In Ch. 20 of his book 'Gravitation and cogravitation' Prof. Jefimenko gives the analysis of the residual precession of Mercury's perihelion. This analysis is based on the expression for the relativistic acceleration of a body (the Mercury) subjected to the action of a gravitation force (of the Sun), Eq. (20-2.1).
As a result, he derives the relativistic correction to the Newtonian force. According to Jefimenko, this correction term should give 14 arc sec per century for the Mercury, or abour 1/3 of the presently accepted value of 43 arc seconds per century.
This correction term is absent in Einstein's solution (in frame of the general relativity). Jefimenko presents own explanation how to obtain the value 43 arc seconds. But my question is not about the theory of gravitaion of Jefimenko but on concordance of the special and general relativity.
Actually, what Jefimenko does is introduction of correction of the mass of the Mercury due to change its velocity (typical relativistic correction).
So why is this dependence of the mass of moving body absent in the general relativity solution?
Some time ago a paper of Wolfgang Engelhardt 'Free fall in gravitational theory' actively discussed at researchagate. Engelhardt stated too that Einstein incorrectly omitted the dependence of the mass of the Mercury on its velocity in the orbit.
The opponents of Engelhardt objected (their main argument) that due to the principle of equivalence, the inertial mass is equal to the gravitational mass. Therefore the mass of the Mercury should be omitted in the equation of motion.
But the dependence of mass on the velocity is being not in the mass itself but in (m dx/ds) - the GR equation of motion of the body should correspond to the second Newton's law. So formally the dependence on the velocity cannot be eliminated from the GR equation of motion.
Initially the dependence of the mass of the attracted body (the Mercury) should be being in the GR equations of motion.
For example, in Sec. 87 'Motion of a particle in a gravitational field' of Landau and Lifshitz 'Theory of field' the authors show how the velocity dependence is eliminated from the equation of motion.
One can find the same procedure in sec. 110 of Moeller's 'Theory of relativity' (the equation below Eq. (20)) so the question is if it is a correct procedure to omit the dependence of md^2x^i/ds^2 on the velocity of the Mercury.
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