Two Systems of Maxwell ’ s Equations and Two Corresponding Systems of Wave Equations in a Rotating Dielectric Medium

In this chapter, on the base of two basic systems of Maxwell ’ s equations for electromagnetic field vectors E ! and B ! in a uniformly rotating dielectric medium, the two corresponding systems of wave equations have been derived (to the first order in an angular velocity Ω ). From their comparative analysis, it can be seen that the structure of the wave equations for electromagnetic field vectors in the first system is asymmetrical with respect to Ω , while the structure of such equations in the second one is symmetrical. On this basis, it can be concluded that if the principle of symmetry is accepted as a criterion for selection, then second system of wave equations (and, therefore, corresponding the second set of Maxwell ’ s equations) for vectors E ! and B ! may be preferred.


Introduction
In order to develop the theory of fiber-optic gyro (or, e.g., ring laser gyro with resonator containing a dielectric medium with index of refraction n), one needs to have a system of Maxwell's equations and corresponding system of wave equations for electromagnetic field vectors E ! and B ! which are written in a frame of reference which uniformly rotates in an inertial frame with angular velocity Ω (Ω ¼ Ω ! ).
Since the module v of vector v ! ¼Ω ! Â r ! of a tangential velocity of such rotating device is much smaller than the speed of light, it is sufficient for these systems to be linear in v or, equivalently, in Ω. From the literature, it can be seen that there are mainly two basic systems of Maxwell's equations for electromagnetic field vectors E ! and B ! derived from the first principles and written for the case of a uniformly rotating dielectric medium. These two systems of equations are in good agreement with the experiments conducted for ring optical interferometers, fiber-optic gyros, and ring laser gyros containing in their arms the gas discharge tubes with Brewster's windows. Both systems are based on the Galilean description of rotation: x ¼ x 0 cos Ω t 0 þ y 0 sin Ω t 0 , y ¼ Àx 0 sin Ω t 0 þ y 0 cos Ω t 0 , z ¼ z 0 , t ¼ t 0 (superscript 0 0 0 refers to the inertial frame).
In the absence of free charges and currents, the first system, which was first obtained in work [1] from the formalism of the theory of general relativity, may be written in the form (we keep the terms only up to first order in Ω): and the second one, which was first obtained in work [2] on the base of the use of the tetrad method in this theory, may be presented as Systems of Maxwell's Eqs. (1)(2)(3)(4) and (5)(6)(7)(8) are written here in units of the SI.
In the above systems, all the quantities are specified by the formulas wherex,ŷ, andẑ are the unit vectors which form an orthogonal coordinate basiŝ ; n ¼ ε r μ r ð Þ 1=2 is the index of refraction of a dielectric medium; and ε r and μ r are the relative permittivity and permeability of a medium, respectively.
Remark about systems (1-4) and (5-8) ⊲ 1. According to the textbook [12], in a uniformly rotating frame of reference with spatial rectangular coordinates x, y, z and time coordinate t, the quadratic form ds 2 may be presented as the nonzero components of a space-time metric tensor (with determinant g ¼ À1) of such rotating frame may be written as g 00 ¼ 1 À v 2 =c 2 , g 11 ¼ g 22 ¼ g 33 ¼ À1, g 01 ¼ g 10 ¼ Àv x =c, g 02 ¼ g 20 ¼ Àv y =c, and g 03 ¼ g 30 ¼ Àv z =c. Then, components κ αβ ¼ Àg αβ þ g 0α g 0β =g 00 of a spatial metric From these formulas for κ αβ and κ, it follows that the spatial metric tensor of a rotating frame of reference, in a linear with respect to Ω approximation, has a diagonal form with nonzero elements κ 11 ¼ κ 22 ¼ κ 33 ¼ 1 and its determinant κ ¼ 1. Therefore, in such approximation, geometry of space in a rotating frame of reference remains Euclidean (flat space). So, spatial rectangular coordinates x, y, and z (or, e.g., cylindrical coordinates ρ, ϕ, z) of the given observation point in this frame have their usual sense, and the operator r are not identical: system (1-4) has asymmetrical structure with respect to Ω in a sense that rotation manifests itself only in the third and fourth equations but not in the first and second ones; system (5-8) has symmetrical structure with respect to Ω because rotation manifests itself in all four equations. The reason of such difference between these two systems is that they were obtained in works [1,2] with the help of two qualitatively different theoretical approaches.
In this situation, one may ask the following questions: (1) what will the form of the two corresponding systems of wave equations for the named vectors in first and second cases be? (2) Which system of such wave equations (first or second) is preferred? The answers to these questions are not given in the literature.
So, the task of this research is to derive the wave equations for vectors E ! and B ! at first on the base of system of Maxwell's Eqs. (1)(2)(3)(4) and then on the base of system (5)(6)(7)(8). The results obtained in both cases must be compared. All calculations must be performed with accuracy approximated to the first order in

Auxiliary relations
In this section, we are going to present some useful formulas for the quantities A. Consider the term r It is known (see, e.g., handbook [13]) that C. Consider the identity r E. Consider the vector r 2 There is the formula Let us first calculate the projection r 2 of this vector onto the axisx.
Taking into account (9), we have or After the calculation, we get Let us add to the right-hand side of (17) the following terms: or Similarly, we may obtain Therefore Finally, taking into account that, to the first order in Ω, Ω Formulas (10)-(13) and (23) will be used in the next sections.

Equation for vector E
Taking into account (10) and (3), we rewrite (24) in the form or, using (1), With the help of (11), we get Taking into account (12) and (3), we have r Consider the last term in (28). In accordance with (13),

Conclusion
In this chapter, on the base of two basic systems of Maxwell's Eqs. (1)(2)(3)(4) and From their comparative analysis, it can be seen that the structure of the wave equations for electromagnetic field vectors in system (67, 68) is asymmetrical with respect to Ω, while the structure of such equations for these vectors in system (69, 70) is symmetrical.
On this basis, it can be concluded that if the principle of symmetry is accepted as a criterion for selection, then system of wave Eqs.