Abstract
The Theory of Special Relativity (and consequently the Theory of General Relativity) would have never be discovered if Maxwell had not formulated the theory of Electrodynamics.
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Notes
- 1.
That is, 3-space rotation and translation or equivalently rigid body motion.
- 2.
They did not call it dark matter.
- 3.
Sometimes it is stated erroneously that this Principle concerns all physical phenomena – see physical quantities. It does not. All Newtonian physical quantities do not obey Einstein’s Relativity Principle whereas they do the Galileo Relativity Principle. For this reason a Newtonian physical quantity (velocity for example) is not a relativistic physical quantity etc. Every theory of Physics has a separate domain of application which comprises a “subset” of physical phenomena. The theory of “everything” is part of the human utopia. See Chap. 2 for details.
- 4.
Electromagnetism is an old subject with applications in the most diverse areas of science and engineering. As a result there is a number of units in terms of which Maxwell equations have been written. In this book we shall use the SI system. In Sect. 13.17 we show how one writes these equations in other systems and especially in the Gauss system.
- 5.
The current j is more general than this but for the time being the conduction current j = ρ v suffices.
- 6.
To be precise equation (13.1) implies that \(\mathbf { \triangledown \cdot B }= \text{constant}\) in K. The requirement that the value of this constant is 0 is an extra assumption whose discussion is outside the scope of this book. For this reason this equation is considered as an extra independent equation compatible with the rest of Maxwell equations.
- 7.
The quantities ε, μ are scalar only for homogeneous and isotropic media. For anisotropic and non-homogeneous materials these quantities are described by second order symmetric tensors.
- 8.
The values assigned to these constants are the following:
$$\displaystyle \begin{aligned} \varepsilon_{0}=8,85 \times 10^{-12} \text{ Farad/meter},\;\;\mu_{0}=1,26 \times 10^{-6}\text{ Henry/meter.} \end{aligned} $$(13.8) - 9.
The operators \(\boldsymbol {\triangledown }\) and \(\frac {d}{dt}\) do not commute, because the operator \(\frac {d}{dt}\) includes variations in space.
- 10.
In order to write Maxwell equations for a homogeneous and isotropic medium it is enough to replace c 2 with the product εμ or with the speed u 2 of the electromagnetic field in the medium.
- 11.
A gauge transformation in a dynamical theory is a transformation of the dynamical fields such that the transformed fields satisfy the same dynamical equations as the original fields.
- 12.
It is interesting to read about the life and the work of D’Alembert. The interested reader can visit the web site http://www-history.mcs.st-andrews.ac.uk/Biographies/D'Alembert.html.
- 13.
We recall at this point the rule for signs of the components when we lower and raise the index of a four-vector. To the contravariant four-vector A i we correspond the covariant four-vector A i with the relation A i = η ij A j where η ij is the Lorentz metric. In case we have Lorentz orthonormal frames (which we assume to be the rule) the Lorentz metric has its canonical form η ij = diag(−1, 1, 1, 1). Therefore in such frames (and only there!) the four-vector \(A^{i}=\left ( \begin {array}{c} B \\ \mathbf {A} \end {array} \right ) _{\Sigma }\) corresponds the four-vector A i = (−B, A)Σ, that is, the sign of the zeroth component changes and the matrix from column becomes row. If the frames is not Lorentz orthonormal then the above simple rule does not apply and one (a) has to compute the components of the Lorentz metric and then (b) multiply the resulting 4 × 4 matrix with the column (or row) matrix defined by the components of the four-vector in the same frame.
- 14.
Indeed let us assume that the four quantity (Bc, A)Σ transforms from a LCF Σ to the LCF Σ′ according to the rule:
$$\displaystyle \begin{aligned} \left( \begin{array}{l} B^{\prime}c \\ {\mathbf{A}}^{\prime} \end{array} \right) _{\Sigma^{\prime}}=\boldsymbol{\kappa}(\upsilon)\left( \begin{array}{l} Bc \\ \mathbf{A} \end{array} \right) _{\Sigma} \end{aligned}$$where κ(υ) is a function of relative velocity v of the frames Σ, Σ′. Then relation (13.34) is preserved in the new frame. Indeed:
$$\displaystyle \begin{aligned} A^{i}\longrightarrow A^{i\prime}=\boldsymbol{\kappa}(\upsilon)A^{i} \end{aligned}$$therefore:
where Q(υ) is some function of the relative velocity resulting from the Lorentz transformation. This condition is not Lorentz covariant except if and only if Q(υ)κ(υ) = 1.
- 15.
It is important to stay for a moment at this point. Indeed so much effort and discussions have been done to the time dilation whereas the charge dilation has passed unnoticed. Obviously this is due to the facts that (a) time is considered to be (unreasonably!) more fundamental than charge and (b) Time concerns everyone therefore it is more familiar to the common reason than charge. Special Relativity proposed a new view of cosmos, because it rejected the absolute character of space and time, hence it effected the concept of “omnipresent mind” which is – in one way or another – at the roots of all social structures. As a result this theory raised much reaction and provoked many “doubts” and paradoxes during the first years of its existence. Later theories of Physics General Relativity and especially Quantum Mechanics where considered to be a matter for physicists only. Indeed the first was concerned with the large scale universe and the second with the very small scale. The realm of application of both theories was therefore too far from our direct sensory experience therefore they did not bother the wide social scale. Thus they were accepted without wide social reaction. However needless to say that the development of Quantum Mechanics today and in the future are controlling most of our everyday activities and our social functioning. Newtonian Physics was founded on the archetypes of the society. After Special Relativity it is the Physics which creates and enforces its archetypes on the society. The solid harmony introduced by ancient Greeks and developed further in Europe was based on the human “structure” and the sensory environment. That died with the appearance of the new relativistic “point of view” and was replaced with technocratic lividness structure which is object oriented, that is, the “I am” has been replaced with “I have”.
- 16.
From this result we can compute the force between two straight parallel conductors of infinite length carrying currents I 1 and I 2. Indeed the first conductor (the I 1 say) exerts to the elementary length dl 2 of the second conductor the force \({\mathbf {F}}_{21}=-\frac { dq_{2}u\mu _{0}I_{1}}{2\pi r}\widehat {\mathbf {e}}_{r}\). But the length dl 2 carries charge dq 2 = I 2 dt so that dq 2 u = I 2 dl 2, therefore the force per unit length on the current I 2 due to the current I 1 is: \(\frac {{\mathbf {F}}_{21}}{dl_{2}}=-\frac {\mu _{0}I_{1}I_{2}}{2\pi r} \widehat {\mathbf {e}}_{r}\). This force is obviously attractive. If the currents have opposite direction of flow that force is repulsive, a phenomenon with important practical applications.
- 17.
In a compact formalism equation (13.13) is written:
$$\displaystyle \begin{aligned} E_{\mu}= -\phi,_{\mu}- cA_{\mu,0} \end{aligned}$$Now (13.43) gives:
$$\displaystyle \begin{aligned} \Omega_{0} = -\frac{\phi}{c},\quad \Omega_{\mu}= A_{\mu}. \end{aligned}$$from which we compute:
$$\displaystyle \begin{aligned} \Omega_{0,\mu} = -\frac 1c \phi,_{\mu}, \quad \Omega_{\mu,0} = A_{\mu,0} \quad \Rightarrow\quad \Omega_{0,\mu} - \Omega_{\mu,0} = -\frac 1c (\phi,_{\mu}+ c A_{\mu,0})=\frac 1c E_{\mu}. \end{aligned}$$For the magnetic field we have from (13.11):
$$\displaystyle \begin{aligned} B^{\mu}= -\epsilon^{ \mu \nu \rho} A_{[\nu \rho ]}=-\epsilon^{ \mu \nu \rho} \Omega_{[\nu \rho]}. \end{aligned}$$ - 18.
According to our convention the first index of a matrix counts rows and the second columns
- 19.
See also Exercise 1.7.3. This practical rule applies to tensors of type (0,2) and (2,0)! It is possible to be generalized to tensors with more indices however, this is outside our interest.
- 20.
It is the so called Frobenius condition which need not worry us further.
- 21.
The tensor η ijkl where i, j, k, l = 0, 1, 2, 3 equals zero if two of the indices have the same value and ± 1 if (ijkl) is an even or an odd permutation of (0123). See Sect. 13.10.1.
- 22.
We are using the identity (A × B) × C = (A ⋅ C)B − (B ⋅ C)A.
- 23.
Recall that the definitions of physical quantities in Special Relativity are made in the proper frame and coincide with the corresponding Newtonian quantities (provided they exist). This is the general rule – strategy for defining physical quantities in Special Relativity and it is justified by the facts (a) that the physical quantities we understand and manipulate in the Newtonian world and (b) Special Relativity must be understood physically in the Newtonian world. The difference introduced by Special Relativity is in the transformation of these quantities from LCF to LCF.
- 24.
In case we have a charge density ρ instead of a single charge, then the four-vector F i is the density of four-force and j i is the density of the four-current of the charge \(j^{i} =\left ( \begin {array}{c} \rho c \\ \mathbf {j} \end {array} \right ) _{\Sigma }\).
- 25.
It is possible to integrate the equations of motion directly and not use the first integrals. This is done as follows. Without restricting the generality we consider the z axis along the direction of the magnetic field (B = B k) in which case the equations of motion in the direction perpendicular to the magnetic field give:
$$\displaystyle \begin{aligned} \left. \begin{array}{rcl} m\gamma_{0} \dot\upsilon_{x} & = & qB\upsilon_{y} \\ m\gamma_{0} \dot\upsilon_{y} & = & - qB\upsilon_{x} \end{array} \right\} \Rightarrow\left. \begin{array}{rcl} \dot\upsilon_{x} & = & \omega\upsilon_{y} \\ \dot\upsilon_{y} & = & - \omega\upsilon_{x} \end{array} \right. \end{aligned}$$where \(\omega = \frac {qB}{m\gamma _{0} }\). To solve this system of simultaneous equations we multiply the second with i and add. It follows:
$$\displaystyle \begin{aligned} \left( \dot{\upsilon}_{x} + i\dot{\upsilon}_{y} \right) = - i \omega\left( \upsilon_{x} + i\upsilon_{y} \right) \Rightarrow \end{aligned}$$$$\displaystyle \begin{aligned} w(t) = w(0)e^{ - i\omega t} \quad \mbox{where} \qquad w(t) =\upsilon_{x} + i\upsilon_{y} . \end{aligned}$$Equating the real and the imaginary parts we find:
$$\displaystyle \begin{aligned} \begin{array}{rcl} \upsilon_{x} (t) &\displaystyle =&\displaystyle \upsilon_{x} (0)\cos\omega t + \upsilon_{y} (0)\sin\omega t \\ \upsilon_{y} (t) &\displaystyle =&\displaystyle \upsilon_{x} (0)\sin\omega t - \upsilon_{y} (0)\cos\omega t. \end{array} \end{aligned} $$Let us assume for simplicity the initial conditions υ x(0) = υ, υ y(0) = 0. Then \(\upsilon _{x} (t) = \upsilon \cos \omega t \quad \upsilon _{y} (t) = \upsilon \sin \omega t.\) Integrating these last relations we find for the motion perpendicular to the direction of the magnetic field:
$$\displaystyle \begin{aligned} x(t) = x_{0} + \frac{\upsilon}{\omega}\sin\omega t \end{aligned}$$$$\displaystyle \begin{aligned} y(t) = y_{0} - \frac{\upsilon}{\omega}\cos\omega t. \end{aligned}$$ - 26.
It will help if we demonstrate the computation of the components of the four-vector for the magnetic field. For example for the B x coordinate we have:
$$\displaystyle \begin{aligned} B_{x}=\frac{1}{2c}\eta_{xjkl}F^{jk}u^{l}=\frac{1}{2c}c\,\eta_{xtjk}F^{jk}=- \frac{1}{2}\eta_{txyz}(F^{yz}-F^{zy})=F^{yz}=B_{x}^{+}. \end{aligned}$$ - 27.
We use semicolon (;) to indicate the partial derivative and not comma (,) as we do for the rest of the book. The reason is that the results we derive hold also in General Relativity where we have the Riemannian covariant derivative, which is indicated with semicolon. These results are important therefore we see no reason for not giving them in all their generality. If the reader finds it hard to follow the formalism he/she can replace semicolon with comma and all results go through without any change.
- 28.
Note that the components are not the same with the coefficients of the 1+3 decomposition because the former are just components whereas the second tensors.
- 29.
We set c = 1.
- 30.
A proof using the classical vector calculus is the following. We take the cross and the inner product of (13.226) with B:
$$\displaystyle \begin{aligned} \mathbf{j\times B}=k(\mathbf{E\times B})+\lambda(\mathbf{j\cdot B})\mathbf{B} -\lambda B^{2}\mathbf{j},\qquad \qquad \mathbf{j\cdot B=}k(\mathbf{E\cdot B}). \end{aligned}$$Subsequently we replace in (13.226) and get the required expression.
- 31.
See Bekenstein and Oron (1978) Phys Rev D 18, 1809.
- 32.
An equivalent definition of the energy momentum tensor for the electromagnetic field is
$$\displaystyle \begin{aligned} T_{ab}=\frac{1}{c}\left[ F_{ac}K_{b}^{.c}-\frac{1}{4}g_{ab}\left( F_{cd}K^{cd}\right) \right] \end{aligned}$$Obviously the two definitions are the same due to the antisymmetry of K ab.
- 33.
Recall that u 0 = c, u 0 = −c.
- 34.
The partial derivative \(\frac {\partial }{\partial t}\) indicates that the volume V is comoving, that is does not change in time.
- 35.
S i is defined in (13.281).
- 36.
We recall that in the Newtonian approach the magnetic field of a current i satisfies two Laws. The Ampére Law \(\oint \mathbf {B}\cdot d\mathbf {l} =\mu _{0} i\) and the Biot–Savart Law \(d\mathbf {B}= \frac {\mu _{0}}{4 \pi } \frac {i}{r^{3}} \, d\mathbf {l} \times \mathbf {r}\) where d l is an elementary length along the conductor and r is the point where one calculates the magnetic field (see Fig. 13.5). Ampére’s Law is used in the cases the magnetic field has high (geometric) symmetry whereas the Biot–Savart Law is used in more general cases in which the magnetic field is computed by integration along the conductor.
- 37.
For the observer on the charge the field at the point P the moment t appears to come from the origin O of Σ. Because in the standard non-relativistic approach to electromagnetism time is understood in the Newtonian approach, the origin O is referred as the retarding point. This terminology has no place in the relativistic approach where time is a mere coordinate and can take any value depending on the frame.
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Tsamparlis, M. (2019). The Electromagnetic Field. In: Special Relativity. Undergraduate Lecture Notes in Physics. Springer, Cham. https://doi.org/10.1007/978-3-030-27347-7_13
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