Smooth Static Solutions of the Einstein-yang/mills Equation

We consider the Einstein/Yang-Mills equations in 3 + 1 space time dimensions with SU(2) gauge group and prove rigorously the existence of a globally defined smooth static solution. We show that the associated Einstein metric is asymptotically flat and the total mass is finite. Thus, for non-abelian gauge fields the Yang/Mills repulsive force can balance the gravitational attractive force and prevent the formation of singularities in spacetime. 1 The only static, i.e., time independent, solution to the vacuum Einstein equations for the gravitational field R¡j \Rgij = 0 is the celebrated Schwarzschild metric that is singular at r = 0 [1]. Despite this defect, this solution has applicability for large r to physical problems, e.g., the perihelion shift of Mercury. Similarly, the Yang/Mills equations d*F = 0, which unify electromagnetic and nuclear forces, have no static regular solutions on R4 [3]. Furthermore, if one couples Einstein's equations to Maxwell's equations, to unify gravity and electromagnetism (1) Ru \R8ij = oTij, d*F = 0 ( Tij is the stress-energy tensor relative to the electromagnetic field Fy ), the only static solution is the Reissner-Nordström metric, which is again singular at the origin [1]. Finally, the Einstein-Yang/Mills (EYM) equations, which unify gravitational and nuclear forces, were shown in [4] to have no static regular solutions in (2+1) space time dimensions for any gauge group G. We announce here that the contrary holds in (3+1) space-time dimensions. Indeed, with SU(2) gauge group (i.e., the weak nuclear force) we prove that the EYM equations (cf. (1), where now F¡j is the su(2)-valued Yang/Mills field), admit Received by the editors November 5, 1991 and, in revised form, January 29, 1992. 1991 Mathematics Subject Classification. Primary 83C05, 83C15, 83C75, 83F05, 35Q75. The first author's research was supported in part by NSF Contract No. DMS-89-05205 and, with the second author, in part by ONR Contract No. DOD-C-N-00014-88-K-0082; the third author was supported in part by DOE Grant No. DE-FG02-88ER25065; the fourth author was supported in part by the U.K. Science and Engineering Council. ©1992 American Mathematical Society 0273-0979/92 $1.00+ $.25 per page 239 240 J. SMOLLER, A. WASSERMAN, S. T. YAU, AND B. McLEOD nonsingular static solutions, whose metric is asymptotically flat, i.e., Minkowskian. (Strong numerical evidence for this conclusion was obtained by Bartnik and McKinnon [2] who also derived the relevant equations.) Thus for nonabelian gauge fields, the Yang-Mills repulsive force can balance gravitational attraction and prevent the formation of singularities in spacetime. Viewed differently from a mathematical perspective, it is the nonlinearity of the corresponding Yang/Mills equations that allows the existence of smooth solutions. The EYM equations are obtained by minimizing the action