DOI: 10.5281/zenodo.13956847
title: Combined Calculation of the Yang-Mills Equation and Einstein Field Equation

description: 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.

 

 
description: This manuscript explores the symbolic and numerical calculations of the Einstein field equation as well as the Yang-Mills equation, both of which are fundamental aspects of theoretical physics. The work specifically focuses on the mass gap problem, one of the Millennium Problems in modern mathematics and physics. The calculations provide valuable insights into the interactions of gluons within the quark-gluon plasma and the confinement phenomenon in quantum chromodynamics. By combining the Einstein field equation with the Yang-Mills theory, the manuscript also opens new perspectives on addressing problems in quantum gravity.

 

"This manuscript explores the profound interactions between gravity and quantum fields, offering groundbreaking insights into the resolution of one of the greatest unsolved problems in physics: the mass gap problem in Yang-Mills theory. The numerical and symbolic calculations yield significant results that enhance our understanding of gluons, quark-gluon plasma, and confinement in quantum chromodynamics. Furthermore, this work proposes new approaches to unify general relativity and quantum field theory, potentially driving future research in quantum gravity."

 

Keywords:

Einstein field equation, Yang-Mills theory, mass gap problem, quantum field theory, gluons, quantum chromodynamics, high-energy physics, quark-gluon plasma, general relativity, cosmological constant, numerical calculation, symbolic calculation, gravitational waves, confinement, particle physics, renormalization, asymptotic freedom, black holes, Millennium Problems, mathematical physics, quantum gravity, Standard Model, LHC, proton-proton collisions, symmetry breaking, gluon interactions, eigenvalue calculation, quarks, cosmology, experimental physics, gauge theory.