Structure of the spacetime under gravitation obtained by purely dynamical method

Attempt to obtain the general relativistic planet's motion by special relativity techniques

“…In carrying it out, it is a problem in which P 0 ‐system and P‐system are the inertial system and the other the accelerating system. Here we notice the fact that no force is felt in the P‐system (at the place of the planet) because it is making a free fall motion while the gravitational force is felt in S‐system and consequently in P 0 ‐system (Kubo 2019).…”

“…The formula in the case where the velocity is not in the direction of z ‐axis but in a general direction represented by $$$ \left({v}_x,{v}_y,{v}_z\right) $$$ is given as follows (Kubo 2019): $$$$ \mathbf{L}=\left(\begin{array}{cccc}1+\left(\gamma -1\right)\frac{v_x^2}{v^2}& \left(\gamma -1\right)\frac{v_x{v}_y}{v^2}& \left(\gamma -1\right)\frac{v_x{v}_z}{v^2}& i\frac{\gamma }{c}{v}...$$…”

“…The equation of motion (18) is solved following the Gauss method in celestial mechanics (Brouwer & Clemence 1961). More detailed derivation of the solution is found in the study by Kubo (2019). First, because the acceleration in Equation (18) is contained in the plane formed by the vectors $$$ \boldsymbol{r} $$$ and $$$ \boldsymbol{v} $$$, the motion occurs in this plane.…”

“…P‐system has acceleration to P 0 ‐system. For two systems that accelerate to each other, the Lorentz transformation of coordinates does not hold, but we have to use the formula introduced in the study by Zhukov (1961) as well as in the study by Kubo (2019). It is explained as follows.…”

“…Kubo (2019) also aimed to obtain the planetary motion as close as possible to the observation or to the result by the general relativity, using solely the method of SR. However, the author followed a quite different approach from those research studies mentioned above.…”