Einstein/Yang-Mills Equations

Abstract

In this paper, we shall describe some recent results, ([6,7]), concerning the existence of smooth, globally defined static solutions of the Einstein /Yang-Mills (EYM) equations. The EYM equations extend Einstein’s celebrated gravitational field equations, to include other “force” fields: electromagnet ism, and the weak and strong nuclear force fields. It turns out that solutions of such coupled systems of equations, in n = 4 space-time dimensions, have some very interesting mathematical and physical properties. Thus for the non-abelian gauge group G = SU (2), we can prove the existence of infinitely many non-singular globally defined solutions - a result that fails to hold if G = U (1), (the coupling of gravity to electromagnetism). Since the (classical) Yang-Mills equations with G = SU (2) correspond to the weak nuclear force, our result indicates that coupling gravity to the weak nuclear force can prevent singularity formation in space-time. Furthermore, these solutions give rise to constants which correspond to masses. In other words, numbers representing masses are a direct consequence of our theory, and need not be prescribed in an ad-hoc manner - as is usually done in physical theories.

Whoever does not from the beginning reject the hypothesis that molecular (small scale) gravitational fields constitute an essential part of matter, will see in the following a strong support for this point of view.

A. Einstein (1915)