We derive the stochastic partial differential equation (PDE) governing shape fluctuations of the horizon of an evaporating Schwarzschild black hole, working within the slowly evolving trapping horizon formalism of Booth and Fairhurst and semiclassical gravity. The horizon is parametrized as r = r0 + ϵh(v, Ω) with r0 = 2M in Eddington-Finkelstein coordinates, andϵ ≪ 1 the slowly evolving parameter. Expanding the marginally trapped surface condition θ(l) = 0 to O(ϵ3) in Booth-Fairhurst gauge, we obtain ∂vh = l(l + 1) − 14M h − ϵ2 2M h(∇h)2 + η, where ∇ denotes the covariant gradient on the unit S2 and η is Hawking noise. The leading nonlinearity is h(∇h)2—not the Kardar-Parisi-Zhang (KPZ) vertex (∇h)2. We show that h(∇h)2 has canonical scaling dimension λ3 = −2 at the Edwards-Wilkinson (EW) fixed point in d = 2, making it irrelevant under renormalization group flow, in contrast to the KPZ vertex which is marginal (λKPZ = 0). The geometric origin of this distinction is the absence of Galilean invariance on S2. A one-loop RG calculation and numerical simulation on an icosahedral mesh confirm the irrelevance quantitatively. The universality class of the Schwarzschild horizon in the pre-Planck regime MP ≪ M ≤ 102MP is therefore Edwards- Wilkinson (χ = 0 logarithmic, z = 2), not KPZ.
ignacio caldini (Tue,) studied this question.