Combined Calculation of the Yang-Mills Equation and Einstein Field Equation

In this paper, we present a comprehensive analysis and calculation of the combined Yang-Mills equation and Einstein field equation. These equations are two of the fundamental building blocks of modern physics. The Einstein field equation describes gravity as the curvature of spacetime, while the Yang-Mills equation describes the dynamic interactions in quantum field theory. By combining these two equations, we aimed to explore a deeper connection between general relativity and quantum field theory. The calculations include symbolic and numerical evaluations of both equations. First, the Einstein field equation was expressed in matrix form to compute the curvature of spacetime and its relationship to the distribution of matter. Subsequently, we symbolically computed the Yang-Mills equation to describe the interactions of the fields. These symbolic expressions were then combined to obtain an extended equation that integrates the effects of the Yang-Mills fields on the curvature of spacetime. Through numerical calculations, the terms of both equations were evaluated using real physical constants. The Einstein field equation demonstrated that the cosmological constant has the most significant influence on the curvature of spacetime, while the gravitational constant, due to its minuscule size, has only a minor effect. The Yang-Mills equation provided numerical values representing the interactions of the fields, particularly regarding the dynamics of gluons in the quark-gluon plasma. Combination of the Yang-Mills Equation with the Einstein Field Equation: By combining both equations, we were able to conduct a detailed analysis of the interactions of quantum fields with the curvature of spacetime. This combination is of particular interest as it could potentially provide new insights into the structure of the universe, dark matter, and the dynamics of particles in extreme gravitational fields. Central Formula of the Work: G_{\mu\nu} + \Lambda g_{\mu\nu} = \frac{8\pi G}{c^4} T_{\mu\nu}, \quad D_{\mu}F^{\mu\nu} = J^{\nu} The numerical calculations were verified using various validation methods, including NumPy, absolute error validation, and the math.isclose function. All calculations passed these tests, confirming the accuracy of the symbolic and numerical expressions. Notably, the calculation of the eigenvalues of the Yang-Mills field strength tensor provides indications of the presence of a mass gap. This is crucial for understanding the behavior of quarks and gluons in extreme states, as they occur in particle accelerators or in the early universe. Our results suggest that the combination of the Yang-Mills equation with the Einstein field equation could provide new insights into fundamental physical processes. This work lays the foundation for future investigations, particularly regarding the simulation of complex interactions and their experimental validation in high-energy experiments.