Hawking radiation for smeared-horizon observers

Determine the form of Hawking radiation—specifically its spectrum, flux, and dependence on the observer’s frame—for the class of smeared-horizon observers defined by the σ-parameter interpolation between the Schwarzschild (σ = 0) and Kerr–Schild/Painlevé–Gullstrand (σ → ∞) frames applied to generalized Schwarzschild/Reissner–Nordström metrics g_{μν}(N(r), S(r)). Ascertain how horizon smearing (finite σ) affects the radiation observed near and far from the black hole and whether the standard thermal spectrum is modified in this observer-dependent description.

Background

The paper introduces smeared-horizon observers (sho), a parameterized class of observers that smoothly interpolate between the stationary Schwarzschild observer and the infalling Kerr–Schild/Painlevé–Gullstrand observer via a parameter σ. This construction regularizes behavior at the event and inner horizons and renders certain field-theoretic quantities, such as the Landau–Lifshitz pseudo-tensor contributions, well-defined across horizons.

Classically, the sho framework leads to distinctive time-of-passage properties for shells: for the ingoing sho, infalling shells cross the horizon in finite time while outgoing shells take infinite time, with the reverse behavior for the outgoing sho. Despite this clarified classical picture and regularized mass definition, the quantum emission process (Hawking radiation) in the sho frames is not derived, motivating the explicit open question about its form.