Abstract
Already within a given, fixed division of four-dimensional spacetime into the space where experiments are performed, and the laboratory time variable, Maxwell’s equations show interesting transformation properties under continuous and discrete spacetime transformations. However, only the action of the whole Lorentz group on them reveals their full symmetry structure. A good example that illustrates the covariance of Maxwell’s equations is provided by the electromagnetic fields of a point charge uniformly moving along a straight line.
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- 1.
The Poincaré lemma applies to any open neighbourhood \(U\subset M\) of the point \(p\in M\) which can be contracted to \(p\) without leaving the manifold \(M\).
- 2.
As the base manifold \(M\) is the flat space \(\mathbb{R}^{4}\) all tangent spaces \(T_{x}\mathbb{R}^{4}\) can be identified with this space. As a consequence, the points \(x\in M\) and the vectors \(v\in T_{x}M\) have the same transformation behaviour.
- 3.
Its geometric role will be clarified in Chap. 5 below.
- 4.
Every smooth finite-dimensional surface that is embedded in a manifold with higher dimension is called a hypersurface.
- 5.
One should notice that on \(\mathbb{R}^{3}\), the Euclidean space, one has \(E_{i}=E^{i}\) and \(B_{k}=B^{k}\). On \(\mathbb{R}^{3}\) and with cartesian coordinates covariant and contravariant indices can be identified and need not be distinguished.
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© 2012 Springer-Verlag Berlin Heidelberg
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Scheck, F. (2012). Symmetries and Covariance of the Maxwell Equations. In: Classical Field Theory. Graduate Texts in Physics. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-27985-0_2
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DOI: https://doi.org/10.1007/978-3-642-27985-0_2
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Publisher Name: Springer, Berlin, Heidelberg
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Online ISBN: 978-3-642-27985-0
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