Logical Analysis of Schwarzschild's Solutions (1)

In this article, I want to compare the different treatments of the issue of the exact solution of the Einstein equation in terms of its logical structure and to clarify the relationships between the results reached by these treatments. This comparison should be accompanied by a criticism of some of the contents of these treatments, especially the incorrect application of the principle of general covariance . It is advisable to start first with the general framework of the solution, which is agreed upon by everyone who dealt with this issue, starting with Schwarzschild, then Droste and Hilbert, and everyone who came after them. Understanding this general framework is very important in this comparison because it distinguishes for us between the real areas of disagreement that we are looking for and the detailed mathematical issues that do not matter for physicists very much like the different methods for solving differential equations and the mathematical tricks used to simplify certain processes. A Schwarzschild Solution (in its general sense, which includes the modified solutions that came later)begins by writing down the functions according to which the parts of the metric are distributed around a center in which there is a mass. These functions are originally 16 functions, but after we apply to them the rules of general symmetry and then the conditions specific to our problem, which are spherical symmetry and stationary conditions we reach a small number of unknown functions, so we try to determine them by applying Einstein's equation and our physical information about the gravitational field. Then, if that is not enough to determine the unknown functions, we ask for help from other sources, which are the ones in which the treatments differed, and let us call them "auxiliary sources". Based on this, we can divide the different treatments of this issue into three sections: The first: Schwarzschild's treatment, the auxiliary source of which was an intelligent guess (but there is no guarantee that it will be correct) of the location of the singularity. Schwarzschild used this source and arrived at a specific solution ( Not that one under his name found in textbooks but the one found in his original paper). Second: The treatment of Droste, Hilbert and others, which is still present in important references by great physicists such as Dirac and Weinberg. The auxiliary source in these treatments is (unfortunately and surprisingly) the incorrect application of the principle of general covariance to the laws of physics. In this treatment, one of the unknown functions is eliminated in the following way: Suppose that one of the unknown functions s f(r), so if you replace it with r , then what you did will not change the problem or its form because you only changed the reference system, and this is correct, but what is not correct is to think that we thus have gotten rid of one of the unknowns. In fact, in this case, we transferred our ignorance of the function to the meaning of variables that we are talking about . After we continue solving the equations and reach the final solution, we find that it is written in quantitative terms similar in symbol to the radius that we all know, but in reality (and this is what the references that rely on this method do not say) it is only unkown function in radius. The third section: The treatment that does not need any auxiliary source, as it takes advantage of all the conditions of spherical symmetry more efficiently than the Schwarzschild and Hilbert treatment, so that the number of unknown functions is only two. This treatment can be found in many modern scientific papers, but I have not been guided and have not yet searched for its first source. The conclusion that we will detail in future articles, God willing, is that Schwarzschild’s solution is completely logically sound, but its result is wrong because his guess about the location of the singularity was not correct, and that Droste and Hilbert’s solution has a problem with incorrect use of the covariance , but nevertheless the result they arrived at is correct. The correct solution in terms of logic, physics of relativity, and the final result is the last section.