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The correct and unusual coordinate transformation rules for electromagnetic quadrupoles
J Gratus1,2, T Banaszek1,2
1Department of Physics, University of Lancaster, Lancaster LA1 4YB, UK.
Proceedings. Mathematical, Physical, and Engineering Sciences
|June 12, 2018
Summary
This study derives correct transformation rules for quadrupoles, enabling their accurate definition in general relativity. These novel rules, involving integrals and second derivatives, overcome limitations of previous methods and allow for coordinate-free definitions.
Area of Science:
- Physics
- General Relativity
- Mathematical Physics
Background:
- Quadrupoles have been historically limited to Cartesian coordinates in flat spacetime.
- Existing transformation rules for quadrupoles are incorrect, hindering their application in advanced physics.
Purpose of the Study:
- To derive correct transformation rules for quadrupoles.
- To enable the accurate definition and application of quadrupoles in general relativity.
- To establish coordinate-free definitions for dipoles and quadrupoles.
Main Methods:
- Derivation of novel coordinate transformation rules for quadrupoles.
- Inclusion of second derivatives and integrals in transformation rules.
- Development of metric-independent and coordinate-free definitions.
Main Results:
- Correct transformation rules for quadrupoles involving second derivatives and integrals were established.
- Quadrupoles can now be correctly defined and utilized within general relativity.
- An example demonstrates unusual behavior of quadrupoles across different coordinate systems (polar vs. Cartesian).
- Metric-independent and coordinate-free definitions for dipoles and quadrupoles were achieved.
Conclusions:
- The derived transformation rules overcome previous limitations, allowing for accurate quadrupole analysis in general relativity.
- Coordinate-free definitions simplify the handling of dipoles and quadrupoles, especially given complex transformations.
- This work opens new avenues for studying multipole expansions in curved spacetimes.
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