S-4_Shadow Quantum Gravity Equation from Unified Spacetime Scale Structure
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2026
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| _version_ | 1866901352354938880 |
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| author | Morita, Kuniyuki |
| author_facet | Morita, Kuniyuki |
| contents | <p>This work (S-4) unifies the time-scale structure developed in S-2 and the spatial scale structure developed in S-3, and derives the Shadow Quantum Gravity Equation. Under three minimal conditions—differentiability, local invertibility, and consistency—the spacetime scale transformation is uniquely generated by</p> <p> K = K_t + K_x = k_t * partial_t + k_x * Delta,</p> <p>which yields a closed spacetime scale flow for any field Phi_ell:</p> <p> d/dell Phi_ell = k_t * partial_t Phi_ell + k_x * Delta Phi_ell.</p> <p>The time direction produces a decay structure, while the spatial direction produces a coarse-graining structure, and both arise from a single unified scale principle.</p> <p>In the Kerr background, the spatial Laplacian is replaced by Delta_Kerr, leading to a scale-dependent radial equation that defines the Shadow Quantum Gravity Equation. This formulation clarifies the geometric origin of the QNM decay structure D(L), and naturally produces the scale-dependent deformation of the Kerr potential and the resulting QNM frequency shifts.</p> <p>This paper completes the mathematical foundation of spacetime scale hierarchies and establishes the framework of shadow quantum gravity. The S-4 equation provides a unified theoretical basis for scale-dependent QNM analysis and opens the path toward the source-side theory (S-5).</p> |
| format | Recurso digital |
| id | zenodo_https___doi_org_10_5281_zenodo_19637085 |
| institution | Zenodo |
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| publishDate | 2026 |
| publisher | Zenodo |
| record_format | zenodo |
| spellingShingle | S-4_Shadow Quantum Gravity Equation from Unified Spacetime Scale Structure Morita, Kuniyuki spacetime scale structure shadow quantum gravity scale-dependent geometry time-scale generator spatial-scale generator Laplacian operator Kerr spacetime quasinormal modes scale-dependent QNM radial equation coarse-graining decay structure operator semigroup mathematical physics quantum gravity foundations <p>This work (S-4) unifies the time-scale structure developed in S-2 and the spatial scale structure developed in S-3, and derives the Shadow Quantum Gravity Equation. Under three minimal conditions—differentiability, local invertibility, and consistency—the spacetime scale transformation is uniquely generated by</p> <p> K = K_t + K_x = k_t * partial_t + k_x * Delta,</p> <p>which yields a closed spacetime scale flow for any field Phi_ell:</p> <p> d/dell Phi_ell = k_t * partial_t Phi_ell + k_x * Delta Phi_ell.</p> <p>The time direction produces a decay structure, while the spatial direction produces a coarse-graining structure, and both arise from a single unified scale principle.</p> <p>In the Kerr background, the spatial Laplacian is replaced by Delta_Kerr, leading to a scale-dependent radial equation that defines the Shadow Quantum Gravity Equation. This formulation clarifies the geometric origin of the QNM decay structure D(L), and naturally produces the scale-dependent deformation of the Kerr potential and the resulting QNM frequency shifts.</p> <p>This paper completes the mathematical foundation of spacetime scale hierarchies and establishes the framework of shadow quantum gravity. The S-4 equation provides a unified theoretical basis for scale-dependent QNM analysis and opens the path toward the source-side theory (S-5).</p> |
| title | S-4_Shadow Quantum Gravity Equation from Unified Spacetime Scale Structure |
| topic | spacetime scale structure shadow quantum gravity scale-dependent geometry time-scale generator spatial-scale generator Laplacian operator Kerr spacetime quasinormal modes scale-dependent QNM radial equation coarse-graining decay structure operator semigroup mathematical physics quantum gravity foundations |
| url | https://doi.org/10.5281/zenodo.19637085 |