Abstract
We shall present the mathematically more sophisticated content of this chapter as simple as possible. We refer to Giulini (2006) for an exposure ‘on an advanced level ...(of) the algebraic and geometric structures that underly the theory of Special Relativity’. Mathematical tools are presented in the Appendix.
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Notes
- 1.
- 2.
The Minkowski space is a physically very important abstract mathematical construction, but it is beyond our practical perception. This concerns both its four-dimensionality and its indefinite metric. This should be kept in mind in all illustrations of examples in this space. For this reason, there also exist different but completely equivalent definitions for the Minkowski space, with different pros and cons, but they can easily be transformed from one to another. For the signature \((+, -, -, -)\) used here, the invariant line element has a direct meaning: \(ds = c\, d\tau \), where \(d\tau \) denotes the eigentime of a clock moving from E to F, cp. also Sect. 2.1.1. An other signature used in literature is \((-, +, +, +)\) and furthermore, if the time is used as fourth coordinate and written at the last place \((+, +, +, -)\) and \((-, -, -, +)\). The last signature is different from our convention only in the kind of numbering. In the covariant representation of electrodynamics, it is important to take care of the counting convention. We shall see further below that also the use of an imaginary fourth coordinate as in (331) can be helpful. However, for the transition to the General Relativity Theory, this construction is unsuitable and will not be used in this book.
- 3.
Einstein (1905a) has already mentioned this statement in a footnote of his most famous paper.
- 4.
- 5.
When we also admit curvilinear coordinates, e.g. spherical polar coordinates in space, or when we even transform to accelerated reference systems, then for keeping the tensor properties, we have to replace the partial by the covariant derivative, cp. Chap. 13 and Appendix B.1.
- 6.
Really the light cone does not depend on the inertial system because of the universality of the speed of light.
- 7.
The identic notion of electric field strength for \({\small \mathbf E}\) and magnetic field strength for \({\small \mathbf H}\) originates from a historically wrong understanding of the magnetic field quantities. Not \({\small \mathbf E}\) and \({\small \mathbf H}\), but \({\small \mathbf E}\) and \({\small \mathbf B}\) are both physically and mathematically related. A relict from this misunderstanding is preserved in the definition of permeability. We write \({{\small \mathbf H} = (1/\mu )\,\small \mathbf B}\), but \({{\small \mathbf D} = \varepsilon \,\small \mathbf E}\), s. Eqs. (487), (491), (497) and (498).
- 8.
As unit for a certain amount of matter we have also introduced the mole as SI-basic unit, cp. footnote 2 of Chap. 6 and Problem 15.
- 9.
In Gauss’ cgs-System the unit for the magnetic induction is one Gauß = 10\(^{-4}\) Tesla.
- 10.
In hypothetical theories attempting to unify the electromagnetic interaction with other fundamental interactions, the possible existence of magnetic monopoles becomes again a theoretical concept. One has been looking for magnetic monopoles in cosmic rays. In accelerator experiments, the very high energies may come into the range for producing magnetic monopoles. Its inclusion into the theory of electromagnetism is quite straight forward. A hypothetical density of magnetic monopoles \(\varrho _m\) and a corresponding current density \(\varsigma _m\) could be introduced with replacements of Eqs. (478) and (483) by \(\mathrm{div}\,\mathbf{B} = \varrho _m\) and \(\mathrm{rot}{} \mathbf{E} + \displaystyle \frac{\partial \mathbf{B}}{\partial t} = -\varsigma _m\). More changes are not necessary.
- 11.
For avoiding the factor \(1/(4\,\pi )\) in the absolute unit system one defines instead of Eq. (486) \(\tilde{\Phi }_D \equiv \int \!\!\!\!\int \limits _{\!\!\!\!\partial K}\tilde{\mathbf{D}}\cdot d\mathbf{S} \, {:}= 4\,\pi \,\int \!\!\!\!\int \limits _{\!\!\!\!K}\!\!\!\!\int \tilde{\rho }\, dxdydz = 4\,\pi \,\sum \tilde{e}_i\). Here we have superscribed all quantities with a tilde. The conversion between both unit systems is given in Eqs. (581).
- 12.
In the cgs-system, the unit of magnetic field is Oerstedt, \(1\,\mathrm{Oe} = (1/4\pi )\cdot 10^{3}\mathrm{A/m}\).
- 13.
Since today the signature \((+, -, -, -)\) is in use, we have also introduced the wave operator with the opposite sign as used in older literature, \((\triangle - \partial ^2\,/\,\partial t^2)\). We remark that this requires at transition to the Klein–Gordon Gleichung with the operator \((\Box - \kappa ^2)\), that \(\kappa \) must be imaginary for getting a real mass.
- 14.
The name is due to honour the danish physicist Ludvig Valentin Lorenz (1829–1891) and should not be mistaken for the dutch physicist Hendrik Antoon Lorentz (1853–1928). There exists even a Lorenz - Lorentz formula, cp. Sommerfeld (1952).
- 15.
The possibility to change the quantities \(\mathbf{A}\) and \(\varphi \) by \(\mathbf{A}\,\,\rightarrow \,\,\mathbf{A}' + {\nabla } \chi \) and \(\varphi \,\,\rightarrow \,\,\varphi ' = \varphi \, - \partial /(\partial _t) \chi \), without changing the physical fields \(\mathbf{E}\) and \(\mathbf{B}\), Eq. (516), is called a gauge transformation. The deeper theory of gauge transformations goes back to Weyl, it got a huge meaning in modern physics.
- 16.
Elementary charges of quarks—the constituents of protons and neutron-do not exist as free particles.
- 17.
Here we disregard a possible three-dimensional tensor character of these quantities in crystalline materials.
- 18.
For the force density \(f^i\), we need no tilde since the pure mechanical quantities in SI-units and the absolute MKS-units are identical.
- 19.
This definition generalises the mechanical angular momentum \(M^{li} = \epsilon ^{lik}L_k\) with \(\mathbf{L} =\mathbf{x} \times \mathbf{p}\), hence \(M^{li} =\epsilon ^{lik}\epsilon _{krs}x^rp^s = x^lp^i - x^ip^l\), cp. Eq. (1645).
- 20.
The source-free Maxwell equations are invariant under four-dimensional translations and under Lorentz transformations in the Minkowski space. The Noether theorem provides the existence of a so-called canonical energy–momentum tensor \(T^{ik}_{can}\) and an angular momentum tensor \(M^{lik}\) fulfilling the differential conservation laws, Eqs. (592) and (593). The canonical energy–momentum tensor \(T^{ik}_{can}\) is not symmetric. The quantity \({\widehat{M}}^{lik} = x^lT^{ik}_{can} - x^iT^{lk}_{can}\) describes the ‘orbital angular momentum’, that does not fulfil a conservation law because of the asymmetry of \(T^{ik}_{can}\). Only the total angular momentum \(M^{lik} =\widehat{M}^{lik} + S^{lik}\) fulfils the conservation law (593). The existence of the eigen angular momentum \(S^{lik}\), the spin tensor of the electromagnetic field, is the reason for the asymmetry of the canonical tensor \(T^{ik}_{can}\). The ‘spin’ is discussed intensively in Chap. 10, Sects. 3–7.
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Günther, H., Müller, V. (2019). Mathematical Formalism of Special Relativity. In: The Special Theory of Relativity. Springer, Singapore. https://doi.org/10.1007/978-981-13-7783-9_9
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