Near-Horizon Scalar-Wave Sampling: Rindler Bounds and Schwarzschild Comparison

Abstract

We study classical reconstruction of an outgoing scalar-wave component from radial-shell observations. Starting from the noncompact-time Rindler metric, we construct Klein--Gordon evolution by closed quadratic forms and reduce the massless characteristic dynamics to translation in optical distance. For an initial optical interval of length $L$, a shell of width $d$, and recording time $T$, the optimal squared observability bounds are $A(T,L)=\min\{d,(T-L)_+\}$ and $B(T,L)=\min\{d,T\}$. At $T=L$, the Rindler observation map is injective with nonclosed range. Discrete shell observations form a spatial fusion frame precisely when their transported regions cover the initial interval; we compute optimal uniform-grid bounds and characterize robustness under profile erasures. Finite-resolution reconstruction separates localization error, approximation error and noise amplification. For fixed Schwarzschild mass and angular momentum, we compare the scalar-wave observation operator with its Rindler limit. Anchored finite-energy preparation permits arbitrary prescribed outgoing $L^2([-L,0])$ profiles without a zero-integral condition. The operator error is explicitly $O(e^{-2κa})$ as an apparatus of fixed optical size is translated toward the horizon; whenever $T>L$, positive Rindler bounds transfer to Schwarzschild for all sufficiently large $a$. A positive Schwarzschild lower bound is impossible at or below the optical travel time, for every translation. These results concern classical observation and reconstruction, not quantum information or dissipative horizon evolution.