Manuscript VII-A-ST@1 · VII-A-ST.tex · section · c39a67a3

The laboratory

Prescribed geometry and the supported Cauchy presentation › The laboratory · lines 328–350

VII-A-ST@1#eq:domain

Choose a regular spatial chart

χ : V ⟶ V \subset R^{3}

whose image is convex and contains $B_{R}$ in its interior. Translate the spatial coordinates if necessary, and set

t = c (T - T_{in}), η = \diag (- 1, 1, 1, 1) .

For $0 < b < R$ , use the shrinking domain

D = {(t, x) : 0 \leq t \leq b, ∣ x ∣ < R - t}, Σ_{0} = {0} \times B_{R} .

Its open positive-time part is denoted $D_{+}^{\circ}$ . Fields are smooth up to the initial surface on neighborhoods of the portions used.

Displayed formulas in this passage

VII-A-ST@1#eq:domain · line 341
$D = {(t, x) : 0 \leq t \leq b, ∣ x ∣ < R - t}, Σ_{0} = {0} \times B_{R} .$
unlabeled · line 331
$χ : V ⟶ V \subset R^{3}$
unlabeled · line 336
$t = c (T - T_{in}), η = \diag (- 1, 1, 1, 1) .$