Paper II: The Kerr Metric from Membrane Dynamics

Within the Quantum Geometrodynamics (QGD) framework, we derive the complete Kerr metric from the dynamics of an elastic S³ hypersurface whose gravitational permittivity εg = 1/(4πG) uniquely fixes the membrane action with no free parameters. The derivation proceeds in four steps: The membrane equation of motion reduces to a Poisson equation ∇²w = −4πGρ/c² for the transverse deformation w; Isoclinic angular momentum conservation (Liso = ℏ) forces a rotating mass to form an effective ring source of radius a = ℏ/(Mc), equal to the Compton wavelength; The Green function on oblate spheroidal coordinates yields w = Rg r/Σ, with Σ = r² + a²cos²θ; The ring geometry determines a null congruence with azimuthal component lφ = −a sin²θ, producing the frame-dragging term gtφ without solving any additional equation. All nonlinear corrections vanish identically due to the null property lμlμ = 0, providing a physical explanation for the Kerr–Schild linearity: the embedding space is flat, and the apparent nonlinearity of Einstein's equations is a projection artefact. The Kerr spin parameter is identified with the Compton wavelength (a = rc), yielding the trade-off relation a · Rg = ℓP² and a natural cosmic censorship bound M ≥ mP. The Newman–Janis complexification r → r + ia cosθ is shown to be the quaternionic norm |r + iwrot|² = Σ, providing the first physical explanation for this long-standing mathematical device. The full nonlinear membrane dynamics is governed by a Dirac–Born–Infeld action whose tension 𝒯 = c⁷/(4πG²ℏ) is uniquely fixed by the gravitational permittivity and the Planck length, imposing a gradient speed limit |∇w| ≤ 1/ℓP that regularises the Kerr ring singularity at the Planck scale. Keywords: Kerr metric, Kerr–Schild construction, frame-dragging, Compton wavelength, membrane dynamics, Dirac–Born–Infeld action, Newman–Janis algorithm, cosmic censorship Related papers: Foundational axioms in Paper I. Dimensional analysis in Paper 0. Gravitational gauge theory in Paper V.