12 The frame S ( l ) 4 will be valid if only the gravitational force given in this coordinate frame could be removed by the inertial force which, in turn, is due to the distortion inertial field potential Eq. (45). Whereas, the two systems S$_{2}$ and S ( l ) 4 can be chosen so that the axis e$_{q}$ of S$_{(2)}$ now lies along ( e$_{q}$ = e$_{f}$ ) the acting net force f = f ( l ) + f g ( l $_{)}$( a ) , while the time coordinates in the two systems are taken the same q 0 = t = X 0 = T. Instead of Eq. (58), we now have tion is involved, we are at once led to seek equations in the form of generally covariant tensor expression us- ing any set of general coordinates which we may desire to introduce. Let the distortion gauge field ( a$_{l}$ ) under- lies gravitation. Then, the generally covariant expres- sion for interval in four-dimensional Riemannian space R$_{4}$ is d ˜ s 2 = g$_{µν}$ ( a ) d ˜ x $^{µ}$d ˜ x $^{ν}$, and Γ λ $_{µν}$( a ) denotes affine connection agreed with the metric g$_{µν}$ ( a ). Although in- ertial forces do not exactly cancel for freely falling sys- tems in an inhomogeneous or time-dependent gravita- tional field, in accordance with the Principle of Equiva- lence, we may still maintain for a sufficiently small re- gion that the effects of gravitation could be removed by the use of local freely falling coordinate frame S ( l ) 4 , having the natural acceleration due to gravity for that region. This implies that the local spacetime structure can be identified with the Minkowski spacetime possess- ing Lorentz symmetry, and that the physical laws in the frame S ( l ) 4 will be the same as in any inertial coordinate frame S$_{(4)}$ . This is similar to assumption of approxi- mate replacement of a curved surface by its tangent plane at a given point, made use of in geometrical considera- tions. Therefore, we can always choose natural coordi- nates X $^{α}$( X, Y, Z, T ) = ( X , T ) with respect to the axes of the system S ( l ) 4 in immediate neighbourhood of any spacetime point ( ˜ x$_{p}$ ) ∈ R$_{4}$ in question, over a differential region taken small enough so that we can neglect the spa- tial and temporal variations of gravity for the range in- volved. In this coordinates, the special relativity formula for interval will be dS 2 = η$_{lk}$dX $^{l}$dXk = d ˜ s $^{2}$, where the components of the metrical tensor η$_{lk}$ assume the special relativity values ( − 1 , 1 , 0) , and the first differential coef- ficients of the η$_{lk}$ with respect to these coordinates will be zero at the point ( p ) . In general, however, the second differential coefficients will not be zero except for the spe- cial case of spacetime that actually is flat. The values of metrical tensor g$_{µν}$ ( a ) and affine connection Γ λ $_{µν}$( a ) at the point ( ˜ x$_{p}$ ) is necessarily sufficient information for deter- mination of the natural coordinates X $^{α}$( ˜ x $^{µ}$) in the small region of neighbourhood of selected point [59]. The whole scheme outlined in the previous subsection will then hold in the frame S ( l ) 4 , provided as a preliminary step we first examine the possibility of re-expressing the special rela- tivity formula for the free ( f ( l ) = 0) test particle dU $^{α}$/dS = d $^{2}$X $^{α}$/dS 2 = 0 , ( α = 1 , 2 , 3 , 0) in generally covariant form as DU $^{α}$/D ˜ s = D $^{2}$X $^{α}$/D ˜ s 2 = 0 , where f ( l ) is the special relativity value of unbalanced relativistic force other than gravitational in the frame S ( l ) 4 and, accord- ing to the general prescription D/D ˜ s is the covariant derivative along the curve ˜ x $^{µ}$( ˜ s ) ∈ R$_{4}$. The relativistic gravitational force f µ g ( ˜ x ) exerted on the test particle of the mass ( m ) is provided e$_{f}$ = ( X / | X | ), v$_{q}$ = ( e$_{f}$ · u ) = | u | , ( γ$_{q}$ = γ ) where ( u ) is the velocity of photon and ( d u /d ˜ t ) is the acceleration, and that, g$_{µν}$ ( a )( d ˜ x $^{µ}$/dT )( d ˜ x $^{ν}$/dT ) = 0 . To obtain some feeling for this, we may turn now to the (PPN) approximation [37–40], which can be regarded as a deformation of a background asymptotically flat Minkowski metric. In this context we calculate the iner- tial force for the photon in gravitating system of particles that are bound together by their mutual gravitational at- traction to order ¯ v 2 ∼ G$_{N}$ ¯ M/ ¯ r of small parameter, where ¯ v, ¯ M and ¯ r are typical , respectively, the average values of their velocities, masses and separations. In doing this, we may expand the metrical tensor to the following or- der: g$_{00}$ = 1+ 2 g$_{00}$ + 4 g$_{00}$ + ..., g$_{ij}$ = − δ$_{ij}$ + 2 g$_{ij}$ + 4 g$_{ij}$ + ..., g$_{i}$$_{0}$ = 3 g$_{i}$$_{0}$ + 5 g$_{i}$$_{0}$ + ...., where N g$_{µν}$ denotes the term of order ¯ v $^{N}$. Taking into account the standard expansions of affine connection Γ σ µν = 2 Γ σ µν + 4 Γ σ µν + ... for the compo- nents Γ i $_{00}$, Γ i $_{jk}$, Γ 0 $_{i}$, and that Γ σ µν = 3 Γ σ µν + 5 Γ σ µν + ... for the components Γ i 0 $_{j}$, Γ 0 $_{00}$, Γ 0 $_{ij}$, where 2 Γ i $_{00}$= 2 Γ 0 $_{i}$= − (1 / 2)( ∂ 2 g$_{00}$ f µ g ( ˜ x ) = m d $^{2}$˜ x µ d ˜ s 2 = − m Γ µ $_{νλ}$( a ) d ˜ x ν d ˜ s d ˜ x λ d ˜ s . (62) f ( in ) = − ( hν c 2 ) e$_{f}$ | ∂X α ∂ ˜ x σ Γ σ $_{µν}$( a ) d ˜ x µ dT d ˜ x ν dT | = − ( hν c 2 ) e$_{f}$ | d $^{2}$˜ t dT $_{2}$) dX α d ˜ t + ( d ˜ t $_{dT}$) 2 ∂X α ∂ ˜ x i du$_{i}$ d ˜ t , (66) f ( in ) = m a in = e$_{q}$ ( − m Γ 1 $_{ab}$( ϱ ) d ˜ q a d ˜ s$_{q}$ d ˜ q b d ˜ $_{s$_{q}$}$) = − m a abs γ$_{q}$ = − e$_{f}$ γ$_{q}$ | f α ( l ) m ∂X α ∂ ˜ x σ Γ σ $_{µν}$( a ) d ˜ x µ dS d ˜ x ν dS | . (65) 1 √$_{2}$ dϱ ds$_{q}$ = 1 m | f α ( l ) + f α g ( l $_{)}$( a ) | . (64) Hence, in general, the RLI can be obtained as In spite of totally different and independent sources of gravitation and inertia, at f ( l ) = 0 when the mass ( m ) is canceled out in Eq.(65), the RLI indeed furnishes justi- fication for introduction of the Principle of Equivalence. A remarkable feature is that the inertial force is of the same nature as gravitational force, i.e., both are due to the distortion of local internal properties of the flat space. The non-vanishing inertial force acting on the photon of energy hν , and that of effective mass ( hν/c 2 ) after in- serting a factor ( c $^{2}$) which so far was suppressed, can be obtained from the Eq. (65), at f ( l ) = 0 , as f α g ( l ) = ∂X α ∂ ˜ x µ f µ g , (63)