where Q ( = Q τ ( λα $_{)}$( a ) is the element of the group Q . This violates the first relation in Eq. (11) because of ˜ O 2 ( λα $_{)}$( a ) = 2æ a$_{(}$$_{λα}$$_{)}$ = 0 for given λ and α . Next, we assume that this induces the distortion transformations of ingredient unit vectors σ$_{β}$ , which, in turn, undergo the rotations: ˜ σ$_{(}$$_{λα}$$_{)}$ ( θ ) = R β ( λα $_{)}$( θ ) σ$_{β}$, where R ( θ ) ∈ SO (3) is the element of the group of rotations of planes in- volving two arbitrary axes around orthogonal third axis in the given ingredient space R 3 $_{λ}$. Then, resulting ba- sis vectors ˜ σ$_{(}$$_{λα}$$_{)}$ ( θ ) of each three-dimensional ingredi- ent space R 3 λ retain the orthogonality condition between themselves, but violate it between the basis vectors of dif- ferent ingredient spaces. That is, < ˜ σ$_{(}$$_{λα}$$_{)}$, ˜ σ$_{(}$$_{τ β}$$_{)}$ $^{>$_{α}$}$̸ = β = 0 , at λ = τ. In fact, distortion transformations of ba- sis vectors ( O ) and ( σ ) are not independent, and rather governed by the spontaneous breaking of distortion sym- metry (see Eq. (24)). To avoid a further proliferation of indices, hereafter we will use upper case Latin ( A ) in in- dexing ( λα ), etc. The infinitesimal transformations then read The nonlinear realization technique [32, 33] provides a way to determine the transformation properties of fields defined on the quotient space G/H . Constructing non- linear realization of the Lie group of distortion G$_{D}$ , first, within the scheme of the GGP we necessarily introduce the language of a conceptual six-dimensional geometry of M$_{6}$ , which is assumed as a toy model underlying the M$_{4}$ . This replacement appears to be indispensable in discussion of the distortion of local internal proper- ties of spacetime continuum, and it mostly manifests its virtues in constructing the relativistic field theory of in- ertia (Sect.4). So, let M$_{6}$ be the smooth differentiable six-dimensional flat space with the decomposition law as follows: The e$_{(}$$_{λα}$$_{)}$ = O$_{λ}$ × σ$_{α}$ ( λ = ± , α = 1 , 2 , 3) are linear independent unit basis vectors at the point (p) of interest of given three-dimensional space R 3 $_{λ}$. The unit vectors O$_{λ}$ and σ$_{α}$ imply where δ$_{αβ}$ is the Kronecker symbol, and ∗ δ$_{λτ}$ = 1 − δ$_{λτ}$. Consequently, three spatial e$_{α}$ = ξ × σ$_{α}$ and three tem- poral e$_{0}$$_{α}$ = ξ$_{0}$ × σ$_{α}$ components are the basis vec- tors, respectively, in spaces R 3 and T $^{3}$, where $^{O}$± = (1 $_{/}$√$_{2)(}$$_{ξ}$$_{0}$ ± ξ ) , ξ 2 0 = − ξ 2 = 1 , < ξ$_{0}$, ξ > = 0 . Within this scheme, we are presumably allowed to perceive di- rectly the three-dimensional ordinary space R $^{3}$, but not the three - dimensional time space T 3 being orthogonal to the former. In using this language it is important to guard a reduction to the space M$_{4}$ which can be achieved in the following way. First, we consider distortion transformations of the in- gredient unit vectors O$_{τ}$ under the distortion gauge field Following the method of phenomenological La- grangians [13, 15, 32–34, 60] and references therein, our where we denote d a $^{′}$A = ω $^{A}$( a, θ ; d a, d θ ) and d θ $^{′}$A = ϑ $^{A}$( a, θ ; d a, d θ ). Then, provided by generators X τ λ = ∗ δ τ λ and I$_{i}$ = σ$_{i}$ 2 , where σ$_{i}$ are the Pauli’s matrices, M$_{αβ}$ = ε$_{αβγ}$I$_{γ}$ and δ ω αβ = ε$_{αβγ}$δ θ$_{γ}$. Transformation matrix D ( a, θ ) = Q ( a ) × R ( θ ) is the element of distortion group G$_{D}$ = Q × SO (3): δ Q τ $_{A}$( a ) = æ δ a$_{A}$X τ λ ∈ Q, δ R ( θ ) = − i $_{2}$M$_{αβ}$ δ ω αβ ∈ SO (3) , (13) d e$_{A}$ = e$_{A}$ d F$_{A}$ = O$_{A}$ ( d ) × σ$_{A}$ + O$_{A}$ × σ$_{A}$ ( d ) = i [ ω $^{A}$( d ) X$_{A}$ + ϑ $^{A}$( d ) I$_{A}$ ] e$_{A}$, (16) where I$_{A}$ ≡ I$_{α}$ at given λ . We may join to each point ( a, θ ) of group space the Descartes’ reper which is equal, in group sense, to the reper joint to point of origin (0 , 0). This is introduced to ensure that the vector ( a, θ ; a + d a, θ + d θ ) has the same analytical expression of the vector (0 , 0; d a ′ , d θ ) ′ : (Eq. (14)) D$_{(}$$_{d a}$ $^{A}$, d θ$^{A}$) = I + d D$_{(}$$_{a}$ $^{A}$, θA $_{)}$, d D$_{(}$$_{a}$ $^{A}$, θ$^{A}$) = i [ d a $^{A}$X$_{A}$ + d θ $^{A}$I$_{A}$ ] , (14) I$_{α}$ at given λ . We may join to each point D$_{(}$$_{a, θ}$$_{)}$ $_{d D}$− 1 ( a, θ ) = i [ ω $^{A}$X$_{A}$ + ϑ $^{A}$I$_{A}$ ] , (15) < O$_{λ}$, O$_{τ}$ > = ∗ δ$_{λτ}$ , < σ$_{α}$, σ$_{β}$ > = δ$_{αβ}$, (11) M$_{6}$ = R 3 + ⊕ R 3 − = R 3 ⊕ T $^{3}$, sgn ( R $^{3}$) = (+ + +) , sgn ( T $^{3}$) = ( − − − ) . (10) ˜ O$_{(+}$$_{α}$$_{)}$ ( a ) = Q τ (+ α $_{)}$( a ) O$_{τ}$ = O$_{+}$ + æ a$_{(+}$$_{α}$$_{)}$$^{O}$−, $^{O$_{(}$}$− α $_{)}$( a ) = τ $^{(}$− α $_{)}$( a ) O$_{τ}$ = $^{O}$− + æ $^{a$_{(}$}$− α $_{)}$O$_{+}$ , (12) 6 III. NONLINEAR REALIZATION OF DISTORTION GROUP G D ( a ): where e$_{A}$ ≡ (exp F$_{A}$ ). The functions ω $^{A}$( d ) and ϑ $^{A}$( d ) will be determined in the next subsection. B. A spontaneous breaking of distortion symmetry A. Distortion of local internal properties of the M 6 2) In case of curved space, the reduction R$_{6}$ → R$_{4}$ can be achieved if we use the projection ( ˜ e$_{0}$ ) of the tempo- ral component ( ˜ e$_{0}$$_{α}$ ) of basis six-vector e ( ˜ e$_{α}$, ˜ e$_{0}$$_{α}$ ) on the given universal direction ( ˜ e$_{0}$$_{α}$ → ˜ e$_{0}$ ) . By this we choose the time coordinate. Actually, the Lagrangian of physi- cal fields defined on R$_{6}$ is the function of scalars such as A$_{(}$$_{λα}$$_{)}$B ( λα ) = A$_{α}$B $^{α}$+ A$_{0}$$_{α}$B 0 $^{α}$, then upon the reduction of temporal components of six-vectors A$_{0}$$_{α}$B 0 α = A 0 α < ˜ e$_{0}$$_{α}$, ˜ e$_{0}$$_{β}$ > B 0 β = A 0 < ˜ e$_{0}$, ˜ e$_{0}$ > B 0 = A$_{0}$B 0 we may fulfill a reduction to R$_{4}$ . 1) In case of free flat space M$_{6}$ , the subspace T 3 is isotropic. And so far it contributes in line element just only by the square of the moduli t = | x $^{0}$| , x 0 ∈ T $^{3}$, then, the reduction M$_{6}$ → M$_{4}$ = R 3 ⊕ T 1 can be readily achieved if for conventional time we use t = | x $^{0}$| .