Quotient Equations and Integrals of Motion for Vector Massless Field

In this article a group-theoretical aspect of the method of dimensional reduction is presented. Then, on the base of symmetry analysis of an anisotropic space geometrical description of dimensional reduction of equations for vector massless field is given. Formula for calculating components of the energymomentum tensor from the variables of the field factor-equations is derived. The covariant equations for massless vector field in the absence of sources are of the form ∇μF μν = 0, ∇μ(∗F ) = 0, (1) where F μν is tensor of vector field, (∗F ) is conjugate magnitude defined by the equation (∗F ) = 1 2 √−g [αβγη]Fγη; [αβγη] is completely antisymmetric tensor with [0123]=1. Because metric tensor (gμν) does not depend on space coordinates, the theory under consideration is invariant under space translations, and consequently, the space transformation of Fourier-components of tensor is rendered possible: F (t,x) = ∫ dk exp(i,k,x)f(t,k), (∗F )(t,x) = ∫ dk exp(ikx)(∗f)(t,k). (2) Substituting (2) in (1), one comes to the equation for f(t,k): −ikjf 0j = 0, − [ 1 √−g∂0 (√ −g f j0 ) + ikjf ji ] = 0. (3) Because f ji = −[jri]Ar(∗f)/ √−g, the previous equation takes the form d dt √−gf 0j ) = −i[jml]klAm(t)(∗f). (4) Similarly, for the conjugate magnitude we have kj(∗f) = 0, d dt √−g(∗f)j0 ) = i[jml]klAm(t)f . (5) To simplify the set of equations (3)–(5), we introduce the new variables S j = √ −g[f 0j ± i(+f)]. (6) This enables us to separate the equations (3)–(5) for S j and S − j : kjS ± j = 0, d dt S j = ∓ [jml] √−g klA 2 mS ± m. (7) The further simplification consists of the introduction spherical coordinates in momentum space according to the relation (k1, k2, k3) = k (sin(δ) cos(ξ), sin(δ) sin(ξ), cos(δ)) . (8) The equation (7) can be automatically satisfied going from (6) to the magnitudes S δ = cos(δ) cos(ξ)S ± 1 + cos(δ) sin(ξ)S ± 2 − sin(δ)S 3 , S ξ = − sin(ξ)S 1 + cos(ξ)S 2 . (9) ¿From the equation (7), rewritten as sin(δ) ( cos(ξ)S 1 + sin(ξ)S ± 2 ) + cos(δ)S 3 = 0, and definition of S δ it is possible to derive the relations S δ = 1 cos(δ) ( cos(ξ)S 1 + sin(ξ)S ± 2 ) , S 3 = −S δ sin(δ). Considering the previous relations and the definition S ξ as a set of equations, we obtain for S 1 and S ± 2 the following coupling relations S 1 = S ± δ cos(ξ) cos(δ)− S ξ sin ξ, S 2 = S ± δ sin(ξ) cos(δ) + S ± ξ cos(ξ), S 3 = −S δ sin(δ). (10) To derive the equations for S δ and S ± ξ , let us differentiate the amgnitude (9) by t: Ṡ ξ = cos(δ) cos(ξ)Ṡ ± 1 + cos(δ) sin(ξ)Ṡ ± 2 − sin(δ)Ṡ 3 , Ṡ ξ = − sin(ξ)Ṡ 1 + cos(ξ)Ṡ 2 . (11) Further we use the equations (7) ±Ṡ 1 = − k √−g ( cos(δ)A2S 2 − sin(δ) sin(ξ)A3S 3 ) , ±Ṡ 2 = + k √−g ( cos(δ)A1S 1 − sin(δ) cos(ξ)A3S 3 ) , ±Ṡ 3 = k √−g ( sin(δ) sin(ξ)A1S 1 − sin(δ) cos(ξ)A2S 2 ) , and then (10) with the result that for S δ and S ± ξ we obtain the equations ± Ṡ δ = −kaS δ − kbS ξ , ±Ṡ ξ = +kcS δ + kaS ξ . (12) The parameters a, b and c can be presented as follows a = cos(δ) cos(ξ) sin(ξ) √−g ( A2(t)−A1(t) ) , b = 1 √−g ( A2(t) cos(ξ) +A1(t) sin(ξ) ) , c = 1 √−g ( A1(t) cos(δ) cos(ξ) +A2(t) cos(δ) sin(ξ) +A3(t) sin(δ) ) . (13) To continue the analysis, it is necessary to discuss the dependence of basis vectors of space on time. The metric gμν defines the natural covariant ”unit” vector eμ with covariant components e α μ = δ α μ and the natural contravariant vector e with contravariant components eα = δ μ α with (eμeν) = gμν . The absence of the off-diagonal terms from metric means that the basis consisting of the vectors eμ is orthogonal, but the length of space basis vectors varies with time according to the relation