Electromagnetic field in rotating frame of reference

Accepted Manuscript Electromagnetic field in rotating frame of reference Z.J. Zawistowski, Ya. Kovivchak PII: DOI: Reference:

S0165-2125(16)30159-7 http://dx.doi.org/10.1016/j.wavemoti.2016.12.001 WAMOT 2130

To appear in:

Wave Motion

Received date: 15 July 2016 Revised date: 7 December 2016 Accepted date: 8 December 2016 Please cite this article as: Z.J. Zawistowski, Y. Kovivchak, Electromagnetic field in rotating frame of reference, Wave Motion (2016), http://dx.doi.org/10.1016/j.wavemoti.2016.12.001 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.

Electromagnetic Field in Rotating Frame of Reference Z.J. Zawistowskia,∗, Ya. Kovivchakb a

Faculty of Applied Informatics and Mathematics, Warsaw University of Life Sciences SGGW, 159 Nowoursynowska Str., 02-776 Warsaw, Poland b Lviv Politechnic National University, 12/110 Bandera Str., Lviv-13, 79013, Ukraine

Abstract The problem of transformation of electromagnetic field from inertial frame of reference to non-inertial rotating frame of reference and vice versa is discussed. By the use of the tetrad method (moving reper method) it is shown that in cylindrical coordinates the usual Lorentz formulas can be applied. It is also shown that in cylindrical coordinates the form of Maxwell equations in rotating frame of reference is is the same as in inertial one. Keywords: electromagnetic field, non-inertial frame of reference, Maxwell equation, transformation to non-inertial frame of reference 1. Introduction The need for finding electromagnetic field in rotating frame of reference appears in designing of electrical motors and generators and in theoretical models in astrophysics. Field methods give the most exact results in designing process of electrical machines. Thus, precise knowledge of distribution of electromagnetic field in the machine is very important. Because the rotator forms a non-inertial frame of reference the serious problem of finding electromagnetic field in this region appears. There are two ways of solving this problem. First, proper transformation formulas from inertial to noninertial frame can be used. Second, a proper form of Maxwell equations in non-inertial rotating frame of reference can be used. These two ways are addressed in this paper. Corresponding author Email addresses: zygmunt [email protected], [email protected] ∗

Preprint submitted to Wave Motion

December 9, 2016

Figure 1: Lorentz boosts.

In the engineering literature the above problem is often unnoticed or ignored. The Lorentz formulas are frequently used without any comments. Other authors have ignored the problem invoking non-relativistic approximation. Rotation in electrical machines is so slow that non-relativistic approximation (neglecting terms proportional to (v/c)2 ) is justified. However for electromagnetic phenomena the situation is not so simple as for mechanical variables. Oversimplified analysis could lead to wrong conclusions such as elimination of electromagnetic induction phenomenon. The problem of the proper form of Maxwell equations is completely unnoticed in engineering literature. Below the Lorentz formulas for field transformation between inertial frames of references and the explicit relativistic form of Maxwell equations are reminded. In Sec. 2 formulas for transformation of electromagnetic field in the case of transformation from inertial to rotating non-inertial frame of reference are obtained. Maxwell equations in the rotating frame are obtained in Sec. 3. For transformations between inertial frames of reference O and O’ (Lorentz boosts, Fig.1) the well known ([1]–[4]) Lorentz formulas hold. Coordinates transform as follows  v  0 0 0 0 t = γ t − 2 x , x = γ(x − vt), y = y, z = z c where

v β= , c

1 γ=p . 1 − β2 2

In relativistic notation x0 = ct, x1 = x, x2 = y, x3 = z these transformations have the form 0

x µ = Λµ ν xν where



γ −γβ  −γβ γ [Λµ ν ] =   0 0 0 0

(1) 0 0 1 0

 0 0  . 0  1

(2)

For general orientation of axes of frames O and O 0 the formula (1) changes according to 0 Λ → L = R ΛR where R is the spatial rotation from actual position of the axis x to the 0 direction of motion and R is the spatial rotation from the direction of motion 0 to the direction of axis x . However we always can use the formula (1) by a proper choice of the orientation of axes. The parallel and orthogonal with respect to the direction of motion components of electric and magnetic fields transform as follow ([1]–[3]) 0

Ek = Ek , 0

Bk = Bk , 0

Dk = Dk , 0

Hk = Hk ,

0

E⊥ = γ(E + v×B)⊥ 1 0 B⊥ = γ(B − 2 v×E)⊥ c 1 0 D⊥ = γ(D + 2 v×H)⊥ c 0 H⊥ = γ(H − v×D)⊥ .

(3) (4) (5) (6)

Apparent asymmetry in the above formulas, 1/c2 in denominators only in equations (4) and (5), stems from using SI system of units. In Gauss system of units (cgs), for example, in all equations for orthogonal components transformations the same coefficient 1/c appears. In non-relativistic approximation, when we omit terms quadratic in β , coordinate transformations take the form of Galileo transformations  v  0 0 0 0 t = t − 2 x ≈ t, x = (x − vt), y = y, z = z c 3

0

where the time approximation t ≈ t follows from c2 in denominator. For relativistic coordinate the term linear in β should remain  v  0 x 0 = x0 − x1 . c

The field transformations (3)–(5) approximate to 0

0

B =B−

E = E + v×B, 1 D = D + 2 v×H, c 0

1 v×E c2

0

(7)

H = H − v×D,

because (v × A)k = 0 for any vector A. We omit arguments of the fields in the equations (3)–(6) and (7). These arguments describe the same physical 0 0 point in a space-time: E (x ), E(x) and so on. The simplest way of obtaining formulas (3)–(6) is to use the explicit relativistic formulation of Maxwell theory. We introduce the 4-vectors of current and potential J µ = (cρ, J),

Aµ = (φ/c, A)

(8)

where ρ is the charge density, J the current vector, φ and A are scalar and vector potential respectively B = ∇×A,

E = −∇φ −

∂A ∂t

(9)

The electromagnetic field is described by the following tensors ([1]–[3])   0 Ex /c Ey /c Ez /c  −Ex /c 0 Bz −By   [F µν ] =  (10)  −Ey /c −Bz 0 Bx  −Ez /c By −Bx 0 and



0  −D x /c [Gµν ] =   −Dy /c −Dz /c

Dx /c 0 −Hz Hy

4

Dy /c Hz 0 −Hx

 Dz /c −Hy  . Hx  0

(11)

In this notation the Maxwell equations ∇· B = 0,

∇· D = ρ, ∂D ∇×H = + J, ∂t B = µH

∂B , ∂t D = E,

∇×E = −

(12)

take the form ∂F µν ∂F νρ ∂F ρµ + + = 0, ∂xρ ∂xµ ∂xν

∂Gµν = J µ. ∂xν

(13)

From the first equation (13) follows that the tensor Fµν can be expressed by 4-potential as ∂Aµ ∂Aν − . Fµν = ∂xν ∂xµ The tensors F µν and Gµν transform in usual way ( [1]–[8]) F and

0 µν

0

0

(x ) = Λµ ρ Λν σ F ρσ (x) 0

G µν (x ) = Λρ µ Λσ ν Gρσ (x). Thus formulas (3)–(6) follow directly from the recipes (10) and (11) ([1]).The above explicit relativistic formulation of the Maxwell theory will be useful in further analysis. 2. Transformation to Non-inertial Rotating Frame For inertial frames of reference the above formulas of Special Theory of Relativity (STR) are sufficient. In the case of non-inertial frame of references we have to use the more general covering theory: General Theory of Relativity (GRT) [3]–[7]. This theory is based on experimental observation of local equivalence of gravity and an acceleration of frame of reference, so this theory gives mathematical tools for description of non-inertial movements of frame of reference. The time-space is modeled as differentiable manifold which is locally flat so we can locally use the above formulas of STR. We use this concept in the following analysis of rotation about fixed axis. Such a rotation is the most important case of non-inertial frame in technological applications (electric motors, generators). 5

Figure 2: Tangent inertial frames of references – tetrads.

Due to the symmetry of the problem the cylindrical coordinates (r, ϕ, z) with z-coordinate along an axis of rotation are most suitable. We relate the inertial frames of reference O 0 to any rotating point (Fig.2). Frames O 0 are comoving with these points and move with constant velocities v = ω × r where the angular velocity ω is directed in positive direction of zcoordinate. We can now perform the family of transformations from the inertial frame of reference to the above family of tangent inertial frames O 0 using formulas (6)–(7). In this way we represent electromagnetic field in each point using the orthonormal bases depicted in Fig.2. This family of local bases (tetrads) coincides with the family of orthonormal bases of rotating cylindrical coordinates (t0 , r0 , ϕ0 , z 0 ) t0 = t,

r0 = r

, ϕ0 = ϕ − ωt,

z 0 = z.

(14)

So in rotating cylindrical coordinates (t0 , r0 , ϕ0 , z 0 ) we can use formulas (3)– (6). For any ω the limit r → ∞ leads to v = ωr → ∞ and thus v > c which is impossible. Therefore we have to assume that the region of nonvanishing electromagnetic field is finite. In practical applications in electrical engineering (electric motors and turbo generators) the following estimation holds: rmax ∼ 1m ω ∼ 314s−1 (50Hz). Thus

β=

v ∼ 10−6 c 6

(15)

so omitting the terms β 2 ∼ 10−12 is justified. However linear in β terms in (6)–(7) should remain because they describe the electromagnetic induction phenomenon. In astrophysical application the problem is much more serious as value of a radius r can be very big. The above procedure is the special example of the general method of moving reper (tetrad) [10] – [13]. In our case the local coordinates generated by local bases in each point are integrable to the global rotating cylindrical coordinates. The inverse transformations are simply obtained by the change of sign of the vector v in formulas (3)–(6). However a new fundamental problem appears: how to find electromagnetic field in a rotating frame – that is how Maxwell equations look in such a frame. We address this problem in the next section. 3. Maxwell Equations in Rotating Frame Because the rotating frame of reference is non-inertial we once again refer to the General Theory of Relativity. The recipe is general and simple (c.f. [3]): , → ; which means that partial derivatives in Maxwell equations should be replaced by covariant derivatives. The covariant derivative of a vector field is defined as follows ∂Aµ DAµ µ µ µ ρ ≡ A = A + Γ + Γµνρ Aρ . A ≡ ;ν ,ν νρ dxν ∂xν

(16)

where Christoffel’s symbols are given by derivatives of the metric tensor gµν   1 µσ ∂gσρ ∂gσν ∂gνρ µ Γνρ = g + − = g µσ Γσνρ . (17) ν ρ σ 2 ∂x ∂x ∂x For our own purposes there is no need to use advanced notions of differential geometry like affine connection or parallel displacement. To explain the meaning of the covariant derivative it is enough to stress that for curvilinear coordinates like cylindrical ones the basis is not constant but depends on values of coordinates – vectors of the basis change with changing of a point. Thus, differentiating a vector field A = Aρ (x)eρ (x) we have to differentiate also basis vectors ∂Aρ ∂eρ ∂A = eρ (x) + Aρ ν . ν ν ∂x ∂x ∂x 7

Then expanding derivatives of basis vectors as linear combination of these basis vectors eα (x) ∂eρ = Γµνρ eµ ν ∂x we get the formula (17). Repeating this procedure for tensors of second order T µν eµ ⊗ eν for each index µ, ν we have T µν ;ρ = T µν ,ρ + Γµηρ T ην + Γνηρ T µη

(18)

and so on for other types of tensors. We want to compare the form of maxwell equations in a non-inertial frame of reference with their form in an inertial frame of reference. To this end we first find the explicit form of Maxwell equations in an inertial frame of reference. In the inertial frame of reference O the metric tensor gµν in cylindrical coordinates is obtained from the linear element (interval) ds2 = gµν dxµ dxν = c2 dt2 − dx2 − dy 2 − dz 2 = c2 dt2 − dr2 − r2 dϕ2 − dz 2 (19) as



c2 0  0 −1 [gµν ] =   0 0 0 0

 0 0 0 0   ≡ diag(c2 , −1, −r2 , −1). 2 −r 0  0 −1

(20)

The inverse g µν tensor has the form

[g µν ] = diag(c−2 , −1, −r−2 , −1).

(21)

From (17), (20), and (21) we find the only non-vanishing Cristoffel’s symbols Γrϕϕ = −r,

Γϕrϕ = Γϕϕr = r−1 .

(22)

There is no summation over repeating indexes (t, r, ϕ, z) and (t0 , r0 , ϕ0 , z 0 ) which are fixed cylindrical coordinates. Using (18) we find the 4-divergence of the electromagnetic field tensor appearing in the second of Maxwell equations (13) (23) Gµν ;ν = Gµν ,ν + Γµην Gην + Γνην Gµη = Gµν ,ν + Γνην Gµη because the tensor Gµν is antisymmetric and Christoffel’s symbols are symmetric in the lower indices. Using (22) we have Γνην Gµη = Γϕrϕ Gµr = r−1 Gµr 8

(24)

Figure 3: Difference between coordinate (left) and normal (right) bases in cylindrical coordinates.

so in the inertial frame O in the cylindrical coordinates the divergence of electromagnetic tensor Gµν in coordinate basis has the form Gµν ;ν = Gµν ,ν + r−1 Gµr = ∂t Gµt + ∂r Gµr + r−1 Gµr + ∂ϕ Gµϕ + ∂z Gµz = ∂t Gµt + r−1 ∂r (rGµr ) + ∂ϕ Gµϕ + ∂z Gµz .

(25)

The coordinate basis is formed by the vectors tangent to coordinate lines obtained by differentiating with respect to coordinates. In general such a basis is neither normalized nor orthogonal. For cylindrical coordinates the difference between these bases and the orthonormal ones is presented in Fig.3. To obtain normalized tangent vectors we have to differentiate with respect to arc length. The only difference between the coordinate basis (er , eϕ , ez ) ˆϕ , e ˆz ) concerns only the vectors eϕ and e ˆϕ : and the normalized basis (ˆ er , e ∂ϕ → ∂(rϕ) = r−1 ∂ϕ

=⇒

ˆϕ = r−1 eϕ . e

Thus (25) takes the form of well known expression for divergence of electromagnetic tensor Gµν in the orthonormal basis in cylindrical coordinates Gµν ;ν = ∂t Gµt + r−1 ∂r (rGµr ) + r−1 ∂ϕ Gµϕ + ∂z Gµz .

(26)

This result is not surprising as the space-time in O is flat and the theory comes down to Special Theory of Relativity. In this case Christoffel’s symbols 9

play the role of Lame coefficients of cylindrical coordinates. In this way we obtained the explicit form of the second Maxwell equations (13) in cylindrical coordinates in inertial frame of reference. The first sourceless equation in (13) does not change because Fµν = Aµ;ν − Aν;µ = Aµ,ν − Aν,µ . due to symmetry of Christoffel’s symbols in the lower indices Γµνρ = Γµρν In non-inertial rotating frame of reference O 0 defined by (14) the linear element has the form ds2 = gµν dxµ dxν = c2 dt2 − dr2 − r2 dϕ2 − dz 2 = c2 dt0 2 − dr0 2 − r0 2 (dϕ0 + ωdt0 )2 − dz 0 2 = (c2 − ω 2 r0 2 )dt0 2 − dr0 2 − 2ωr0 2 dt0 dϕ0 − r0 2 dϕ0 2 − dz 0 2 so the metric tensor is non-diagonal this time  2 (c − ω 2 r0 2 ) 0 −ωr0 2 0  0 −1 0 0 [gµν ] =   −ωr0 2 0 −r0 2 0 0 0 0 −1 and so the inverse tensor  c−2 0  0 −1 [g µν ] =   −ωc−2 0 0 0

   

 −ωc−2 0 0 0  . −2 0 −2 2 2 02 −c r (c − ω r ) 0  0 −1

(27)

(28)

(29)

Non-vanishing Christoffel’s symbols have now the form 0

0

Γrt0 t0 = −ω 2 r0 0 ω Γϕt0 r0 = 0 r

0

Γrt0 ϕ0 = −ωr0 0 1 Γϕr0 ϕ0 = 0 r

Γrϕ0 ϕ0 = −r0

so (no summation over fixed cylindrical coordinates t0 and ϕ0 ) 0

Γνην = Γϕr0 ϕ0 =

1 r0

(30)

and the expression for Γνην in divergence of the tensor Gµν 0

0

0

0

0

Γνη0 ν 0 = Γtη0 t0 + Γrη0 r0 + Γϕη0 ϕ0 + Γzη0 z0 , 10

(31)

where on the right side there is no summation over repeated indices, reduces to 0 0 Γνη0 ν 0 = Γϕη0 ϕ0 = r−1 . (32) The same result may be obtained in simpler way from the formula ([3],[4], [6], [7]) 1 ∂g 0 0 (33) Γνη0 ν 0 = 0 η0 . 2g ∂x where g 0 is the determinant of the metric tensor (28) g 0 = det(gµ‘ν‘ ) = −c2 r02 .

(34)

Using (30) in (23) we obtain the following formula for divergence of Γνην in coordinate basis of rotating cylindrical coordinates (t0 , r0 , ϕ0 , z 0 ) 0 0

Gµ ν

;ν 0

0 0

0 0

0

0

0 0

= ∂t0 Gµ t + r0−1 ∂r0 (rGµ r ) + ∂ϕ0 Gµ ϕ + ∂z0 Gµ z .

(35)

which is exactly of the same form as (25) in the case of an inertial nonrotating frame. We can find that formula in an easier and more direct way from ([3],[4],[7]) √ 1 ∂( −gGµ‘ν‘ ) µ0 ν‘ . (36) G ;ν = √ −g ∂xν which follows from (33) and (23). Of course in the orthonormal basis of cylindrical coordinates we obtain the same formula as (26). The first Maxwell equation in (13) does not change and from (35) and (25) we see that the form of the second equation in (13) is also the same, therefore, we proved that in cylindrical coordinates Maxwell equations have the same form (with partial derivatives) in inertial and non-inertial rotating frame of references. The form of Maxwell equations is also the same in both frames if we use the Cartesian coordinates (dx0 = cdt, x1 = x, x2 = y, x3 = z). The nonvanishing components of the metric tensor read g00 = 1 − ω 2 (x2 + y 2 )/c2 , g11 = g22 = g33 = −1.

g01 = g10 = yω/c,

g02 = g20 = −xω/c,

(37)

√ The determinant of this tensor is equal to g = −1, so −g = 1 and the covariant 4-divergence in (36) comes down to the usual coordinate divergence which means that the form of Maxwell equations does not change. Using (10), (11) we can obtain corresponding equations for vector fields E, D, H, B. 11

4. Discussion and Conclusions We have proved that in cylindrical coordinates in rotating frame of reference we can use Maxwell equations in traditional form, the same as in inertial frame, as well as the usual Lorentz formulas for transforming electromagnetic field from inertial to rotating frame of references and vice versa. The form of Maxwell equations does not change also if we use the Cartesian coordinates. Of course we have to fulfill the condition v = ωr < c so the region of non-vanishing electromagnetic field must be finite. This condition is trivially fulfilled in technical applications in which, in fact, the non-relativistic approximation holds (15). In astrophysics applications the problem is much more difficult. There is no rigid body in relativity theory and so there is no rigid rotation (14). In physically realizable rotation angular velocity should depend on radius: ω(r) → 0 for r → ∞. Asymptotically we have the limitation ω(r) ∼ c/r so ω(r)r → c at infinity and ω(r) → ω(0) ≡ ω for r → 0. It is an interesting question whether the maximally rigid rotation exist in such a class of admitted rotations v = ωr < c. By the maximally rigid rotation we mean the one for which dependence ω(r) on r follows only from kinematic limitations (dynamics and so forces can be ideal). This problem will be addressed in a future paper. It is interesting to consider more general form of accelerated motion of the observer and look for possible effects of the so called Mashhoon spinrotation coupling. Especially on quantum level for photons as in the papers [14]–[21] and citation therein. Comparing our result with that of the Hehl and Ni paper [15] it should be stressed that we consider much more simple movement of frame of reference and above all the differential operator in Maxwell equations is much simpler than that in Dirac equation where Dirac gamma matrices appear. This is the reason for absence of Mashhoon spinrotation term in Maxwell equations in our case of pure rotation. However, such a term appears in Dirac equation even in our case of pure rotation. In Cartesian coordinates from metrics (37) follows the Christoffel symbols Γ102 = Γ120 = ω/c,

Γ201 = Γ210 = −ω/c,

which through the formula (12) of [15] give iω iω i i − (σ 12 Γ120 + σ 21 Γ210 ) = − (σ 12 − σ 21 ) = − σ 12 = − ω · S 4 4c 2c }c where } ω = ωez , S = (σ 23 , σ 31 , σ 12 ) 2 12

(38)

Thus, spin-rotation term (38) appears in the hamiltonian of Dirac equation in the case of pure rotation about fixed axis. The possible way to analyze the problem of Mashhoon term in Maxwell equations in the case of general accelerated movement may be based on transformation of Maxwell equations into Schr¨odinger-like (Dirac-like) form with a hamiltonian containing expressions with the spin-like operator as, for example, Kramers form [16], [22], Majorana form [23], Moses form [24], and then examination of behaviour of these terms in different noninertial frames of reference. However, such considerations are beyond the scope of our paper which is restricted to pure uniform rotation about fixed axis and classical electrodynamics so we leave them for the future paper. 5. References [1] J. Jackson, Classical Electrodynamics, Wiley and Sons, New York, 1975. [2] D. Griffiths, Introduction to Electrodynamics, Prentice Hall, Upper Saddle River, New Jersey, 1981. [3] J. Foster, J. Nightingale, A Short Course in General Relativity, Longman, London and NY, 1979. [4] L.D. Landau, E.M. Lifshic, Field Theory (in Russian), Fizmatlit, Moscow, 2001. [5] J.B. Hartle, Gravity. An Introduction to Einstein’s General Relativity, Addison Wesley, San Francisco, 2003. [6] B. Shutz, A First Course in General Relativity, Cambridge University Press, Cambridge, 1985. [7] C.W. Misner, K.S. Thorn, J.A. Wheeler, Gravitation, W.H. Freeman and Company, NY, 1997. [8] J. Schouten, Ricci Calculus, Springer, Berlin-Heidelberg, 1954. [9] C. Pellegrini, J. Pleba´ nski, Tetrad fields and gravitational fields, J. Mat. Fys. Skr. Vid. 2, No 4(1963) 1–38. [10] O. Ivanitskaja, Generalized Lorentz Transformations and their Applications (in Russian), Izd. Nauka i Tekhnika, Minsk, 1969. 13

[11] O. Ivanitskaja, Lorentz Basis and Gravitational Effects in Einstein Theory of Gravitation (in Russian), Izd. Nauka i Tekhnika, Minsk, 1974. [12] C. Moller, On the crisis in the theory of gravitation and a possible solution, Danske Vidensk. Selsk., Mat-fys. Meddr 39, No 13 (1978) 1– 31. [13] J.J. Slawianowski, GL(n,R) as a candidate for fundamental symmetry in field theory, Nuovo Cimento 106 B (1991) 645–668. [14] B. Mashhoon, Phys. Rev. Lett. 61, 2639, (1988). [15] F.W. Hehl and Wei-Tou Ni, Phys. Rev. D42, 2045,(1990). [16] J.C. Hauck and B. Mashhoon, Ann. Phys. 12, 275, (2003). [17] B. Mashhoon, Ann. Phys. 17, 705, (2008). [18] B. Mashhoon, Acta Phys. Polonica B, Proceedins Supplement, 1,113, (2008). [19] B. Mashhoon, Phys. Rev. A79: 062111. [20] P.J. Mohr, https://archiv.org/pdf/0910.1874v4.pdf [21] U.D. Jentschura and J.H. Noble, J. Phys. A47, 04502, (2014). [22] I. Biaynicki-Birula and Z. Biaynicka-Birula, Quantum Electrodynamics, Ch. 3, p.123-124, Pergamon Press, Oxford, (1974). [23] R. Mignoni, E. Recami and M. Badlo, Lett. Nuovo Cim. 11, 568, (1974). [24] H.E. Moses, Nuovo Cim. Suppl. No 1, 1, (1958).

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Field transformation to noninertial frame of reference Maxwell equations in noninertial frame of reference