Concerning the Paper by A. Einstein “Explanation of the Perihelion Motion of Mercury from the General Theory of Relativity”

It is shown that in A. Einstein’s 1915 paper “Explanation of the perihelion motion of Mercury from the general theory of relativity” an error was actually incurred in the integration of the equation \( \phi =\left[1+\upalpha \left({\upalpha}_1+{\upalpha}_2\right)\right]\underset{\upalpha_1}{\overset{\upalpha_2}{\int }}\frac{dx}{\sqrt{-\left(x-{\upalpha}_1\right)\left(x-{\upalpha}_2\right)\left(1-\upalpha x\right)}} \), and in the result instead of \( \phi \approx \uppi \left[1+\frac{5}{4}\upalpha \left({\upalpha}_1+{\upalpha}_2\right)\right] \) (if we limit ourselves to first-order terms in the small quantity (α₁ + α₂)) the value \( \phi \approx \uppi \left[1+\frac{3}{4}\upalpha \left({\upalpha}_1+{\upalpha}_2\right)\right] \) was obtained, where α₁ and α₂ are the inverse values of the maximum and minimum distances of Mercury from the Sun, \( \upalpha =\frac{2G{m}_0}{c^2} \) is the gravitational radius, G is the gravitational constant, m₀ is the mass of the Sun, c is the velocity of light (i.e., the Chinese mathematician Hua Di was actually right). And, as a result, for the precession of the orbit of Mercury in the gravitational field of a spherical Sun in the general theory of relativity (after 100 years) one obtains not ~43”, as A. Einstein obtained, but ~71.63”. Exactly this latter result is obtained by direct numerical modeling of the precession of the perihelion of Mercury’s orbit in the gravitational field of the Sun within the framework of the general theory of relativity if the fitting coefficient α in the equation of motion of Mercury (not to be confused with α in the above equations) is set equal to zero. The result \( \phi \approx \uppi \left[1+\frac{3}{4}\upalpha \left({\upalpha}_1+{\upalpha}_2\right)\right] \), obtained by A. Einstein if it is obtained, happens only upon integration of either the equation \( \phi =\left[1+\frac{\upalpha}{2}\left({\upalpha}_1+{\upalpha}_2\right)\right]\underset{\upalpha_1}{\overset{\upalpha_2}{\int }}\frac{dx}{\sqrt{-\left(x-{\upalpha}_1\right)\left(x-{\upalpha}_2\right)\left(1-\upalpha x\right)}} \) (if we also limit ourselves to first-order terms in the small quantity (α₁ + α₂)), i.e., in front of the integral in brackets the coefficient 1/2 should stand in front of the quantity α, or the equation \( \phi =\left[1+\upalpha \left({\upalpha}_1+{\upalpha}_2\right)\right]\underset{\upalpha_1}{\overset{\upalpha_2}{\int }}\frac{dx}{\sqrt{-\left(x-{\upalpha}_1\right)\left(x-{\upalpha}_2\right)\left(1-\upalpha x\right)}} \) (if we also limit ourselves to first-order terms in the small quantity (α₁ + α₂)), i.e., in the denominator under the differential in the integral, inside the square root\ a plus sign should stand in front of the quantity αx.