The exterior and dimensionless estimates

In physical units,

The time generator is normalized at infinity.

For the PDE estimates we measure lengths in units of $r_{\mathrm{h}}$:

The packet rescales as

This preserves both packet normalization and its annihilator. After this paragraph, tildes are suppressed:

The occupation integrals are unchanged by the rescaling. All constants in the estimates are dimensionless in this convention.

The physical coupling associated with a dimensionless profile $ϵ\rho$ is ${r_{\mathrm{h}}}^{-2}ϵ\rho$. A dimensionless scalar smearing is converted to its physical smearing with the corresponding factor ${r_{\mathrm{h}}}^{-3}$. Thus the profile choices below do not equate quantities of different physical dimensions.

The exterior is globally hyperbolic, with $t=\mathrm{constant}$ Cauchy surfaces. Indeed, in the conformally related optical metric, a causal curve has $|\mathrm{d}x/\mathrm{d}t|\le 1$. On every bounded $x$-interval the angular optical geometry is regular and compact. An inextendible causal curve cannot escape the spatial geometry at finite $t$, and therefore crosses every time slice once.