9 provided, the Eq. (38) and Eq. (39) give the relation dg$_{00}$ dr = dg$_{00}$ d ˜ r 1+ ω ( F ) 1+ r g 2 r = r$_{g}$ r $_{2}$(1 − r$_{g}$ 2 $_{r}$) , (40) F = (4 r 2 $_{g}$æ $^{2}$F$_{mn}$ F$_{mn}$ ) 1 / 4 = r$_{g}$/r << 1 , (41) A. The case of unbalanced net force other than gravitational IV. THE RELATIVISTIC FIELD THEORY OF INERTIA As we mentioned in Sect.2, in the proposed theory of gravitation, the preferred systems and group of transfor- mations of the real-curvilinear coordinates relate only to the real gravitational fields. This prompts us to intro- duce separately the distortion inertial fields , which have other physical source than that of gravitation, and con- struct the relativistic field theory of inertia. The latter, similarly to gravitation theory, treats the inertia as a dis- tortion of local internal properties of flat M$_{2}$ space. The geometry of Sect.3 is a language which is almost indis- pensable for the treatment of this problem. for determining the function ˜ r ( r ) . We must now turn to the actual correspondence between the expression (39) for the line element surrounding an attracting central body and the observational facts of astronomy. The in- vestigating methods are so well known that it will be sufficient for our purposes merely to indicate the classi- cal tests of GR conducted in solar-system dealing only with the shape of the trajectories of photons and plan- ets. All these tests are carried out in empty space and in gravitational fields that are to a good approximation static and spherically symmetric. Therefore, in sufficient approximation, at great distances from the central body we may take expansion of function ω ( F ) = λ$_{1}$F + λ$_{2}$F $^{2}$+ · · · , and that from Eq. (40) we obtain First, we shall discuss the inertia effects in particu- lar case when the relativistic test particle accelerated in the flat space under unbalanced net force other than gravitational. Let us concentrate our attention on the first observer in two-dimensional Minkowski flat space M$_{2}$ = R 1 (+) ⊕ R 1 $^{(}$−) = R 1 ⊕ T $^{1}$, being regarded as in a state of rest or uniform motion. Suppose this unaccel- the tests of gravity via astrometric VLBI observations. Detailed analysis of VLBI data have yielded a consis- tent stream of improvements γ = 1 . 0000 . 003 [47, 48], γ = 0 . 99960 . 0017 [45], γ = 0 . 999940 . 00031 [43] and γ = 0 . 999830 . 00045 [50] resulting in the accuracy of bet- ter than ∼ 0 . 045%. The major advances in several dis- ciplines notably in microwave spacecraft tracking, high precision astrometric observations, and lunar laser rang- ing (LLR) suggest new experiments. LLR, a continuing legacy of the Apollo program, provided improved con- straint on the combination of parameters 4 β − γ − 3 [52– 55]. The analysis of LLR data [53] constrained this com- bination as 4 β − γ − 3 = (4 . 0 ± 4 . 3) × 10 − $^{4}$, leading to an ac- curacy of ∼ 0 . 011% in verification of general relativity via precision measurements of the lunar orbit. A significant improvement was reported in 2003 from Doppler tracking of the Cassini spacecraft while it was on its way to Sat- urn [41], with a result γ − 1 = (2 . 1 ± 2 . 3) × 10 − $^{5}$. This was made possible by the ability to do Doppler measurements using both X-band (7175 MHz) and Ka-band (34316 MHz) radar, thereby significantly reducing the dispersive effects of the solar corona. In addition, the 2002 superior conjunction of Cassini was particularly favorable: With the spacecraft at 8.43 astronomical units from the Sun, the distance of closest approach of the radar signals to the Sun was only 1 . 6 $^{R}$⊙ . This experiment has reached the current best accuracy of ∼ 0 . 002% [42]. Keeping in mind aforesaid, the best fit for satisfactory agreement be- tween the proposed theory of gravitation and observation can be reached at λ$_{1}$ − 1 / 4 = (2 . 95 ± 3 . 24) × 10 − $^{5}$.