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Universal Newton Time in Classical Electrodynamics. Elements of Physical Interpretation

It is shown that the universal Newton time may be introduced in the classical electrodynamics. The statement results from an existence of the generalized symmetry of Maxwell equations with respect to Galilei transformations (physics/9701006). In the case of the extended Galilei transformations the postulate of invariance of the speed of light may be made compatible with the concept of the universal Newton time. Some physical consequences of the extended Galilei symmetry are considered.


Introduction
The postulate on the existence of the universal time was entered into physics by Newton in 1687. Up to the end of the XIXth century this concept perfectly corresponded to the known set of experimental facts, described by the equations of movement in the classical mechanics both on the Earth and within the Sun system. It will suffice to mention one of the remarkable achievements of this concept, namely, the prediction of the Neptune's orbit parameters on the basis of the Uran's perturbation study by Le Verrier, 1846. The difficulties have begun at the end of the XIXth century in attempting to make the universal time concept compatible with Maxwell electrodynamics. It is known from a formally mathematical side that the idea of the universal time is realized in Galilei transformations: x ′ = x − V t; y ′ = y; z ′ = z; t ′ = t.
Here x, y, z are the space variables, t is the time, V = (V, 0, 0) is the velocity of the inertial reference K ′ relative to K. While the Newton mechanics equation ma = F is in accordance with these transformations (invariant with respect to Galilei transformations), Maxwell equations did not seem to have the similar property. In other words, Maxwell equations, by their mathematical nature were presented as Galilei non-invariant equations. This point of view is popularized everywhere. One can find this in any university course of general physics. Moreover, it is from resolving the difficulty of Galilei approach in electrodynamics that the special relativity theory (SR) and the modern relativization of space-time representations have arisen. And now about 90 years after publishing the fundamental works of Voigt, Larmor, Lorentz, Einstein and Poincaré we return to seemingly already resolved problems again. It was found that a situation is more complicated here, than it is commonly supposed, and the concept of the Newton time may be also accorded with Maxwell equations. It is important to determine what we mean by this. For definiteness, by the words "the concept of the Newton time may be also accorded with Maxwell equations" we shall mean that vacuum (microscopic) Maxwell equations possess not only the property of the Lorentz invariance, but also the property of the invariance with respect to Galilei transformations in some generalized sense. The Galilei transformations contain the time transformation t ′ = t, and the universal Newton time will be thereby entered automatically into Maxwell equations.
It should be noted that although the relativistic concept was dominant, the research of Galilei symmetry of Maxwell electrodynamics were carried out after arising SR and are continued at the present time. As example it is possible to note the publications of Le Bellac and Levi-Leblond [1] and Fushchich and Nikitin [2]. The authors have established that besides the well known relativistic and conform symmetry, the first and second pair of Maxwell equations, considered separately, have also the property of invariance with respect to Galilei transformations. The various linear representations of Galilei group corresponding to the transformations of the fields E ′ = E, H ′ = (H − V x E); E ′ = (E + VxH), H ′ = H are realized on the solutions of these equations [1], [2]. (Here the speed of light c is believed equal to unit; E, H are the electrical and magnetic fields). So, the situation has considerably been cleared, but the attempt at establishing the property of Galilei symmetry of Maxwell equations in the total set has not been quite successful. It was made later in the works [3] - [5], which we shall follow below.

Galilei symmetry of Maxwell equations
Let us introduce the free Maxwell equations: As it is known, they may be brought to a single equation of the second order namely to the D'Alembert equation with φ ∈ (E, H): The algebraic equation of the light cone may be brought in correspondence with the differential D'Alembert equation: Any university course of general physics contains the statement on its Galilei non-invariance. Let us show by direct calculation that it is not so. For this purpose we use the transformations (1) and Galilei theorem of velocities addition, recorded for the speed of light: Let us put (1) and (5) into the equation of light cone in frame '. We have: Here it is taken into account that x = c x t = cn x t. It follows from here that the Galilei transformations map the sphere x ′2 = c ′2 t ′2 with center in the point x ′ = y ′ = z ′ = 0 onto the sphere x 2 = c 2 t 2 with the center in the point x = y = z = 0 just as the Lorentz transformations, which accords with the relativity principle. The failure in the former considerations consists in the fact that space and time were transformed according to Galilei but the speed of light was transformed according to Lorentz. Such inconsistency has resulted to incorrect conclusion that the relativity principle is violated at joint consideration of the ratios (1) and (4). In the fact the light cone equation is invariant with respect to Galilei transformations. This circumstance prompts to consider the symmetric properties of D'Alembert equation more carefully.
In accordance with theoretical-algebraic definition of the symmetry [6], let us consider the commutational ratios of the Lie algebra generators of Galilei group with the D'Alembert operator. We have [3]: Here p 0 = i∂ t , p k = −i∂ k J k = (xxp) k , H k = −tp k , k = 1, 2, 3, x 1,2,3 = x, y, z. It follows from here that the Lie algebra of Galilei group is the invariance algebra of the free Maxwell equations. Hence, the set of field transformations should exist which will transform into themselves the equations (2) in combination with the Galilei transformations (1). The task is to find them. For this we record the sought transformations as: where Φ(x, t) is some weight function. Let us express the Galilei transformations in variables where γ = (1 − 2βn 1 + β 2 ) 1/2 , n = c/c = (n 1 , n 2 , n 3 ), β = V /c, and produce the replacement of the variables in the equations (2), (3) taking into account the relations (8) and the relationships between the derivatives: Let us use the concept of the generalized symmetry [7] and require that the initial equations should be transformed into themselves due to the compatibility of the set of engaging equations (the conditions of transformation into themselves and the final equations), where the conditions of transformation into themselves are obtained by means of replacement of the variables (8) and (10) in Maxwell equations (2).

Initial Equations Conditions of Transformation into Themselves. Final Equations
The weight function Φ entering into this set may be found from the condition of transformation of D'Alembert equation into himself [3]:

Initial Equation Condition of Transformation into Himself . Final Equation
For the plane waves where l and m are the polarization vectors; k · x = k 0 x 0 − kx, k 0 = ω/c, k = k 0 n, ω is the electro-magnetic frequency; n is the guiding vector of wave front, the weight function Let us put the function Φ into the set of equations (11), taking into account that Φφ = As a result we have the eight algebraic equations of the second order for finding the five unknown group parameters a, e 23 , e 32 , h 23 , h 32 : By analogy with relativistic theory we will find the solution of the set (15) from the requirements These relations will be true if the group parameters possess the following properties: Putting (17) into the set (15), we obtain four equations for finding the unknown parameter a: From this and the set (17) we have As a result the formulas (8) take the form: (For comparison, in relativistic theory we have Φ = 1, . It can be shown that the transformations (20) form a group by virtue of the Galilei theorem of velocities addition β" = β + γβ ′ and transformation properties of the guiding cosines of a wave front In this case the group parameters and the weight function Φ have the following transformation properties: which correspond to the matrix law of multiplying the matrices of field transformations in going to primed variables. The formulas (22) are completely analogous to the formula of the relativistic theorem of velocities addition and also arise in multiplying the matrices from the Lorentz group. From this and due to the presence of the local weight function Φ(x, t) (14) it can be concluded that the matrices of the field transformations (20) form the projective representation of the Lorentz group [8] on the solutions as plane waves of D'Alembert equation. It is similar to the projective Galilei group representations realizing on the solutions as plane waves of Schrödinger equation in Quantum theory [9], [10], and it is also the unexpected circumstance in the symmetry theory of Maxwell equations. But the similar cases are known.
For example, as early as 1909, Cunningham [11] showed that the inversion group I: in Minkowski space induces the electromagnetic field transformations, which can be described through a matrix of Lorentz group representation D(L) with the local parameters β = 2x 0 r/(x 0 2 + r 2 ), where r 2 = x 2 + y 2 + z 2 , x 4 = (x 0 2 − x 2 ) 2 ; φ p ∈ (E, H), p, q = 1, 2, · · · , 6. In 1970 Isham, Salam and Strathdee generalized this result for the special conformal group C 4 : x ′µ = (x µ − a µ x 2 )/σ, σ = (1 − 2a · x + a 2 x 2 ) and fields of different conformal dimensions [12]. We write their result, as applied to 4 -potential (A 0 , A) of electromagnetic field where φ ∈ (A 0 , A), p, q = 0, 1, 2, 3, l = −1 is the conformal dimension of 4 -potential. (The conformal dimension is equal −2 for the electromagnetic fields E, H in the case of Cunningham). The same result (l = −1) may be also received by the method [7]. Now let us turn to the our case. One can see that the formulas (20) for electromagnetic field transformations may be written as: with the weight function Ψ and the group parameters a, e 23 , h 23 from the expressions (14) and (19).
We have an analogy with the result [11] exception that the weight function x 4 in the expression (25) has a universal character, but in the formula (27)  We also note that if the guiding vector is n = (1, 0, 0), the field transformations is reduced to formulas Considerable for the theory is the question on the invariants of the space-time and field transformations. Let us write them in comparison with the relativistic case.
It is not difficult to see that the part of Galilei-invariants coincides with Lorentz-invariants. However, the distinctions are also observed. They are caused by the distinction in transformational properties of the appropriate values. For example, formulas of transformation of the zero vector components (k 0 , k) in the Galilean and relativistic variants of the theory are different: They differently describe the Doppler effect in electrodynamics. Let us consider the approximation of the small speeds. At the small speeds, when β << 1, for both the relativistic and Galilean cases the common formulas of field transformations are realized: They are known as the Galilean limit. In the present work this term has a conditional character, as the formulas are the limit both for the relativistic and nonrelativistic field variables transformations. The situation here is analogous to the case with independent variables: These transformations being neither Galilean, nor relativistic are the limit ratios both for the Galilei and relativistic cases.
We also note, that in the relativistic theory the transformation parameters of the field variables do not depend on the kind of field. Owing to this fact these transformations have global nature in the space of solutions. In Galilean case, the transformation parameters and the weight functions in space of solutions are determined by the concrete field. In this sense the Galilean field transformations are non linear. Just this circumstance is the reason why Galilean symmetry and the associated concept of the Newton time in electrodynamics so long slipped off from researchers, as for all the cases, except [3] - [5], the field transformations were being searched in the class of the linear transformations.
These are the mathematical consequences from the introductions of the Newton time in electrodynamics. It is natural the question of the physical consequences arises as well. We consider only some of them.
If we admit in the framework of Galilei transformations, that the speed of light is equal 3.10 10 cm/s with respect to an emitter (Ritz hypothesis [14]), according to the Galilean theorem of velocities addition the speed of light should depend on the velocity of movement of the emitter. However such effects have not been found in the specially carried out experiments. (Bonch-Bruevitch and Molchanov, 1956;Alväger, Nilsson and Kjellman, 1963;James and Sternberg, 1963;Sadeh, 1963;Filippas and Fox, 1964;Alväger, Farley, Kjellman and Wallin, 1964;Babcock and Bergman, 1964). It is possible to overcome the marked difficulty, when we assume that the speed of light is equal 3.10 10 cm/s not with respect to an emitter, but with respect to an observer [3]. For this case, the relationship between the speed of light and the velocity of the emitter is eliminated, but it is necessary to assume that any emitter beforehand knows, with which the speed it should radiate the light that the speed of this light relative to any observer would have the required value 3.10 10 cm/s. This hypothesis and the experiments connected with it have been discussed in the paper [3]. Here we consider an another approach.

The extended Galilei transformations
The extended Galilei group G 11 may be defined as the set of linear space-time transformations where R is the matrix of the 3-dimensional spatial rotations; V is the velocity of the inertial reference frame K ′ relative to K; ̺ is the scale parameter. The transformations induced are defined in 4-dimensional real space E 3 T 1 with the metric Bearing this in mind, let us choose the parameter ̺ as with s = V/V , n = c/c, β = V /c. Then the extended Galilei transformations (34) will be compatible both with the concept of the universal Newton time t ′ = t and with the postulate of invariance of the speed of light because we may conclude from the theorem of velocities addition that c ′ = c in the given case. Owing to this fact, for the particular case when R kl = δ kl , V = (V, 0, 0), b = 0, the transformations of space and time are [5] x where group parameters have the following properties:

Invariants of the extended transformations
We note invariants of extended transformations induced once again.
The speed of light. We have from the transformational properties of velocity The equation of propagation of a spherical wave. Suppose c ′2 t ′2 − x ′2 = 0. Then where x = cn x t, n ′ x = (n x − β)/γ, n ′ y = n y /γ, n ′ z = n z /γ. Hence we obtain from c ′2 t ′2 − x ′ 2 = 0 that c 2 t 2 − x 2 = 0.
The radius of spherical wave. Suppose R 2 = x 2 + y 2 + z 2 = c 2 t 2 , R = ct. Then The sum of square of guiding cosines. Suppose n 2 = 1. Then Hence, in accordance with the principle of relativity the extended Galilei transformations (38) transform the sphere of radius R ′ = c ′ t ′ with center x ′ = y ′ = z ′ = 0 to the sphere with the same radius R = ct = c ′ t ′ = R ′ and center x = y = z = 0. The speed of light c ′ transforms to the same speed of light c = c ′ . The sum of square of guiding cosines turns to unit. We obtain an analogy with SR. We also note that in the framework of extended Galilei transformations (38) the consecutive explanation of the basic relativistic experiments is possible. Let us consider these explanations following the papers [3], [5].

The elements of physical interpretation 4.1 Similar to SR interpretation of observations
The Michelson experiment. Let the test run with the terrestrial light source, System K. Then by virtue of isotropy of 3-space the speed of light c will be the same in all directions. Therefore, for a terrestrial observer immobile with respect to an interferometer, the interference pattern will not be changed when the interferometer is rotated. Then let the test run with an extraterrestrial source of light, System K'. In this case, due to the transformational property c'=c -inv the speed of light in the system K, i.e. on the Earth, turns out to be equal c and will coincide with the speed of light from the terrestrial source. Therefore, experiments with terrestrial and extraterrestrial sources of light are indistinguishable physically, and the interference pattern will not be changed in our case as in SR.
Test to check independence of the speed of light from velocity of the light source. As in SR, all these experiments will lead to negative results since c ′ = c.
The Fizeau experiment. Let us suppose that the speed of light in moving medium (water) is equal where n is the refractive index, V is the velocity of water. Then according to the velocity addition theorem (following from the transformations (38) ) in the laboratory frame the speed of light turns out to be at n 1 = 1 in agreement with the experiment. Hence, the explanation of the Fizeau experiment is conventional. For example, as compared with the formula (44) we have v ′ = c/n in SR, in classical physics -v ′ = (c/n − V /n 2 ) [14].
Aberration of light. Let K ′ be the Sun, K be the Earth. For explanation the effect, as in SR, we assume that V = (V, 0, 0) is the velocity of the Sun and that the light propagate in the direction n ′ = (0, 1, 0) in frame K ′ . Then we have from the transformational properties of the guiding vector n of the speed of light Since the half of aberration angle is equal to α = (π/2−θ) we obtain sinα = β, α ≈ β = V /c = 10 −4 in agreement with the experiment. In more general case with n ′ = (cosθ ′ , sinθ ′ , 0) the angle of view on a star may be found from the expression tgθ ′ = sinθ/(cosθ − β). This formula coincides with the relativistic one tgθ ′ = sinθ · 1 − β 2 /(cosθ − β) [14] with an accuracy β 2 [3].
The Doppler effect. Suppose that an emitter moves with a velocity V along the x -axis relative to an observer K. Let us attach the frame K ′ to the emitter, and take into account the formulas k 0 ′ = γk 0 , k 1 ′ = (k 1 − βk 0 ), k 2 ′ = k 2 , k 3 ′ = k 3 (instead of the the formulas (30)). Then we have: From this we obtain where ω 0 ′ = ω 0 is the proper frequency and λ 0 is the proper wavelength of radiation; ω, λ are respectively the frequency and wavelength, measured experimentally. From the results (48) we have the following frequencies for the longitudinal (n 1 ≈ 1) and transversal (n 1 = 0) Doppler effect: For comparison, in the relativistic case the analogous formulas are: These formulas coincide with the formulas (48) and (49) for the longitudinal and the transverse Doppler effects in Galilean case with an accuracy β 2 and β 4 [3].

Distinction from SR interpretation of observations
We note some of them.
The twin paradox. In Galilean approach the twin paradox do not exist because of universal character of time.
Tests to measure lifetimes of fast nonstable particles. In the framework of transformations (38) it is necessary to admit, that instead of the effect of the time retardation the movement with superrelativistic velocity should exist. For example, the velocity of atmospheric µ -mesons should be equal to ∼ 6 · 10 6 /2 · 10 −6 = 3 · 10 12 cm/s [3].
Superrelativistic objects with real mass. In the relativistic physics such objects can not exist.
In the Galilean case their existence is not forbidden. Besides the fast nonstable particles, such objects may be space objects with a large value of the redshift In the case of a radial movement the parameter of the redshift is z ≃ β(β − n 1 )/(β + 1) ≃ β when n 1 ≃ −1, and for z > 1 the superrelativistic velocity of object will be equal v ≃ zc > c [5]. For the transversal movement when n 1 = 0, the redshift is z = 1 + β 2 − 1 and the velocity of object will be equal v ⊥ = z(z + 2) c. Then the superrelativistic movements may be if z > √ 2 − 1 = 0.414. The quasar NRAO 140 with redshift z=1.258 may be an object of such type. In the framework of Galilei approach the radial calculated velocity of this object may be equal v ≃ zc = 1.258c; the transversal calculated velocity may be equal v ⊥ ≃ z(z + 2) = 2.02c. The surprising thing is the fact that the value 2.02c is close to the lower limit of the superrelativistic expansion velocity of QSO NRAO 140 in the framework of the Friedmann cosmological model: from 3c to 10c, depending upon assumptions [15].
Let us also note, that the considered example is not the only case when a quasar has the value of redshift z > 1. According to [16], the number of these quasars is great enough, for example, the quasars LB8796 z=1.320, PKS z=2.170, PHL957 z=2.690, OQ172 z=3.530, etc. It is known more than 100 objects with redshift z > 3 too [17], among which the quasar Q1158+4635 with z=4.73 is present. From standpoint of the work all these objects may move with superrelativistic velocities v ≃ zc and superrelativistic motion is an ordinary phenomenon in astrophysics. Hubble's distance for these objects is equal D H ≃ zc/H and may be connected with Friedmann's distance (Robertson-Walker metric) by the relation: D F = D H {q 0 + (q 0 − 1)[(1 + 2q 0 z) 1/2 − 1]/z}/q 0 2 (1 + z), where H is the Hubble constant, q 0 is the deceleration parameter [18]. For the case of q 0 = 1 we have D F = D H /(1 + z) = zc/H(1 + z) and the horizon c/H, which is absent in the Galilean approach, occurs in Friedmann's model. If q → 0, the Friedmann's distance D F → z 2 c/H(1 + z) is close to the Galilean value D H ≃ zc/H for the great values of the red shift parameters z ≫ 1.
The limiting speed of propagation of interactions. According to the addition velocities theorem, we have the following value v for movement along of the x -axis at n = (1, 0, 0): Here the value v ′ = c has the invariance property only for the case of such phenomenon as the propagation of the light. If v ′ = c, the velocity v may be any amount great by appropriate value V . Therefore the limiting speed of propagation of all interactions in the approach (38) do not exist.
The interactions having the nature other than the electromagnetic one may propagate with velocity distinct from the value c = 3 · 10 10 cm/s.

Conclusion
It is shown that in the frame of the generalized symmetry approach [7] Maxwell equations are invariant with respect to Galilei transformations and admit the existence of the universal Newton time. In the case of the extended Galilei transformations the postulate of the universal Newton time may be made compatible with the concept of invariance of the speed of light. The Galilei symmetry of Maxwell equation means that the Galilei relativity principle may be realized not only in the classical mechanics but in the classical electrodynamics too.