Abstract
As we saw, the fundamental equations of electrodynamics are invariant under Poincaré transformations and, in addition, they entail the conservation of the four-momentum and the four-dimensional angular momentum. At first sight these two aspects – relativistic invariance and the presence of conservation laws – do not seem to have anything to do with each other. Noether’s theorem – which intertwines them in an inextricable way – relies, actually, heavily on a fundamental paradigm of theoretical physics, that we have not yet introduced: the variational method. In fact, the fundamental equations of electrodynamics own a property that at this stage of the presentation appears still hidden, namely, they descend from a variational principle: It is precisely this peculiar characteristic which ensures the above link between symmetries and conservation laws.
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Notes
- 1.
\(\delta I\) is a functional of the 2N functions q and \(\delta q\), and so, actually, it should be denoted by \(\delta I[q,\delta q]\).
- 2.
For further details on hypersurfaces, the properties of the induced metric (3.23), and the infinitesimal volume element \(\sqrt{g} \,d^3\lambda \), see Sect. 19.3.2, where it is in particular shown that the latter is unaffected by arbitrary changes of the parameters \(\varvec{\lambda }\), as it must be.
- 3.
The invariant \(\mathcal{L}_0=\varepsilon ^{\mu \nu \rho \sigma }F_{\mu \nu }F_{\rho \sigma }=-8\mathbf {E}\cdot \mathbf {B} \) is actually a pseudoscalar – in that under parity and time reversal it changes sign, see (1.68) – while the Lagrangian of electrodynamics is a scalar, see (4.17). The inclusion of \(\mathcal{L}_0\) would therefore violate the invariance of electrodynamics under the complete Lorentz group O(1, 3).
- 4.
To keep the notation simple we maintain for the infinitesimal variations of the fields and the coordinates the same symbol \(\delta \) as for the finite ones.
- 5.
In General Relativity, Einstein’s equations equal a certain two-index symmetric tensor, built with the metric \(g_{\mu \nu }(x)\) and its derivatives, to the energy-momentum tensor. These equations would thus be inconsistent, would the latter not be a symmetric tensor, see Sect. 9.2.
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Lechner, K. (2018). Variational Methods in Field Theory. In: Classical Electrodynamics. UNITEXT for Physics. Springer, Cham. https://doi.org/10.1007/978-3-319-91809-9_3
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DOI: https://doi.org/10.1007/978-3-319-91809-9_3
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