The complete solution of the Yang-Mills equations for centrally symmetric metric in the presence of electromagnetic field

Abstract

Previously, we found the complete solution of Yang-Mills equations for a centrally symmetric metric in 4-dimensional space of conformal torsion-free connection in the absence of the electromagnetic field. Later, in another article, we found a solution of the Yang-Mills equations for the same metric in the presence of an electromagnetic field of a special type, suggesting that its components depend not on the four, but only on two variables. There we compared the resulting solutions with the well-known Reissner-Nordstrom solution and indicated the reason why these solutions do not match. In this paper, we do not impose any prior restrictions on the components of the electromagnetic field. This greatly complicates the derivation of the Yang-Mills equations. However, all computational diffculties were overcome. It turned out that the solutions of these equations all the same depend only on two variables and new solutions, in addition to previously obtained, do not arise. Consequently, we have found all the solutions of the Yang-Mills equations for a centrally symmetric metric in the presence of an arbitrary electromagnetic field, agreed with the Yang-Mills equations in the torsionfree space (i.e., without sources). These solutions are expressed in terms of the Weierstrass elliptic function.

Keywords

• curvature of the connection
• hodge operator
• einstein equations
• maxwell's equations
• yang-mills equations
• centrally symmetric metric
• weierstrass elliptic function
• 4-manifold with conformal connection