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Relativistic optics with spacetime algebra using the vacuum form reduction

Ribeiro, M. ; Paiva, C. R.

Relativistic optics with spacetime algebra using the vacuum form reduction, Proc IEEE AP-S/URSI International Symp., Charleston, United States, Vol. --, pp. -- - --, June, 2009.

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Abstract
A metric-free approach to the foundations of classical electrodynamics, through the calculus of exterior differential forms, enables to write the two Maxwell equations (homogeneous and inhomogeneous) in a topological framework. In fact, a topological framework, based on a naked 4-dimensional smooth manifold – that allows for a foliation into 3-dimensional hypersurfaces – where no connection and no metric are explicitly introduced can be developed (F. W. Hehl and Yu. N. Obukhov, Foundations of Classical Electrodynamics: Charge, Flux, and Metric, Birkhäuser, Boston, 2003). This metric-free approach renders clear that the metric should be relegated to a peripheral region within the whole theoretical framework of electrodynamics. Indeed its role should be identified with the spacetime constitutive relation. Maxwell equations are then valid in any metric context: either in the rigid (flat) Lorentz metric of Minkowski spacetime or in the flexible (curved) Riemannian metric of general relativity that changes from point to point according to Einstein’s field equation. By using the grammar of Clifford’s geometric algebras and the technique of vacuum form reduction, it is also possible to give a geometric (i.e., an intrinsic and coordinate-free) interpretation of what we have called the equivalence principle (M. A. Ribeiro and C. R. Paiva, Metamaterials, Vol. 2, Issues 2+3, pp. 77-91, September 2008): a given electromagnetic medium creates an effective geometry and a given geometry creates an effective medium. This mathematical construction, stemming from the specificity of the grammar of geometric algebra, allows for a rather transparent formulation of the equivalence between the topological and the materials interpretations that are useful in studying both transformation and moving media (M. A. Ribeiro and C. R. Paiva, chapter in: S. Zouhdi et al., Eds., Metamaterials and Plasmonics: Fundamentals, Modelling, Applications, Springer, Dordercht, 2009, pp. 63-74).

In this communication, we restrict our attention to spacetime algebra in Minkowski spacetime (C. R. Paiva and M. A. Ribeiro, J. Electromagn. Waves and Appl., Vol. 20, No. 7, pp. 941-953, July 2006). However, we intend to show that it is also possible to use the vacuum form reduction technique to study relativistic optics in Minkowski spacetime with a geometric perspective: several results concerning relativistic optics of moving media can be easily handled by introducing this transformation that leads to a fictitious spacetime. By this fictitious spacetime medium we mean: a spacetime constitutive relation that has the same mathematical structure as vacuum, although we are analyzing an isotropic medium that is moving in the laboratory frame and hence, as is well-known, it is actually a nonreciprocal bi-anisotropic medium in that lab frame. In summary: by using the technique of vacuum form reduction we can circumvent the usual spacetime algebra approach to special relativity by performing our calculations for a nonreciprocal bi-anisotropic medium in a fictitious – albeit completely equivalent – spacetime that has, from the mathematical point of view, the same intrinsic simplicity as vacuum itself.