APPEARED IN BULLETIN OF THE

arXiv:math/9210226v1 [math.AP] 1 Oct 1992RESEARCH ANNOUNCEMENTAPPEARED IN BULLETIN OF THEAMERICAN MATHEMATICAL SOCIETYVolume 27, Number 2, October 1992, Pages 239-242SMOOTH STATIC SOLUTIONSOF THE EINSTEIN-YANG/MILLS EQUATIONJ. Smoller, A. Wasserman, S. T. Yau, and B. McLeodAbstract.

We consider the Einstein/Yang-Mills equations in 3 + 1 space time di-mensions with SU(2) gauge group and prove rigorously the existence of a globallydefined smooth static solution. We show that the associated Einstein metric is asymp-totically flat and the total mass is finite.Thus, for non-abelian gauge fields theYang/Mills repulsive force can balance the gravitational attractive force and preventthe formation of singularities in spacetime.1The only static, i.e., time independent, solution to the vacuum Einstein equationsfor the gravitational field Rij −12Rgij = 0 is the celebrated Schwarzschild metricthat is singular at r = 0 [1].

Despite this defect, this solution has applicabilityfor 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’sequations to Maxwell’s equations, to unify gravity and electromagnetism(1)Rij −12Rgij = σTij,d∗F = 0(Tij is the stress-energy tensor relative to the electromagnetic field Fij), the onlystatic solution is the Reissner-Nordstr¨om metric, which is again singular at theorigin [1]. Finally, the Einstein-Yang/Mills (EYM) equations, which unify gravita-tional 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 thecontrary holds in (3 + 1) space-time dimensions.

Indeed, with SU(2) gauge group1991 Mathematics Subject Classification. Primary 83C05, 83C15, 83C75, 83F05, 35Q75.Received by the editors November 5, 1991 and, in revised form, January 29, 1992The 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 thirdauthor was supported in part by DOE Grant No.

DE-FG02-88ER25065; the fourth author wassupported in part by the U.K. Science and Engineering Council.c⃝1992 American Mathematical Society0273-0979/92 $1.00 + $.25 per page1

2J. SMOLLER, A. WASSERMAN, S. T. YAU, AND B.

McLEOD(i.e., the weak nuclear force) we prove that the EYM equations (c.f. (1), wherenow Fij is the su(2)-valued Yang/Mills field), admit nonsingular static solutions,whose metric is asymptotically flat, i.e., Minkowskian.

(Strong numerical evidencefor this conclusion was obtained by Bartnik and McKinnon [2] who also derivedthe relevant equations.) Thus for non-abelian gauge fields, the Yang-Mills repulsiveforce can balance gravitational attraction and prevent the formation of singular-ities in spacetime.

Viewed differently from a mathematical perspective, it is thenonlinearity of the corresponding Yang/Mills equations that allows the existence ofsmooth solutions.The EYM equations are obtained by minimizing the actionZ(−R + |F|2)√g dx,over all metrics gij having signature (−, +, +, +). These equations becomeRij = 2FikF kj −12|F|2gij.Here R is the scalar curvature associated to the metric gij and F is the Yang-Millscurvature.

These formidable equations become more tractible if we consider staticsymmetric solutions.2The problem of finding static, symmetric nonsingular solutions of the EYM equa-tions with SU(2) gauge group can be reduced to the study of the following systemof ordinary differential equations(2a)r2Aw′′ + Φw′ + w(1 −w2) = 0,(2b)rA′ + (2w2 + 1) = 1 −(1 −w2)2r2,(2c)2raT ′ + (2w′2A + Φ/r)T = 0.Here Φ(r) = r(1 −A) −(1−w2)2r, A and T are the unknown metric coefficients,ds2 = −T −2(r)dt2 + A−1(r)dr2 + r2(dθ2 + sin2 θdφ2), and w is the “connectioncoefficient” relative to the sought-for connection α = wτ1dθ+[cosθτ3+w sin θτ2]dφ,τ1, τ2, and τ3 being the generators of the Lie algebra su(2). The associated curvatureF −dα + α ∧α isF = w′τ1dr ∧dθ + w′τ2dr ∧(sin θdφ) −(1 −w2)τ3dθ ∧(sin θdφ).If < τi, τj >= −2trτiτj denotes the Killing form on su(2), and if |F|2 = gijgklFijFkl,then an easy calculation gives|F|2 = 2w′2/r2 + (1 −w2)2/r4.

EINSTEIN-YANG/MILLS EQUATIONS3Figure 1In order that our solution has finite mass, i.e., that limr→∞r(1 −A(r)) < ∞werequire that(3)limr→∞(w(r), w′(r)) be finite.Furthermore, asymptotic flatness of the metric means that(4)limr→∞(A(r), T (r)) = (1, 1).Finally, the conditions needed to ensure that our solution is nonsingular at r = 0arew(0) = 1,w′(0) = 0,A(0) = 1,T ′(0) = 0.One sees from (2) that the first two equations do not involve T . Thus we firstsolve these for A and w, subject to the above initial and asymptotic conditions.3We prove that under the above boundary conditions, every solution is uniquelydetermined by w′′(0); w′′(0) = −λ is a free parameter.

We seek a λ > 0 such thatthere exists an orbit (w(r, λ), w′(r, λ)) that “connects two rest points.” It is thennot very difficult to prove that (4) will also hold.A major difficulty is to show that the equations (2a), (2b) actually define anonsingular orbit; i.e., that w′(r, λ) is bounded and that A(r, λ) remains positive.Our first result isTheorem 1. If 0 ≤λ ≤1, then in the regionΓ = {w2 ≤1, w′ ≤0},A(r, λ) > 0 and w′(r, λ) is bounded from below.On the other hand, we can also prove (see Figure 1)Theorem 2.

If λ > 2, then the solution of equations (2a), (2b), with initial con-ditions (5) blows up in Γ; i.e., w′(r) is unbounded.If λ is near zero, then by rescaling we can show that the orbit (w(r, λ), w′(r, λ))exits Γ through the line w = −1. Furthermore, for λ = 1, numerical approximations

4J. SMOLLER, A. WASSERMAN, S. T. YAU, AND B. McLEODindicate that w′ becomes positive in the region −1 < w < 0.If this could beestablished rigorously, we could assert the existence of some λ, 0 < λ < 1, forwhich the corresponding orbit stays in Γ for all r ≥0, thereby proving (3).

Itwould then be possible to prove that(5)limr→∞(w(r, λ)), w′(r, λ)) = (−1, 0),and as a consequence, that (4) would also hold.4We can give a completely rigorous proof of the existence of a connecting orbitwith λ < 2, which we now outline. First Theorem 2 and the fact that for λ near0 the corresponding orbit exits Γ through the line w = −1 implies that there is asmallest λ = λ for which the orbit (w(r, λ), w′(r, λ)) does not exit Γ through thisline.

Thus only the following two possibilities can arise:(P1) There is a real number r > 0 such that either (a) w′(r, λ) = 0, or (b)A(r, λ) = 0, or (c) w′(r, λ) is unbounded for r near r.(P2) For all r > 0, w(r, λ) > −1, w′(r, λ) < 0, and A(r, λ) > 0.In the case that (P2) holds, we can show, as above, that both (6) and (7) hold.In order to rule out possibility (P1), we consider several cases. The crucial caseoccurs when A(r, λ) = 0, w′(r, λ) is unbounded near r = r, and Φ(r, λ) = 0.

Nowset w = limrրr w(r, λ). If w < 0, then defining v(r, λ) = (Aw′)(r, λ), we show thatv satisfies a first order ode, and we can prove that for λ < λ, λ near λ, there is anr = r(λ) such that v(r, λ) = 0 and w(r, λ) > −1.

This violates the definition of λ.Similarly, if w > 0, we can reduce this case to the previous one. Finally, the casewhere w = 0 is dealt with by extending our solution into the complex plane andusing the fact that the pair of functions (w(r), A(r)) = (0, 1 + 1/r2 −c/r) is alwaysa solution of (2a) and (2b).References1.

R. Alder, M. Bazin and M. Schiffer, Introduction to general relativity, 2nd ed., McGraw-Hill, New York (1975).2. R. Bartnik and J. McKinnon, Particlelike solutions of the Einstein-Yang Mills equations,Phys.

Rev. Lett.

61 (1988), 141–144.3. S. Soleman, In New Phenomena in Subnuclear Physics (A. Zichichi, ed.

), Plenum, NewYork 1975.4. S. Deser, Absence of static solutions in source-free Yang-Mills theory, Phys.

Lett. B 64(1976), 463–465.(J.

Smoller and A. Wasserman) Department of Mathematics, University of Michigan,Ann Arbor, Michigan 48109-1003E-mail address: Joel Smoller@ub.cc.umich.eduE-mail address :Arthur Wasserman@um.cc.umich.edu(B. McLeod) Department of Mathematics, University of Pittsburg, Pittsburgh,Pennsylvania 15260(S. T. Yau) Department of Mathematics, Harvard University, Cambridge, Mas-sachusetts 02198E-mail address: styau@math.nthu.edu.tw

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