UDC 538.3

PHYSICS

A. VIGLIN, V. KASHIN

ELECTROMAGNETIC POTENTIALS AND THEIR GAUGE IN AN ANISOTROPIC, DISPERSIVE, AND MOVING MEDIUM

(Presented by Academician B. P. Konstantinov on 26 V 1966)

If the electromagnetic-field equation in four-dimensional form for the 4-tensors \(F_{ik}(\mathbf E,\mathbf B)\) and \(\Phi_{ik}(\mathbf D,\mathbf H)\) (see (1), § 33) is written for the Fourier amplitudes of these quantities, \(f_{ik}\) and \(\varphi_{ik}\), respectively, then in a Galilean coordinate system with metric tensor

\[ g_{ik}= \begin{pmatrix} 1&0&0&0\\ 0&1&0&0\\ 0&0&1&0\\ 0&0&0&-1 \end{pmatrix} \]

they have the form

\[ \varphi^{ik}k_k=-i\frac{4\pi}{c}j^i, \tag{1} \]

\[ \bar f^{ik}k_k=0, \tag{2} \]

\[ \varphi^{ik}=S^{ik}_{\ \ sr}\bar f^{sr}, \tag{3} \]

where \(\bar f^{ik}\) is the pseudo-4-tensor dual to \(f_{ik}\) and related to it by the relation \(\bar f^{ik}=\,^{1}\!/_{2}\varepsilon^{ikmn}f_{mn}\); \(k_i\) is the wave 4-vector; \(j^i\) is the Fourier amplitude of the 4-vector of current density; \(S^{ik}_{\ \ sr}\) is the material pseudo-4-tensor of permeability, antisymmetric in the upper pair and in the lower pair of indices, first introduced by I. E. Tamm \((^2)\). Here and below, indices denoted by letters of the Latin alphabet take the values \(1,2,3,0\) and number all coordinates of 4-space; summation is implied over twice repeated indices; \(\varepsilon^{iksr}\) are the contravariant components of the unit completely antisymmetric pseudo-4-tensor \(((^3), § 83)\), whose covariant components are such that \(\varepsilon_{1230}=1\) and \(\varepsilon^{iksr}=-\varepsilon_{iksr}\).

Equation (2) can be satisfied by putting

\[ \bar f^{ik}=iN^{iks}k_s, \tag{4} \]

where \(N^{iks}\) is a pseudo-4-tensor antisymmetric in all three indices. In the definition of \(N^{iks}\) there is an ambiguity: to \(N^{iks}\) one may, without changing \(\bar f^{ik}\), add \(T^{iksr}k_r\), where the pseudo-4-tensor \(T^{iksr}\) is arbitrary and completely antisymmetric, and therefore one may always put \(T^{iksr}=T\cdot\varepsilon^{iksr}\), where \(T\) is a true 4-scalar. We introduce the 4-vector \(a_r\), dual to the pseudo-4-tensor \(N^{iks}\) and being the Fourier amplitude of the 4-vector potential:

\[ N^{iks}=\varepsilon^{iksr}a_r. \tag{5} \]

Since the pseudo-4-tensor \(N^{iks}\) is defined ambiguously, one may impose on the components \(N^{iks}\), and consequently also on the components \(a_r\), one entirely arbitrary scalar condition, which merely restricts the arbitrariness in the choice of the 4-scalar \(T\).

Substituting (5) into (4), forming \(\varphi^{ik}\) with the aid of (3), and using (1), we obtain the equation for the 4-vector potential

\[ T^{in}a_n=-\frac{4\pi}{c}j^i, \tag{6} \]

where

\[ T^{in}=S^{ik}_{\cdot\cdot sr}\varepsilon^{srmn}k_k k_m=R^{ikmn}k_k k_m \tag{7} \]

is a true 4-tensor. Here the 4-tensor

\[ R^{ikmn}=S^{ik}_{\cdot\cdot sr}\varepsilon^{srmn}, \tag{8} \]

has been introduced; it is antisymmetric in the first pair and in the second pair of indices and establishes the relation between \(\varphi^{ik}\) and \(f_{mn}\)

\[ \varphi^{ik}=1/2 R^{ikmn}f_{mn}. \tag{9} \]

The relation inverse to (8) has the form

\[ S^{ik}_{\cdot\cdot sr}=-1/4 R^{ikmn}\varepsilon_{mnsr}. \tag{10} \]

For stationary media the components of the 4-tensor \(R^{ikmn}\) are expressed in terms of the components of the 3-tensors of the electric \(\varepsilon^{\alpha\beta}\) and magnetic \(\mu^{\alpha\beta}\) permeabilities

\[ R^{\alpha\beta\gamma 0}=0,\qquad R^{\alpha\beta\gamma\sigma}=\varepsilon^{\alpha\beta\lambda}\mu_{\lambda\nu}^{-1}\varepsilon^{\nu\gamma\sigma},\qquad R^{\alpha 0\beta\gamma}=0,\qquad R^{\alpha 0\sigma 0}=-\varepsilon^{\alpha\sigma}. \tag{11} \]

Indices denoted by letters of the Greek alphabet take the values 1, 2, 3 and number the spatial coordinates. In three-dimensional space the unit completely antisymmetric pseudo-3-tensor \(\varepsilon^{\alpha\beta\gamma}\) and \(\varepsilon_{\alpha\beta\gamma}\) is related to the analogous 4-tensor in four-dimensional space by the relations

\[ \varepsilon^{\alpha\beta\gamma}\equiv \varepsilon^{\alpha\beta\gamma 0},\qquad \varepsilon_{\alpha\beta\gamma}\equiv \varepsilon_{\alpha\beta\gamma 0}. \]

Introduce the 4-tensor

\[ G_{ik}=1/4\varepsilon_{ibsr}k_l R^{lspr}k_a R^{abmt}\varepsilon_{mtpk} \tag{12} \]

and multiply both sides of equation (6) by it, summing over the index \(i\). Then we obtain

\[ T^{in}a_n G_{ik}=-\frac{4\pi}{c}j^i G_{ik} \]

or

\[ C_k{}^n a_n=-\frac{4\pi}{c}j^iG_{ik}, \tag{13} \]

where

\[ C_k{}^n=G_{ik}T^{in}. \tag{14} \]

Using (7) and (8), for \(C_k{}^n\) we obtain

\[ C_k{}^n=A_{vwg}^{pmt}\varepsilon_{mtpk}\varepsilon^{vwgn}, \tag{15} \]

where the 4-tensor

\[ A_{vwg}^{pmt}=1/4\varepsilon_{ibsr}k_l R^{lspr}k_a R^{abmt}S^{if}_{\cdot\cdot vwk}k_f k_g, \tag{16} \]

has been introduced; it is antisymmetric in the first pair of lower and in the second pair of upper indices. By virtue of the identity

\[ \varepsilon_{mtpk}\varepsilon^{vwgn} \equiv \varepsilon_{mtps}\varepsilon^{vwgs}\delta_k^n+ \varepsilon_{mtps}\varepsilon^{vwsn}\delta_k^g+ \varepsilon_{mtps}\varepsilon^{vsgn}\delta_k^w+ \varepsilon_{mtps}\varepsilon^{swgn}\delta_k^v \tag{17} \]

equality (15) takes the form

\[ C_k{}^n= A_{vwg}^{pmt}\varepsilon_{mtps}\varepsilon^{vwgs}\delta_k^n+ A_{vwk}^{pmt}\varepsilon_{mtps}\varepsilon^{vwsn} -2A_{kwg}^{pmt}\varepsilon_{mtps}\varepsilon^{nwgs}. \tag{18} \]

The last term in (18) can be transformed using the relation

\[ \varepsilon_{mtps}\varepsilon^{nwgs} =-\left( \delta_m^n\delta_t^w\delta_p^g+ \delta_m^w\delta_t^g\delta_p^n+ \delta_m^g\delta_t^n\delta_p^w -\delta_m^n\delta_t^g\delta_p^w -\delta_m^w\delta_t^n\delta_p^g -\delta_m^g\delta_t^w\delta_p^n \right) \tag{19} \]

and the symmetry properties of the 4-tensor \(A_{vwg}^{pmt}\) noted above. As a result we obtain

\[ -2A_{kwg}^{pmt}\varepsilon_{mtps}\varepsilon^{nwgs} =-4A_{kwg}^{gwn}+4A_{kwg}^{nwg}+4A_{wkg}^{wng}. \tag{20} \]

It is not difficult to verify that the first term in (20) is equal to zero

\[ -4A_{kwg}^{gwn}=\varepsilon_{ibsr}k_l R^{lsrg}k_g k_a R^{abwn}k_f S^{if\cdot\cdot}_{\cdot\cdot kw}=0. \tag{21} \]

For this, instead of \(k_lR^{lsrg}k_g\), one must substitute, in accordance with (8), \(k_lS^{ls\cdot\cdot}_{\cdot\cdot vm}\varepsilon^{vmrg}k_g\), and expand the product \(\varepsilon_{ibsr}\varepsilon^{vmrg}\) by formula (19); then in each of the 6 terms there will be present the product of a symmetric 4-tensor of rank 2 with a 4-tensor antisymmetric in the same indices.

It can be shown that the 3rd term in (20)

\[ 4A_{wkg}^{wng}=\varepsilon_{ibsr}k_l R^{lswr}k_f S^{if\cdot\cdot}_{\cdot\cdot wk}k_a R^{abng}k_g=-C_k^n . \tag{22} \]

To convince oneself of this, note that the last three factors in (22) form, according to (7), the 4-tensor \(T^{bn}\), and, if the factor \(S^{if\cdot\cdot}_{\cdot\cdot wk}\) is transformed by formula (10), then all the factors standing before \(T^{bn}\) form, according to (12), \(G_{bk}\), and on the basis of (14)

\[ -G_{bk}T^{bn}=-C_k^n . \]

Let us transform the second term in (20)

\[ 4A_{kwg}^{nwg}=\varepsilon_{ibsr}k_l R^{lsnr}k_a R^{abwg}k_g k_f S^{if\cdot\cdot}_{\cdot\cdot kw}. \tag{23} \]

If in (23) we use the equality \(R^{abwg}=S^{ab\cdot\cdot}_{\cdot\cdot vm}\varepsilon^{vmwg}\) (see (8)), and then the identity analogous to (17):

\[ \begin{aligned} \varepsilon_{ibsr}\varepsilon^{vmwg}\equiv{}& \varepsilon_{pbsr}\varepsilon^{vmwp}\delta_i^g +\varepsilon_{ibsp}\varepsilon^{vmwp}\delta_r^g,\\ &+\varepsilon_{ibpr}\varepsilon^{vmwp}\delta_s^g +\varepsilon_{ibsp}\varepsilon^{vmwp}\delta_r^g, \end{aligned} \tag{24} \]

then it is obvious that, of the four terms formed in (23), three turn into zero because in each there is present the product of a 4-tensor symmetric in two indices with a 4-tensor antisymmetric in the same indices: in one, \(k_a k_b S^{ab\cdot\cdot}_{\cdot\cdot vm}\), in another, \(k_l k_s R^{lsnr}\), and in the third, \(k_i k_f S^{if\cdot\cdot}_{\cdot\cdot kw}\). Then

\[ 4A_{kwg}^{nwg}=\varepsilon_{ibsp}k_a S^{ab\cdot\cdot}_{\cdot\cdot vm}\varepsilon^{vmwp}k_f S^{if\cdot\cdot}_{\cdot\cdot lw}k_l R^{lsnr}k_r . \]

If one notes that, according to (8), \(S^{ab\cdot\cdot}_{\cdot\cdot vm}\varepsilon^{vmwp}=R^{abwp}\), then the resulting expression does not differ from that which stands in (22) after the first equals sign. Consequently,

\[ 4A_{kwg}^{nwg}=-C_k^n . \tag{25} \]

Taking into account (21) and (25), formula (18) takes the form

\[ C_k^n=V\delta_k^n+k_k P^n, \tag{26} \]

where the 4-scalar \(V=\frac{1}{3}A_{wwg}^{pmt}\varepsilon_{mtps}\varepsilon^{vwgs}\) and the 4-vector \(P^n\) are introduced, the latter having the form

\[ P^n=\frac{1}{3}G_{is}R^{ifsn}k_f . \tag{27} \]

Substituting (26) into (13), we obtain

\[ V\delta_k^n a_n+k_k P^n a_n=-\frac{4\pi}{c}j^i G_{ik}. \tag{28} \]

Since one scalar condition may be imposed on \(a_n\), choosing it in the form \(P^n a_n=0\), we obtain from (28) the solution for the 4-vector potential

\[ a_k=-\frac{4\pi}{c}\frac{j^i G_{ik}}{V}. \tag{29} \]

The equality \(P^n a_n=0\) is a generalization of the Lorentz condition in a material medium that is anisotropic and dispersive with respect to its electric and magnetic properties. Using (11), we obtain the components of the 4-vector \(P^n\) for stationary media:

\[ P^0=\frac{1}{3}G_{is}R^{if s0}k_f =\frac{1}{2}k_0T^{00}\varepsilon^{\sigma\nu}\mu_{\sigma\nu}^{-1} -\frac{1}{2}k_0g^\alpha\mu_{\alpha\nu}^{-1}p^\nu -k_0^3|\varepsilon^{\alpha\beta}|; \]

\[ P^\alpha=\frac{1}{3}G_{is}R^{if s\alpha}k_f = \]
\[ =-\frac{1}{2}\frac{k_\sigma\mu^{\sigma\nu}k_\nu}{|\mu^{\alpha\beta}|}p^\alpha +\frac{1}{2}k_0^2\varepsilon^{\sigma\nu}\mu_{\sigma\nu}^{-1}p^\alpha -\frac{1}{2}k_0^2\mu_{\sigma\nu}^{-1}p^\nu\varepsilon^{\sigma\alpha} -\frac{1}{2}T^{00}\frac{k_\sigma\mu^{\sigma\alpha}}{|\mu^{\alpha\beta}|}, \tag{30} \]

where the notations \(p^\alpha=k_\beta\varepsilon^{\beta\alpha}\), \(g^\alpha=\varepsilon^{\alpha\beta}k_\beta\), \(T^{00}=k_\sigma\varepsilon^{\sigma\nu}k_\nu\) have been introduced; \(|\mu^{\alpha\beta}|\) and \(|\varepsilon^{\alpha\beta}|\) are determinants composed of the components of the 3-tensors \(\mu^{\alpha\beta}\) and \(\varepsilon^{\alpha\beta}\), respectively.

For an isotropic medium the equality \(P^n a_n=0\) becomes

\[ \frac{\varepsilon}{\mu} \left( \varepsilon k_0^2-\frac{k_\alpha k^\alpha}{\mu} \right) \left(k^\alpha a_\alpha-\varepsilon\mu k_0a_0\right)=0, \]

which is satisfied under the condition

\[ k^\alpha a_\alpha-\varepsilon\mu k_0a_0=0, \]

or, taking into account that \(k_0=-\omega/c\), \(a_0=-\varphi\),

\[ \mathbf{k}\cdot\mathbf{a}-\frac{\omega\varepsilon\mu}{c}\varphi=0. \tag{31} \]

Since \(\mathbf{a}\) is the Fourier amplitude of the 3-vector potential of the electromagnetic field \(\mathbf{A}\), and \(\varphi\) is the Fourier amplitude of the scalar potential \(\Phi\), equality (31) obviously corresponds to the condition

\[ \operatorname{div}\mathbf{A}+\frac{\varepsilon\mu}{c}\frac{\partial\Phi}{\partial t}=0. \]

The solution (29) was first obtained by one of the authors by a more cumbersome route without formulating the gauge condition for the 4-potential [4]. Of course, the solution presented here, based on the use of the 4-tensor \(G_{ik}\), and the gauge condition could have been found only after carrying out the aforementioned cumbersome solution, since the method of constructing the 4-tensor \(G_{ik}\) is not evident in advance.

Ural Polytechnic Institute
named after S. M. Kirov

Received
16 V 1966

CITED LITERATURE

1. V. Pauli, Theory of Relativity, 1947.
2. I. Tamm, Zhurn. Russk. fiz.-khim. obshch., 56, issue 2, 3, 248 (1924).
3. L. D. Landau, E. M. Lifshitz, The Classical Theory of Fields, 1962.
4. A. S. Viglin, ZhETF, 50, issue 1, 85 (1966).