Abstract
Full Text
PHYSICS
Academician of the Academy of Sciences of the Armenian SSR A. G. IOSIF’YAN, K. P. STANYUKOVICH, G. A. SOKOLIK
ANALYSIS OF QUASI-MAXWELLIAN EQUATIONS DESCRIBING COMPENSATING FIELDS
1. In this paper we study general quasilinear equations for an arbitrary field, first introduced for the case of gravitation in the work of one of us \((^1)\), in the form
\[ \operatorname{rot} R = s v + \frac{\partial N}{\partial t}; \qquad \operatorname{div} N = \rho; \qquad L = \frac{4 \pi G}{c^2} R; \]
\[ \operatorname{rot} \Gamma = - \frac{\partial L}{\partial t}; \qquad \operatorname{div} L = 0; \qquad N = \frac{\Gamma}{4 \pi G}, \]
where the vectors \(R;\ \Gamma;\ L;\ N\) fully characterize the gravitational-inertial field; \(\rho\) is the density; \(v\) is the velocity of motion of the medium; \(G\) is Newton’s gravitational constant. Equations of this type are naturally derived from Noether’s theorem \((^{2-5})\), generalized to the case of localized groups of transformations of reference systems.
2. In this case we have:
\[ L_T = L(\psi, \nabla_\mu \psi) + L_0(A_\mu^a; A_{\mu,\nu}^a), \]
where \(L\) and \(L_0\) correspond to the dynamical and field parts of the Lagrangian, and \(\nabla_\mu = \partial_\mu - I_a A_\mu^a\), with
\[ \delta A_\mu^a = \varepsilon^b f_{bc}^a A_\mu^c + \partial_\mu \varepsilon^a . \tag{1} \]
The form of the covariant derivative \(\nabla_\mu\) and the transformation law of the compensating field \(A_\mu^a\) \((^{2-5})\) can be obtained from the condition of covariance of the wave equation for particles of arbitrary spin 3 with respect to the local gauge group: \(\delta \psi = \varepsilon^a(x) I_a \psi\), where
\[ [I_a I_b] = f_{ab}^c I_c, \qquad \varepsilon^a = \varepsilon^a(x). \]
Let us formulate Noether’s theorem for local gauge transformations.
Varying \(L_T\) under the condition that
\[ \frac{\partial L}{\partial \psi} - \nabla_\mu \frac{\partial L}{\partial \nabla_\mu \psi} = 0, \]
\[ \frac{\partial L_0}{\partial A_\mu^a} - \partial_\nu \frac{\partial L_0}{\partial A_{\mu,\nu}^a} = J_a^\mu, \tag{2} \]
where
\[ J_a^\mu = \frac{\partial L}{\partial \nabla_\mu \psi} I_a \psi, \qquad \frac{\partial L_0}{\partial A_\mu^a} + \frac{\partial L_0}{\partial A_{\nu,\mu}^b} f_{ac}^b A_\nu^c = 0, \]
and, moreover,
\[ \nabla_\mu \frac{\partial L}{\partial \nabla_\mu \psi} = \partial_\mu \frac{\partial L}{\partial \nabla_\mu \psi} + A_\mu^a \frac{\partial L}{\partial \nabla_\mu \psi} I_a, \]
we obtain from \(\delta L_T = 0\) the conservation law
\[ \partial_\mu \left(J_a^\mu + \overset{0}{J}{}_a^\mu\right) = 0, \qquad \overset{0}{J}{}_a^\mu = \frac{\partial L_0}{\partial F_{\mu\nu}^b} f_{ac}^b A_\nu^c, \tag{3} \]
where
\[ F_{\mu\nu}^a = \partial_\mu A_\nu^a - \partial_\nu A_\mu^a - \frac{1}{2} f_{bc}^a \left(A_\mu^b A_\nu^c - A_\nu^b A_\mu^c\right). \]
The formula for the field strength \(A_\mu^a\), through which
\[ L_0=f_{am}^{\,l} f_{bl}^{\,m} F_{\mu\nu}^{a} F^{b;\mu\nu}, \]
is expressed, follows from the generalized condition of gradient invariance \(L_0[F_{\mu\nu}^{a}]\), since \(\delta F_{\mu\nu}^{a}=\varepsilon^{b} f_{bc}^{a}F_{\mu\nu}^{c}\).
From the extremum condition (2) for \(A_\mu^a\) there follows the quasi-Maxwell system of equations
\[ f_{am}^{\,l}f_{bl}^{\,m}\partial_\nu F^{b;\mu\nu}=J_a^\mu+\overset{0}{J}{}_a^\mu . \tag{4} \]
In the particular case of an Abelian gauge group
\[ \psi'=e^{i\alpha(x)}\psi,\qquad \text{where } f_{bc}^{a}=0, \tag{4′} \]
the system reduces to Maxwell’s equations.
- Analyzing the compensating field \(A_\sigma^a\) by the method given in (4), one can show that in \(A_\sigma^a\) longitudinal and transverse parts are distinguished:
\[ A_\sigma^a=\overset{T}{A}{}_\sigma^a+\overset{L}{A}{}_\sigma^a, \]
so that \(F_{\mu\nu}^{a}(\overset{L}{A}{}_\sigma^a)=0\). In this case \(F_{\mu\nu}^{a}\) for the complete \(A_\sigma^a\) has the form:
\[ F_{\mu\nu}^{a}=\omega_\lambda^{a\tau}R_{\mu\nu;\tau}^{\cdots\lambda}, \qquad \omega_\lambda^{a\tau}=\langle \Omega^\tau \mid I^a \mid \Omega_\lambda\rangle, \]
where \(\Omega^\tau,\ \Omega_\lambda\) define a basis of the representation of the local group specified by the generators \(I_a\).
It follows from this that, in the general case, the quasi-Maxwell equation (4) reduces to the form:
\[ f_{am}^{\,l}f_{bl}^{\,m}\partial_\nu\bigl(\omega_\lambda^{b\tau}R^{\mu\nu\cdot\lambda}_{\cdot\cdot\tau}\bigr) = J_a^\mu+\overset{0}{J}{}_a^\mu . \tag{5} \]
- To analyze the physical meaning of the quasi-Maxwell equations, let us separate the spatial and temporal components of the fields. To this end we write:
\[ F_{\mu\nu}^{a}=-\xi_p H_p^a-i\eta_p E_p^a, \]
where \(\xi_p,\ \eta_p\) correspond to spatial and temporal infinitesimal rotations. Then
\[ H^a=\operatorname{rot}A^a-f_{bc}^{a}[A^b\vee A^c], \]
\[ E^a=-\operatorname{grad}\varphi^a-\frac{1}{c}\frac{\partial}{\partial t}A^a -\frac{1}{2}f_{bc}^{a}(\varphi^b A^c-A^b\varphi^c). \tag{6} \]
In this case the transformations of \(A_\sigma^a\) preserving (4) have the form
\[ \varphi^{a'}=\varphi^a-\frac{1}{c}\frac{\partial\varepsilon^a}{\partial t} +\varepsilon^b f_{bc}^{a}\varphi^c, \]
\[ A^{a'}=A^a+\operatorname{grad}\varepsilon^a+\varepsilon^b f_{bc}^{a}A^c, \]
where \(\{\varphi^a;A^a\}=A_\sigma^a\), with \(\varphi^a,A_\sigma^a\) being generalizations of the scalar and vector potentials, respectively.
Substituting (6) into (4), we obtain the three-dimensional notation for the field equations:
\[ \operatorname{rot}E^a+\frac{1}{c}\frac{\partial H^a}{\partial t} = f_{bc}^{a}\{[A^c\vee \operatorname{grad}\varphi^b]-\varphi^b\operatorname{rot}A^c\}, \tag{7} \]
\[ \operatorname{div}H^a = f_{bc}^{a}\{A^b\cdot H^c-A^c\cdot H^b\} + 4f_{cb}^{a}f_{b''b}^{c}A^b\cdot[A^{b'}\vee A^{b''}], \]
\[ \operatorname{div}E^a=s^a+\overset{0}{s}{}^a,\qquad \operatorname{rot}H^a-\frac{1}{c}\frac{\partial E^a}{\partial t} = G^a+\overset{0}{G}{}^a, \]
where \(G^{a}_{0}, G^{a}_{0}, S^{a}\) and \(S^{a}_{0}\) are the spatial and temporal components of the currents \(J^{\mu}_{a}\) and \(J^{\mu}_{a\,0}\). In the particular case of the local Lorentz group (3), equations (7) take the form of equations for the gravitation tensor in orthogonal components
\(R_{\mu\nu}(i,k)=\Omega^{\tau}(i)\Omega_{\lambda}(k)R_{\mu\nu\tau}^{\ \ \ \lambda}\); for this it is sufficient to substitute into (7) the structure of the Lorentz group (3):
\[ f^{ik}_{lm;i'k'}=g^{k}_{k'}\mu^{lm}_{ii'}+g^{i}_{i'}\mu^{lm}_{kk'}, \]
where
\[ \mu^{lm}_{ii'}=\frac{1}{2}\left(g^{l}_{i}g^{m}_{i'}-g^{l}_{i'}g^{m}_{i}\right). \]
Then
\[ H^{ik}=\operatorname{rot} A^{ik}-8\,[A^{ip}\vee A^{pk}], \tag{8} \]
\[ E^{ik}=-\operatorname{grad}\varphi^{ik}-\frac{1}{c}\frac{\partial A^{ik}}{\partial t} -4\,[\varphi^{ip}A^{pk}-A^{ip}\varphi^{pk}]. \]
Thus we have arrived at the “second pair” of quasi-Maxwell equations for the case of a gravitational field.
The first pair has the form:
\[ \operatorname{rot} H^{ik}-\frac{1}{c}\frac{\partial E^{ik}}{\partial t} =J^{ik}+J^{ik}_{0}, \]
\[ \operatorname{div} E^{ik}=\rho^{ik}+\rho^{ik}_{0}. \]
The equations obtained, close to those published in \((^{1})\), are of particular interest, since their quasi-Maxwell character is preserved also in a strong gravitational field, while the nonlinearity is completely contained in the current \(J_{0}\).
It is also interesting to note that in the case of an arbitrary field \(A^{a}_{\mu}\), (4) reduces to (5), i.e. to an equation for \(R_{\mu\nu}(i,k)\) close to (8).
Thus we arrive at a unified description of fields based on quasilinear equations of type (4).
In conclusion, let us note that the nonlinear term \(J^{\mu}_{a\,0}\), entering into the extended conservation law (3), can probably be used in describing gravitational waves, as a term corresponding to an essentially strong gravitational field.
The authors express their gratitude to N. P. Konopleva, whose conversations and ideas contributed to the writing of the present work.
Received
15 IV 1964
CITED LITERATURE
- A. G. Iosifyan, Questions of a Unified Theory of Electromagnetic and Gravitational-Inertial Fields, Yerevan, 1959.
- R. Utiyama, Phys. Rev., 101, 1957 (1956).
- G. A. Sokolik, DAN, 148, No. 3 (1963).
- G. A. Sokolik, N. P. Konopleva, DAN, 154, No. 2 (1964).
- A. M. Brodsky, D. D. Ivanenko, G. A. Sokolik, ZhETF, 41, 10 (1961).