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Determination of electromagnetic medium from the Fresnel surface
- Publication Year :
- 2011
- Publisher :
- arXiv, 2011.
-
Abstract
- We study Maxwell's equations on a 4-manifold where the electromagnetic medium is described by an antisymmetric $2\choose 2$-tensor $\kappa$. In this setting, the Tamm-Rubilar tensor density determines a polynomial surface of fourth order in each cotangent space. This surface is called the Fresnel surface and acts as a generalisation of the light-cone determined by a Lorentz metric; the Fresnel surface parameterises electromagnetic wave-speed as a function of direction. Favaro and Bergamin have recently proven that if $\kappa$ has only a principal part and if the Fresnel surface of $\kappa$ coincides with the light cone for a Lorentz metric $g$, then $\kappa$ is proportional to the Hodge star operator of $g$. That is, under additional assumptions, the Fresnel surface of $\kappa$ determines the conformal class of $\kappa$. The purpose of this paper is twofold. First, we provide a new proof of this result using Gr\"obner bases. Second, we describe a number of cases where the Fresnel surface does not determine the conformal class of the original $2\choose 2$-tensor $\kappa$. For example, if $\kappa$ is invertible we show that $\kappa$ and $\kappa^{-1}$ have the same Fresnel surfaces.<br />Comment: 23 pages, 1 figure
- Subjects :
- Statistics and Probability
Physics
Surface (mathematics)
Antisymmetric relation
Lorentz transformation
General Physics and Astronomy
Physics::Optics
FOS: Physical sciences
Statistical and Nonlinear Physics
Cotangent space
Mathematical Physics (math-ph)
Null (physics)
Mathematics::Logic
symbols.namesake
Mathematics - Analysis of PDEs
Modeling and Simulation
symbols
FOS: Mathematics
Tensor
Tensor density
Hodge dual
78A25, 83C50, 53C50, 78A02, 78A05
Mathematical Physics
Mathematical physics
Analysis of PDEs (math.AP)
Subjects
Details
- Database :
- OpenAIRE
- Accession number :
- edsair.doi.dedup.....36f42333195bc1f20136f208b41aea58
- Full Text :
- https://doi.org/10.48550/arxiv.1103.3118