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Electrodynamics and the Gauss linking integral on the 3-sphere and in hyperbolic 3-space

J. Math. Phys. 49, 023504 (2008); doi:10.1063/1.2827467

Published 8 February 2008

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Dennis DeTurck and Herman Gluck
University of Pennsylvania, Philadelphia Pennsylvania 19104, USA
In this first of two papers, we develop a steady-state version of classical electrodynamics on the 3-sphere and in hyperbolic 3-space, including an explicit formula for the vector-valued Green's operator, an explicit formula of Biot–Savart type for the magnetic field, and a corresponding Ampere's law contained in Maxwell's equations. We then use this to obtain explicit integral formulas for the linking number of two disjoint closed curves in these spaces. These formulas, like their prototypes in Euclidean 3-space, are geometric rather than just topological because their integrands are invariant under orientation-preserving isometries of the ambient space. In the second paper, we obtain integral formulas for twisting, writhing, and helicity and prove the theorem LINK=TWIST+WRITHE in the 3-sphere and in hyperbolic 3-space. We then use these results to derive upper bounds for the helicity of vector fields and lower bounds for the first eigenvalue of the curl operator on subdomains of these two spaces. An announcement of most of these results and a hint of their proofs can be found in e-print arXiv:math.GT/0406276, while an expanded version of this paper, containing many details which are here left to the reader, can be found at e-print arXiv:math.GT/0510388. ©2008 American Institute of Physics
History: Received 26 March 2007; accepted 28 November 2007; published 8 February 2008
Permalink: http://link.aip.org/link/?JMAPAQ/49/023504/1
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KEYWORDS and PACS

Keywords
PACS
  • 03.50.De
    Classical electromagnetism, Maxwell equations
  • 02.30.Rz
    Integral equations
  • YEAR: 2008

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ISSN:
0022-2488 (print)   1089-7658 (online)
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REFERENCES (18)
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  4. Cantarella, J., DeTurck, D., and Gluck, H., “Upper bounds for the writhing of knots and the helicity of vectors fields,” in Knots, Braids, and Mapping Class Groups-Papers dedicated to Joan S. Birman, edited by J. Gilman, X.-S. Lin, and W. Menasco, AMS/IP Studies in Advanced Mathematics, Volume 24, International Press, Cambridge, MA (2002).
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  8. Cantarella, J., DeTurck, D., Gluck, H., and Teytel, M., “Influence of geometry and topology on helicity,” Geophys. Monogr. 111, 17–24 (1999).
  9. Cantarella, J., DeTurck, D., Gluck, H., and Teytel, M., “Isoperimetric problems for the helicity of vector fields and the Biot-Savart and curl operators,” J. Math. Phys. 41, 5615–5641 (2000).
  10. DeTurck, D., and Gluck, H., e-print arXiv:math.GT/0406276.
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