On Corrections to a Planet Movement

Abstract

Significant improvement in observational data on the motion of test bodies in the gravitational field will answer two main questions. First, is the basic assumption that the field is determent by differential equations, and not by propagators from the known sources, valid? Second, does the general relativity correctly describe the simplest nonlinear source, i.e., a 3-graviton vertex? The article discusses the corrections with which it is necessary to compare observational data of appropriate accuracy. Corrections to a planet movement are obtained in standard and harmonic coordinate systems. These include corrections to minimal and maximal radii and to half-periods in terms of the world and proper times. The method used permits the calculations not only the first correction but also “ correction to correction”.

6. APPENDIX

For reader’s convenience I give here some integrals occurring in the text.

$$\begin{gathered} \int\limits_{{{u}_{ - }}}^{{{u}_{ + }}} {\frac{{du}}{{{{u}^{2}}}}} \frac{1}{{\sqrt {(u - {{u}_{ - }})({{u}_{ + }} - u)} }} = \pi \frac{{{{u}_{ - }} + {{u}_{ + }}}}{{2{{{({{u}_{ - }} - {{u}_{ + }})}}^{{3/2}}}}}, \\ \int\limits_{{{u}_{ - }}}^{{{u}_{ + }}} {\frac{{du}}{u}} \frac{1}{{\sqrt {(u - {{u}_{ - }})({{u}_{ + }} - u)} }} = \pi \frac{1}{{\sqrt {{{u}_{ - }}{{u}_{ + }}} }}, \\ \end{gathered} $$

$$\begin{gathered} \int\limits_{{{u}_{ - }}}^{{{u}_{ + }}} {du} \frac{1}{{\sqrt {(u - {{u}_{ - }})({{u}_{ + }} - u)} }} = \pi , \\ \int\limits_{{{u}_{ - }}}^{{{u}_{ + }}} {} duu\frac{1}{{\sqrt {(u - {{u}_{ - }})({{u}_{ + }} - u)} }} = \frac{\pi }{2}({{u}_{ - }} + {{u}_{ + }}). \\ \end{gathered} $$