Geometry and units

In physical units,

The time generator is normalized at infinity.

For the estimates, measure length in units of $r_{\mathrm{h}}$. Thus, after rescaling,

A physical frequency is ${r_{\mathrm{h}}}^{-1}\omega$, and a physical energy is ${r_{\mathrm{h}}}^{-1}$ times the displayed dimensionless energy. Packet normalization is preserved by

A dimensionless bulk smearing used below corresponds to a physical smearing with factor ${r_{\mathrm{h}}}^{-3}$.

The right exterior is globally hyperbolic. We use

The free scalar action is

Let $Y_0=(4\pi {)}^{-1/2}$. In the s-wave sector,

For a real reduced test $f\in {C_c}^{\infty }({\mathbb{R}}^2)$, put

Then

The factor follows from $\mathrm{d}{\mathrm{vol}}_g=r^2\mathsf{f}(r)\mathrm{d}t\mathrm{d}x\mathrm{d}\Omega$.