{2 ( 1 + z _{α} ) - b ^{2}}, tan θ_{k} = \frac{S _{α} ^{2} - b ^{2}}{1 + z _{α}} = \frac{4}{5} tan \frac{α}{2} .$
unlabeled · line 2320
$κ_{P}^{-} = 2 π - 3 arccos (- 4/5), κ_{P}^{+} = 2 π - 6 arctan (4/5) .$