Differential Electromagnetic Forms in Rotating Frames

Differential forms are completely antisymmetric homogeneous r-tensors on a differentiable n-manifold 0 ≤ r ≤ n belonging to the Grasssman algebra [1] and endowed by Cartan [2] with an exterior calculus. These differential forms found an immediate application in geometry and mechanics; introduced by Deschamps [3,4] in electromagnetism, they have known in parallel with the expansion of computers, an increasing interest [5-9] because Maxwell’s equations and the constitutive relations are put in a manifestly independent coordinate form. In the Newton (3+1) space-time, with the euclidean metric ds


Introduction
Differential forms are completely antisymmetric homogeneous r-tensors on a differentiable n-manifold 0 ≤ r ≤ n belonging to the Grasssman algebra [1] and endowed by Cartan [2] with an exterior calculus. These differential forms found an immediate application in geometry and mechanics; introduced by Deschamps [3,4] in electromagnetism, they have known in parallel with the expansion of computers, an increasing interest [5][6][7][8][9] because Maxwell's equations and the constitutive relations are put in a manifestly independent coordinate form. In the Newton (3+1) space-time, with the euclidean metric ds 2 = dx 2 + dy 2 + dz 2  and, they also have the differential form representation (∂ τ =1/c∂ t ) [5,7] d ∧ E +∂ τ B = 0, d ∧ B = 0 (2a) d ∧ H − ∂ τ D = 0, d ∧ D = 0 (2b) d = dx∂ x + dy∂ y +dz∂ z is the exterior derivative, E , H the differential 1-forms E = E x dx + E y dy + E z dz, H = H x dx + H y dy + H z dz (3a) and B , D the differential 2-forms B = B x (dy∧dz) + B y (dz∧dx) +B z (dx∧dy) , D = −[ D x (dy∧dz) + D y (dz∧dx) +D z (dx∧dy)] (3b) We are interested here, for reasons to be discussed in Sec. (6) in a Frenet-Serret frame rotating around oz with a constant angular velocity requiring a relativistic processing, We shall prove that this situation leads to an Einstein space-time with a riemannian metric. As an introduction to this problem, we give a succcinct presentation of differential electromagnetic forms in a Minkowski space-time with the metric ds 2 = dx 2 + dy 2 + dz 2 −c −2 ∂ t 2 .

Differential forms in Minkowski space-time [7]
In absence of charge and current, the Maxwell equations have the tensor representaation [10,11] the greek (resp.latin) indices take the values 1,2,3,4 (resp.1,2,3) with x 1 = x, x 2 = y, x 3 = z, x 4 = ct, ∂ j = ∂/∂x j , ∂ 4 = 1/c∂/∂ t and the summation convention is used. The components of the tensors F μν and F μν are with the 3D Levi-Civita tensor ε ijk and in vacuum Let d be the exterior derivative operator d = (∂ x dx +∂ y dy +∂ z dz +∂ t dt )∧ (6) and F =E + B be the two-form in which : Then the Maxwell equations (4a) have the differential 3-form representation d F = 0. Similarly for G =D + H with : the differential 3-form representation of Maxwell's equations (4b) is d G = 0. To manage the constituive relations (5a) the Hodge star operator [6,9] is introduced Applying the Hodge star operator to F gives *F = *E + *B and one checks easily the relation G = λ 0 *F with λ 0 =(ε 0 /μ 0 ) 1/2 so that the Maxwell equations in the Minkowski vacuum, have the diffrerential 3-form representation

Electromagnetidsm in a Frenet-Serret rotating frame
We consider a frame rotating with a constant angular velocity Ω around oz. Then, using the Trocheris-Takeno relativistic description of rotation [12,13], the relations between the www.intechopen.com Differential Electromagnetic Forms in Rotating Frames 57 cylindrical coordinates R,Φ,Z,T and r,φ,z,t in the natural (fixed) and rotating frames are with ß = ΩR /c R= r, Φ = φ coshβ − ct/r sinhβ Z = z , cT = ct coshβ − rφ sinhβ (11) and a simple calculation gives the metric ds 2 in the rotating frame A = ß sinhß ct/r + ß coshß φ + sinhß φ, B = ß sinhß φ +ß coshß ct/r − sinhß ct/r (12a) Using the notations x 4 = ct, x 3 = z, x 2 = φ, x 1 = r, we get from (12) ds 2 = g μν dx μ dx ν with g 44 = 1, g 33 = −1, The determinant g of g μν is g = g 33 [ g 11 g 22 g 44 − g 12 2 g 44 − g 14 2 g 22 ] = R [g 11 − g 12 2 r −2 − g 14 2 ] (14) but g 12 2 r −2 + g 14 2 and, taking into account the expression (13) of g 11 , we get finally So, the rotating Frenet-Serret frame defines an Einstein space-time with the riemannian metric ds 2 = g μν dx μ dx ν , and in this Einstein space-time the Maxwell equations have the tensor representation [14,15] in which, using the cylindrical coordinates r,φ.z ,t with x 1 = r, the components of the electromagnetic tensors are To work with the differential forms, we introduce the exterior derivative (underlined expressions mean that they are defined with the cylindrical coordinates r, φ,z,t) and the two-forms F =E + B with and writing |g| 1 Then, the Maxwell equatios have the 3-form representation A simple calculation gives The Hodge star operator needed to take into account the constitutive relations (5a) in vacuum is defined by the relation Applying (22) to F gives *F = *E + *B and it is easily checked that G = λ 0 *F with λ 0 =(ε 0 /μ 0 ) 1/2 so that in vacuum dF = 0 , d *F = 0.

Wave equations in vacuum 4.1 Minkowski space-time
The wave equations satisfied by the electromagnetic field (in absence of charges and currents) are obtained from differential forms with the help of the Laplace-De Rham operator [6,8] www.intechopen.com Differential Electromagnetic Forms in Rotating Frames 59 requiring the Hodge star operators for the n-forms, n = 1,2,3. They are given in Appendix A where, using the exterior derivative (6), and assuming E x = E y = 0 so that the two-form (7) becomes E z = E z (dz∧cdt), we get A similar relation exists for E x, E y and for the components of the B-field so that, we get finally for the 2-form F: so that the wave equation has the 2-form representation L F = 0 .
Now, it is proved [6] that the Laplace-De Rham operator L = d*d* + *d*d applied to the two form (7) written in which dx 1 = dx, dx 2 = dy, dx 3 = dz, dx 4 = cdt gives for the component F i4 L E = 0 gives the 2-form representation of the wave equation in the Einstein space-time with cartesian coordinates A similar result is obtained for B writing −1/2F ij (dx i ∧dx j ) the B magnetic two-form (7) so that Summing (30) and (30a) gives In the Minkowski cartesian frame where g ij = δ ij , g 44 = −1, Eq.(31) reduces to (25).

Frenet-Serret frame
In the Frenet-Serret frame, the Laplace-De Rham operator is defined with the exterior derivative operator (18) and to get a relation such as (30) on the components of the electric field requires some care. First with the greek indices associated to the polar coordinates as previously, one has first to get the Christoffel symbols needed to define the covariant derivative and according to (28), the Riemann curvature and Ricci tensors, a job performed in Appendix B, we are now in position to transpose (30) to a rotating cyindrical frame. To this end, the electric two-form (19a) with is written but E r , rE φ , E z are the G i4 components of the G μν tensor (17) and we get so that the components of the electric field are solutions of the two-form equation L E = 0 in the Frenet-Serret rotating frame. For the other components of the electromagnetic field, it comes We have only considered the two-form F because in vacuum G =λ 0 * F .

To solve differential form equations
The local 2-form representation (2) of Maxwell's equations follows, as a consequence of the Stokes's theorem, from the Maxwell-Ampère and Maxwell-Faraday integral relations. Then, coming back to these theorems, to solve differential form equations is tantamount to perform the integrals in which ω is a n-form, for instance F ou G, and M an oriented manifold with the same ndimension as the degree of the ω form [5].
In the 3D-space, the numerical evaluation of (37) is based on the finite element technique, largely used [17,18] in the simulation of partial differential equations. The manifold M is described by a chain of simplexes made for instance of triangular surfaces, tetrahedral vo-lumes… on which the Whitney forms [5,19,20] gives a manageable description of the nform ω. A simple example may be found in [19] and a through discussion of the technique in [20]. These solutions may be called weak in opposition to the strong solutions of the Maxwell's equations (1).
The numerical process just described is limited to the 3D-space but in the 4D space-time, in particular for the Frenet-Serret frame, ω depends on dt so that M has to be defined in terms of 2-cells, 3-cells, 4-cells of space-time [21] and the Whitney forms must be generalized accordingly. It does not seem that computational works have been made in this domain.

Discussion
Differential electromagnetic forms are usually managed in a Newton space and more rarely in a Minkowski space-time although, in this case, the comparison between tensors and differential forms is very enlightning [7]. This formalism is analyzed here in an Einstein spacetime with a Riemann metric, particularly that of a Frenet-Serret frame. From a theoretical point of view, except for some more intricate relations due to Riemann, Ricci tensors and Christoffel symbols there is no difficulty to go from Newton to Einstein differential forms.. The situation is different from a computational point of view, since as mentionned in Sec.5, an important worrk has still to be performed to get the solutions of the differential form equation in an Einstein space-time. This work may be considered as a first step in a complete analysis of electromagnetic differential forms in an Einstein space-time. The subjects to be discussed go from the presence of charges and currents (left aside here) to boundary conditions with between the introduction of potentials, the energy conveyed by the electromagnetic field and so on. This extension could be performed in the syle used in [7] to analyze the electromagnetic differential forms in a Minkowski space-time. In addition, it would make possible an interesting comparison (al-ready sketched in Sec.4.1) with the electromagnetic tensor formalism of the General Relati-vity [15]. Now, why to take an interest in rotating frames? A first response could have been ''Universe'' assumed cylindrical. But, although Einstein and Romer (also Levi Civita) have obtained some exact cylindrical wave solutions of the general relativity equations [22], this cosmos has been superseded by a spherical world (nevertheless, because of its particular properties, some works are still devoted to the Levi Civita world [23]. A second response comes from the analysis of the Wilsons' experiments in which was measured the electric potential between the inner and outer surfaces of a cylinder rotating in an external axially directed magnetic field: an analysis with many different approaches [6,23,24,25] (the Trocheris-Takeno des-cription of rotations is used in [25]). Finally, a third response is provided by the increasing at-tention paid to paraxial optical beams with an helicoidal geometrical structure [26], [27] lea-ding to a discussion of light propagation in rotating media: a problem object of some dispu-tes [28][29][30][31][32].The relativistic theory of geometrical optics [15] is still a challenge to which it would be interesting to see what could be the differential form contribution.