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February 26, 20260 citationsOpen Access

Paper II: The Kerr Metric from Membrane Dynamics

Key Points

The aim is to derive the complete Kerr metric from the dynamics of an elastic S³ hypersurface in quantum geometrodynamics.
Utilized a Poisson equation to describe transverse deformation of the membrane.
Applied conservation laws to define angular momentum and characterize mass formation.
Developed a Green function in oblate spheroidal coordinates to parameterize deformations.
Analyzed frame-dragging effects through the geometry of a null congruence.
Identified the Kerr spin parameter with the Compton wavelength, indicating a fundamental relationship in gravitational theory.
Established a trade-off relation connected to the Planck length through the gradient speed limit for membrane dynamics.
Demonstrated that nonlinear effects vanish, simplifying the interpretation of Einstein's equations as projections.

Abstract

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.

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