dorsal/arxiv
View SchemaOn Two Complementary Types of Total Time Derivative in Classical Field Theories and Maxwell's Equations
| Authors | R. Smirnov-Rueda |
|---|---|
| Categories | |
| ArXiv ID | physics/0510013 |
| URL | https://arxiv.org/abs/physics/0510013 |
| DOI | 10.1007/s10701-005-6515-8 |
| Journal | Foundations of Physics 35(10) (2005) |
Abstract
Close insight into mathematical and conceptual structure of classical field theories shows serious inconsistencies in their common basis. In other words, we claim in this work to have come across two severe mathematical blunders in the very foundations of theoretical hydrodynamics. One of the defects concerns the traditional treatment of time derivatives in Eulerian hydrodynamic description. The other one resides in the conventional demonstration of the so-called Convection Theorem. Both approaches are thought to be necessary for cross-verification of the standard differential form of continuity equation. Any revision of these fundamental results might have important implications for all classical field theories. Rigorous reconsideration of time derivatives in Eulerian description shows that it evokes Minkowski metric for any flow field domain without any previous postulation. Mathematical approach is developed within the framework of congruences for general 4-dimensional differentiable manifold and the final result is formulated in form of a theorem. A modified version of the Convection Theorem provides a necessary cross-verification for a reconsidered differential form of continuity equation. Although the approach is developed for one-component (scalar) flow field, it can be easily generalized to any tensor field. Some possible implications for classical electrodynamics are also explored.
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"abstract": "Close insight into mathematical and conceptual structure of classical field\ntheories shows serious inconsistencies in their common basis. In other words,\nwe claim in this work to have come across two severe mathematical blunders in\nthe very foundations of theoretical hydrodynamics. One of the defects concerns\nthe traditional treatment of time derivatives in Eulerian hydrodynamic\ndescription. The other one resides in the conventional demonstration of the\nso-called Convection Theorem. Both approaches are thought to be necessary for\ncross-verification of the standard differential form of continuity equation.\nAny revision of these fundamental results might have important implications for\nall classical field theories. Rigorous reconsideration of time derivatives in\nEulerian description shows that it evokes Minkowski metric for any flow field\ndomain without any previous postulation. Mathematical approach is developed\nwithin the framework of congruences for general 4-dimensional differentiable\nmanifold and the final result is formulated in form of a theorem. A modified\nversion of the Convection Theorem provides a necessary cross-verification for a\nreconsidered differential form of continuity equation. Although the approach is\ndeveloped for one-component (scalar) flow field, it can be easily generalized\nto any tensor field. Some possible implications for classical electrodynamics\nare also explored.",
"arxiv_id": "physics/0510013",
"authors": [
"R. Smirnov-Rueda"
],
"categories": [
"physics.class-ph",
"physics.flu-dyn"
],
"doi": "10.1007/s10701-005-6515-8",
"journal_ref": "Foundations of Physics 35(10) (2005)",
"title": "On Two Complementary Types of Total Time Derivative in Classical Field Theories and Maxwell\u0027s Equations",
"url": "https://arxiv.org/abs/physics/0510013"
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