Abstract
In this chapter we present the fundamental equations that govern the dynamics of a system of charged particles interacting with the electromagnetic field, namely the Maxwell and Lorentz equations, clarifying their role and analyzing their general characteristics. For the reasons explained in the previous chapter we will write these equations in a manifestly covariant form, as well as in the standard three-dimensional notation. We will in particular highlight their distributional nature and derive the main conservation laws they imply. A considerable part of the book will then be devoted to a detailed analysis of the solutions and physical consequences of these equations. We begin the chapter with a description of the kinematics of a relativistic particle.
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Notes
- 1.
In textbooks the time-dependent position vector of a particle is usually denoted by the symbol \(\mathbf {x}(t)\). We prefer the notations \(\mathbf {y}(t)\) and \(y^\mu (\lambda )\) – in place of \(\mathbf {x}(t)\) and \(x^\mu (\lambda )\) – in order to avoid confusion with the symbol \(x^\mu =(t,\mathbf {x})\) denoting a generic space-time point, as for instance the argument x of the electromagnetic field \(F^{\mu \nu }(x)\).
- 2.
- 3.
In a sense, the only purpose of the Maxwell equation (2.22) is to qualify the singularities of \(F^{\mu \nu }\) along the world lines of the particles, given that in their complement \(\mathcal{C}\) the current is zero. In fact, in \(\mathcal{C}\) the electromagnetic field satisfies the trivial equation \(\partial _\mu F^{\mu \nu }=0\).
- 4.
In this section the indices \(\mu \), \(\nu \) etc. assume the values \(1,\ldots , D\), or equivalently \(0,1,\ldots ,D-1\).
- 5.
With respect to the spatial variables \((x^1,\cdots ,x^{D-1})\) the function \(\widehat{\varphi }(k)\) in (2.87) corresponds, actually, to the inverse Fourier transform. In a Minkowski space-time this choice has the advantage to preserve the manifest Lorentz invariance. For example, if \(\varphi (x)\) is a scalar field, and \(k^\mu \) is considered as a four-vector, then also \(\widehat{\varphi }(k)\) is a scalar field, see Sect. 5.2.
- 6.
Setting, in the general case, \(\mathbf {E}(t,0,0,0)\) equal to zero would, in particular, eliminate tout court the fundamental energy-loss phenomenon of accelerated particles.
- 7.
A priori \(T^{\mu \nu }_\mathrm{em}\) could also include terms involving the Levi-Civita tensor, as for instance
$$ \widetilde{T}^{\mu \nu }_\mathrm{em}=\eta ^{\mu \nu } \varepsilon ^{\alpha \beta \gamma \delta }F_{\alpha \beta }F_{\gamma \delta }= -8\eta ^{\mu \nu }\mathbf {E}\cdot \mathbf {B}. $$However, since \( T^{\mu \nu }_\mathrm{em}\) must be a tensor, and \(\widetilde{T}^{\mu \nu }_\mathrm{em}\) is a pseudotensor, see Sect. 1.4.3, such terms would violate the parity and time-reversal invariances of electrodynamics.
- 8.
In (2.140) we made the implicit assumption that the components of the tensor \(F^{\mu \nu }(x)\) are sufficiently regular functions, so that the derivatives of \(T^{\mu \nu }_\mathrm{em}\) could be computed by applying Leibniz’s rule. Actually we will see that – for point-like particles – the tensors \(F^{\mu \nu }(x)\) solving Maxwell’s equations are so singular that the components of \(T^{\mu \nu }_\mathrm{em}\) are not even distributions, so that eventually it makes no sense at all to compute \(\partial _\mu T^{\mu \nu }_\mathrm{em}\). For the moment we will ignore this fundamental problem and continue with our formal analysis, postponing its solution to Chap. 16.
- 9.
In Sect. 3.4 we will show that in any relativistic theory which is based on a variational principle there exists a conserved as well as symmetric energy-momentum tensor.
- 10.
Since in electrodynamics \(T^{\mu \nu }\) at large distances decreases as \(1/r^4\), see (2.143), the tensor \(M^{\mu \alpha \beta }\) in (2.156) decreases only as \(1/r^3\). Nevertheless, it can be seen that in electrodynamics – as in all other fundamental theories – the integrals (2.159) converge anyway, thanks to the peculiar behavior of the constituent fields at large distances.
- 11.
A function f is said to be asymptotically polynomially bounded if there exist a number \(L>0\) and a positive polynomial \({\mathcal P}(x)\), such that for every x for which \(\sqrt{(x^1)^2+\cdots +(x^D)^2}>L\) the function f satisfies the bound \(|f(x)|\le {\mathcal P}(x)\).
References
M. Reed, B. Simon, Methods of Modern Mathematical Physics - I Functional Analysis (Academic Press, New York, 1980)
W. Rudin, Functional Analysis (McGraw-Hill, New York, 1991)
E.G.P. Rowe, Structure of the energy tensor in the classical electrodynamics of point particles. Phys. Rev. D 18, 3639 (1978)
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Lechner, K. (2018). The Fundamental Equations of Electrodynamics. In: Classical Electrodynamics. UNITEXT for Physics. Springer, Cham. https://doi.org/10.1007/978-3-319-91809-9_2
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