State and the one imported existence theorem

We prescribe independent incoming boundary states:

1. on the complete past Kruskal horizon, the vacuum defined by its affine coordinate $U$;

2. on past null infinity ${\mathcal{I}}^{-}$, the vacuum defined by positive Killing frequency in $v$.

The product of these states is the Unruh boundary prescription Unruh1976. On the exterior part of the past horizon,

Here $\kappa$ is calculated from (eq:metric) in Lemma lem:geometry; its value is not an independent parameter.

For the massless scalar on Schwarzschild, the past-horizon and past-null-infinity characteristic maps give a symplectic bulk-to-boundary embedding. Pulling back the product of the affine horizon vacuum and the Killing-frequency vacuum on ${\mathcal{I}}^{-}$ gives a quasifree Unruh state. This state is Hadamard on the exterior and on its stated extension through the future horizon.

The external input is the construction and regularity result of Dappiaggi, Moretti, and Pinamonti, specifically their bulk-to-boundary results and Theorem 4.1 DMP2011. We do not reproduce their global microlocal proof. Thus the calculation below is self-contained at the radiation-boundary level, while its identification with a regular bulk state uses this declared theorem.

The normalized horizon field for a partial wave is $r_{\mathrm{h}}\varphi$; the normalized field at null infinity is $r\varphi$. Both have the one-dimensional canonical normalization used below.