--- { "id": "OP-EM-005-EN", "path": "/en/open-problems/electromagnetism/longitudinal-transverse-coupling", "title": "Longitudinal–Transverse Coupling", "title_en": "Longitudinal–Transverse Coupling", "category": "Electromagnetism", "summary": "Carry the chapter’s nonlinear coupling candidate into Space Dynamics and test stability, source normalization and backreaction.", "template": "problem", "version": "1.0.0", "updated": "2026-10-24", "canonical": false, "source_tier": "research-notes", "theory_status": "OPEN", "evidence_status": "UNTESTED", "locale": "en", "module_key": "electromagnetism-research", "module_navigation": [ { "path": "/en/theory/electromagnetism", "label": "Overview", "en": "Overview" }, { "path": "/en/theory/electromagnetism/coulomb-interaction", "label": "Coulomb", "en": "Coulomb" }, { "path": "/en/theory/electromagnetism/magnetic-force", "label": "Magnetic force", "en": "Magnetic force" }, { "path": "/en/theory/electromagnetism/moving-charge-field", "label": "Moving charge", "en": "Moving charge" }, { "path": "/en/theory/electromagnetism/vector-potential", "label": "Vector potential", "en": "Vector potential" }, { "path": "/en/theory/electromagnetism/magnetic-field", "label": "Magnetic field", "en": "Magnetic field" }, { "path": "/en/theory/electromagnetism/electromagnetic-waves", "label": "EM waves", "en": "EM waves" }, { "path": "/en/tests/electromagnetism", "label": "Tests", "en": "Tests" } ], "canonical_scope": "Research task; no new canonical result", "source_chapter": "Chapter 3", "source_reference": "§VIII.1—2 (3.74)—(3.77); §IX, item 4", "source_version": "ASD-BOOK-ddbade997b45", "source_sha256": "dace8ff491b4e4ae8a59b2b98e2a6c9c10e689079021ce0bc117985baba68e86", "source_kind": "manifest", "source_records": [ { "id": "ASD-BOOK-ddbade997b45", "title": "各向异性空间动力学（最终合订稿）", "filename": "各向异性空间动力学.docx", "format": "docx", "size_bytes": 503564, "sha256": "ddbade997b45913248e588953b1d27ccb9c7c206c3c8d992d3aef2364a64836e" } ], "source_manifest": "final-manuscript-source-manifest.json", "source_manifest_url": "/downloads/ASD_Final_Manuscript_Source_Manifest.json", "last_reviewed": "2026-09-15", "depends_on": [ "/en/theory/electromagnetism/electromagnetic-waves", "/en/theory/gravity" ], "related": [ "/en/theory/electromagnetism", "/en/theory/electromagnetism/coulomb-interaction", "/en/theory/electromagnetism/magnetic-force", "/en/theory/electromagnetism/moving-charge-field", "/en/theory/electromagnetism/vector-potential", "/en/theory/electromagnetism/magnetic-field", "/en/theory/electromagnetism/electromagnetic-waves", "/en/tests/electromagnetism", "/en/theory/space-dynamics/longitudinal-transverse-coupling" ], "open_problems": [ "/en/open-problems/electromagnetism/moving-charge-field", "/en/open-problems/electromagnetism/field-representation", "/en/open-problems/electromagnetism/maxwell-effective-limit", "/en/open-problems/electromagnetism/high-gamma-discriminator" ], "equations": [ "em-lt-coupling", "em-lt-source", "em-lt-bound" ], "sources": [ { "label": "Final combined manuscript — adopted scope documented on this page", "kind": "final-combined-manuscript", "url": "/downloads/ASD_Final_Manuscript_Source_Manifest.json" }, { "label": "Previous source: CH3-d9ead146f6ac · 第三章(1).pdf", "kind": "historical-source" } ], "outline": [ "Task", "Established baseline", "Local energy bound and induced expansion", "Work to complete", "Deliverables and decision criteria", "Review boundary and source", "Formal continuation in Space Dynamics" ], "seo_title": "Longitudinal–Transverse Coupling | Anisotropic Space Dynamics", "canonical_url": "https://asd-physics.org/en/open-problems/electromagnetism/longitudinal-transverse-coupling", "translation_path": "/open-problems/electromagnetism/longitudinal-transverse-coupling", "mathematical_review": "OPEN REVIEW", "review_path": "/en/open-problems/electromagnetism/longitudinal-transverse-coupling#review-notes", "review_note": "Derivations use the constitutive inputs and domain stated on this page; a formal source designation is distinct from experimental validation.", "evidence_note": "Evidence is assigned per observable in the EM test matrix; no new experimental fit is claimed.", "translation_of": "/open-problems/electromagnetism/longitudinal-transverse-coupling", "translation_version": "1.0.0", "problem": { "expertise": [ "Field dynamics, measurement and mathematical modeling" ], "difficulty": "Open research", "missing_result": "Determine coupling coefficients in a common longitudinal–transverse action, with positive quadratic and gradient energies and boundaries. Study spectra, energy exchange and stable finite-energy structures, separating zero-background free waves from finite-amplitude coupling.", "deliverable": "A complete candidate action and equations; dimensional and conservation checks; boundedness and stability spectra; a quantitative solution connecting density and transverse structure.", "current_state": "The manuscript supplies a rotational scalar cubic coupling, mutual response terms and a locally bounded quartic completion. These remain candidate constitutive choices." }, "final_source": { "source_id": "ASD-BOOK-ddbade997b45", "chapter": "Chapter 3", "reference": "§VIII.1—2 (3.74)—(3.77); §IX, item 4", "locators": [ "p1240", "p1241", "p1243", "p1244", "p1245", "p1246", "p1249", "p1250", "p1251", "p1252", "p1284" ], "adopted_scope": "Candidate coupling, variations in both fields, local lower bound and induced dilation; retain the original OPEN status.", "conditions": [ "Vanishing boundary variations; constant g_LT, K_L and λ_T.", "K_L>0 and λ_T≥2g_LT²/K_L", "Minimize over η at fixed transverse intensity." ], "unresolved": [ "Full gradient energy, microscopic parameters, long-range sources and stable configurations remain research tasks." ], "migration_status": "reviewed", "manuscript_sha256": "ddbade997b45913248e588953b1d27ccb9c7c206c3c8d992d3aef2364a64836e", "source_manifest": "final-manuscript-source-manifest.json", "body_review": "updated" }, "previous_source": { "source_chapter": "Chapter 3 · Electromagnetism in ASD", "source_reference": "Chapter 3 / 第三章 · §VIII.1–2; (3.74)–(3.77), PDF pp. 17–18", "source_version": "EM-f06476f55cae", "source_sha256": "f06476f55cae728d712f553b338c513eb182e144c2c5cc4cd5ae5ed87ed1d1c6", "source_kind": "manifest", "source_records": [ { "id": "CH3-d9ead146f6ac", "title": "第三章(1).pdf", "sha256": "d9ead146f6ac3335b8663b23cea963319306f549ea4fcd5e3ee34fb7f7ab9413", "pages": 20 } ], "source_manifest": "electromagnetism-source-manifest.json", "source_manifest_url": "/downloads/ASD_Electromagnetism_Source_Manifest.json", "sources": [ { "label": "CH3-d9ead146f6ac · 第三章(1).pdf", "kind": "supplied-manuscript" } ] }, "section_anchors": [ "coupling-task", "coupling-baseline", "coupling-energy-bound", "coupling-required-work", "coupling-deliverables", "coupling-review", "coupling-space-continuation" ], "legacy_anchors": { "s0": "coupling-task", "s1": "coupling-baseline", "s2": "coupling-required-work", "s3": "coupling-deliverables", "s4": "coupling-review", "s5": "coupling-space-continuation" } } --- ## Task **OP-EM-005 · OPEN** Continue the established local variational model toward normalized sources, complete fields and stable finite-energy configurations.
## Established baseline This task continues the nonlinear longitudinal–transverse candidate in final Chapter 3, Section VIII; it does not start by inventing a coupling. Under compatible boundaries, split displacement into curl-free $\mathbf u_L$ and divergence-free $\mathbf u_T$. Define the dimensionless $\eta=-\nabla\cdot\mathbf u_L$ and $\boldsymbol\Omega_T=\nabla\times\mathbf u_T$. Positive $\eta$ means local compression; negative $\eta$ means expansion. Adopt the rotational-scalar coupling The coefficient $g_{LT}$ has energy-density units. This term makes transverse torsion intensity a local input to longitudinal volumetric strain. The coupling is a chosen nonlinear constitutive relation; its variations are results that can be checked within that choice. For variations vanishing at temporal endpoints and spatial boundaries, use $\delta\eta=-\nabla\cdot\delta\mathbf u_L$: $$ \delta_L S_{LT}=g_{LT}\int|\boldsymbol\Omega_T|^2\nabla\cdot\delta\mathbf u_L\,d^3x\,dt =-g_{LT}\int\nabla|\boldsymbol\Omega_T|^2\cdot\delta\mathbf u_L\,d^3x\,dt. $$ For the transverse variation, $\delta\boldsymbol\Omega_T=\nabla\times\delta\mathbf u_T$. Curl integration by parts preserves its sign under these boundary conditions, so $$ \delta_T S_{LT}=-2g_{LT}\int\eta\boldsymbol\Omega_T\cdot(\nabla\times\delta\mathbf u_T)\,d^3x\,dt =-2g_{LT}\int\nabla\times(\eta\boldsymbol\Omega_T)\cdot\delta\mathbf u_T\,d^3x\,dt. $$ Vary together with the free fields and place the terms on the right-hand sides to obtain reciprocal generalized force densities: Their units are N/m³. They are a longitudinal gradient source and a divergence-free transverse source, not electric current density $\mathbf J$. The coupling is cubic about a uniform zero background and leaves the lowest-order free-wave spectrum unchanged; finite backgrounds or amplitudes allow mutual driving. ## Local energy bound and induced expansion A cubic term alone need not bound local energy below. Add a positive longitudinal quadratic term and a transverse quartic term, with constant $K_L>0$ and $\lambda_T$, both having pressure units. Completing the square gives A sufficient condition is $\lambda_T\geq2g_{LT}^2/K_L$. Minimizing with respect to $\eta$ at fixed transverse intensity gives $$ K_L\eta+g_{LT}|\boldsymbol\Omega_T|^2=0, \qquad \eta_{\rm eq}=-\frac{g_{LT}}{K_L}|\boldsymbol\Omega_T|^2. $$ Thus positive $g_{LT}$ makes transverse torsion induce expansion and density deficit. Local strain $\eta$ is not the exterior long-range gravitational scale potential. Connecting the local response to a long-range source requires longitudinal constitutive relations, complete gradient energy and a boundary solution. In a closed conservative system, including the coupling potential preserves total energy. A loss in one channel can be a gain in another. Exchange is not irreversible dissipation, and a local lower bound is not an already obtained complete stable particle solution. ## Work to complete Starting from this local candidate and both variations, determine coefficients and normalization in one longitudinal–transverse action and specify complete gradients, sources and boundaries. Solve a finite-energy configuration, perturbation spectrum about a nonzero background, channel exchange or dynamical-stability problem. The next result should distinguish assumed parameters, derived coefficients and computable response functions. ## Deliverables and decision criteria Provide the complete equations of one action, boundary conditions, dimensional and conservation checks, a reproducible configuration or spectrum calculation, and an explicit map from local density response to a long-range source. If only the local square completion is tested, state its lower-bound scope without claiming electron or proton masses or stability in every perturbation direction. ## Review boundary and source
Based on final Chapter 3, equations (3.74)–(3.77) and the fourth scope item in Section IX. Variables, reciprocal sources, a local lower bound and induced expansion have been obtained. Microscopic coefficients and complete stable localized structures remain research tasks. The existing problem identity OP-EM-005 is retained. ## Formal continuation in Space Dynamics The [coupling page](/en/theory/space-dynamics/longitudinal-transverse-coupling), [OP-SD-001](/en/open-problems/space-dynamics/unified-dynamics) and [OP-SD-006](/en/open-problems/space-dynamics/source-coupling) continue this task. They should calculate sources, fields and stable structures for the same candidate without reclassifying the completed local variations as absence of dynamics.