Numerical Recovery of the Kerr Spacetime via Saturated Quaternionic Tetrads in the Scalar–Tensor–Observable Hierarchy
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2026
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| _version_ | 1866901691461271552 |
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| author | CATRAMBONE, EUGENE |
| author_facet | CATRAMBONE, EUGENE |
| contents | <p>This work presents a numerical reconstruction of the Kerr spacetime within the Scalar–Tensor–Observable (STO) hierarchy, a reformulation of gravitational dynamics based on a universal scalar clock and quaternionic selector geometry.</p> <p>Instead of treating the spacetime metric as fundamental, the model reconstructs it from an orthonormal tetrad that is dynamically deformed by two primary components: (1) an anisotropic expansion-locking scalar, Theta, and (2) a quaternionic selector field q in SU(2), whose phase dynamics generate rotational structure. A hyperbolic tangent (tanh) saturation law is introduced as a nonlinear stability mechanism, ensuring bounded behavior in strong-field regions.</p> <p>A numerical parameter sweep across spin values (a in [0, 0.99M]) and radial distances (r in [1.5M, 10M]) demonstrates high-fidelity agreement with the Kerr solution, with median Einstein tensor residuals of 9.96 × 10^-3. In the non-rotating limit (a = 0), the Schwarzschild solution is recovered to machine precision (~10^-16), confirming the accuracy of the numerical engine.</p> <p>The framework also provides a reinterpretation of key features of rotating spacetimes. The r → 0 limit is replaced by a bounded “phase-zero” state, eliminating curvature singularities through saturation of the scalar clock. The ergosphere is interpreted as a phase-precession boundary rather than a region of spacetime dragging. Frame-dragging itself emerges as a projection of internal quaternionic phase dynamics into the observable tetrad structure.</p> <p>Small deviations from standard Kerr behavior are predicted in the high-spin, near-horizon regime, including anisotropies in black hole shadow geometry and potential modifications to frame-dragging decay. These effects provide possible observational tests using VLBI imaging and gravitational wave data.</p> <p>An accompanying appendix provides full numerical implementation details, including grid construction, tetrad formulation, selector evolution, and residual definitions, enabling reproducibility.</p> <p>Overall, this work establishes a bridge between quaternionic internal geometry and macroscopic gravitational structure, suggesting that spacetime curvature may be understood as a saturated projection of an underlying scalar field rather than a fundamental primitive.</p> <p> </p> |
| format | Recurso digital |
| id | zenodo_https___doi_org_10_5281_zenodo_19122085 |
| institution | Zenodo |
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| publishDate | 2026 |
| publisher | Zenodo |
| record_format | zenodo |
| spellingShingle | Numerical Recovery of the Kerr Spacetime via Saturated Quaternionic Tetrads in the Scalar–Tensor–Observable Hierarchy CATRAMBONE, EUGENE Kerr spacetime, general relativity, tetrads, quaternion, SU(2), frame-dragging, numerical relativity, black holes, ergosphere, singularity resolution, scalar field, holonomy, spacetime geometry <p>This work presents a numerical reconstruction of the Kerr spacetime within the Scalar–Tensor–Observable (STO) hierarchy, a reformulation of gravitational dynamics based on a universal scalar clock and quaternionic selector geometry.</p> <p>Instead of treating the spacetime metric as fundamental, the model reconstructs it from an orthonormal tetrad that is dynamically deformed by two primary components: (1) an anisotropic expansion-locking scalar, Theta, and (2) a quaternionic selector field q in SU(2), whose phase dynamics generate rotational structure. A hyperbolic tangent (tanh) saturation law is introduced as a nonlinear stability mechanism, ensuring bounded behavior in strong-field regions.</p> <p>A numerical parameter sweep across spin values (a in [0, 0.99M]) and radial distances (r in [1.5M, 10M]) demonstrates high-fidelity agreement with the Kerr solution, with median Einstein tensor residuals of 9.96 × 10^-3. In the non-rotating limit (a = 0), the Schwarzschild solution is recovered to machine precision (~10^-16), confirming the accuracy of the numerical engine.</p> <p>The framework also provides a reinterpretation of key features of rotating spacetimes. The r → 0 limit is replaced by a bounded “phase-zero” state, eliminating curvature singularities through saturation of the scalar clock. The ergosphere is interpreted as a phase-precession boundary rather than a region of spacetime dragging. Frame-dragging itself emerges as a projection of internal quaternionic phase dynamics into the observable tetrad structure.</p> <p>Small deviations from standard Kerr behavior are predicted in the high-spin, near-horizon regime, including anisotropies in black hole shadow geometry and potential modifications to frame-dragging decay. These effects provide possible observational tests using VLBI imaging and gravitational wave data.</p> <p>An accompanying appendix provides full numerical implementation details, including grid construction, tetrad formulation, selector evolution, and residual definitions, enabling reproducibility.</p> <p>Overall, this work establishes a bridge between quaternionic internal geometry and macroscopic gravitational structure, suggesting that spacetime curvature may be understood as a saturated projection of an underlying scalar field rather than a fundamental primitive.</p> <p> </p> |
| title | Numerical Recovery of the Kerr Spacetime via Saturated Quaternionic Tetrads in the Scalar–Tensor–Observable Hierarchy |
| topic | Kerr spacetime, general relativity, tetrads, quaternion, SU(2), frame-dragging, numerical relativity, black holes, ergosphere, singularity resolution, scalar field, holonomy, spacetime geometry |
| url | https://doi.org/10.5281/zenodo.19122085 |