Geometry, reduction, and measures

In physical units the background is the right Schwarzschild exterior,

Killing time is normalized at infinity.

For the estimates we measure lengths in units of $r_{\mathrm{h}}$. Suppressing tildes,

Physical frequencies and energies are ${r_{\mathrm{h}}}^{-1}$ times the displayed values. A dimensionless bulk scalar smearing corresponds to physical smearing with factor ${r_{\mathrm{h}}}^{-3}$. Packet normalization is preserved by

The free real scalar has

Let $Y_0=(4\pi {)}^{-1/2}$. The selected s-wave field is

with reduced operator

For a real compact reduced test $f$, define

Because $\mathrm{d}{\mathrm{vol}}_g=r^2\mathsf{f}(r)\mathrm{d}t\mathrm{d}x\mathrm{d}\Omega$,

All finite compact reduced supports give compact supports strictly outside the horizon.

We use complete stationary massless-scalar scattering on Schwarzschild, the free normally hyperbolic Cauchy and Green-operator theory, and the bulk Hadamard identification of the Unruh boundary prescription. The horizon input is centered quasifree with

the infinity input is vacuum, and the inputs are independent.

The classical scattering and Green inputs are Dimock,BGP. The selected bulk state construction is DMP. They do not supply the new source dynamics or a coherent receiver.