The Weak Field Approximation of General Relativity, Retardation, and the Problem of Precession of the Perihelion for Mercury

In this paper we represent a different approach to the calculation of the perihelion shift than the one presented in common text books. We do not rely on the Schwarzschild metric and the Hamilton Jacobi technique to obtain our results. Instead we use a weak field approximation, with the advantage that we are not obliged to work with a definite static metric and can consider time dependent effects. Our results support the conclusion of Krizek \cite{Krizek} regarding the significant influence of celestial parameters on the indeterminacy of the perihelion shift of Mercury's orbit. This shift is thought to be one of the fundamental tests of the validity of the general theory of relativity. In the current astrophysical community, it is generally accepted that the additional relativistic perihelion shift of Mercury is the difference between its observed perihelion shift and the one predicted by Newtonian mechanics, and that this difference equals $43"$ per century. However, as it results from the subtraction of two inexact numbers of almost equal magnitude, it is subject to cancellation errors. As such, the above accepted value is highly uncertain and may not correspond to reality.Comment: 23 pages, 2 figures. arXiv admin note: substantial text overlap with arXiv:2108.0468