P H Y S I C A L R E V I E W D V O L U M E 1 2 , N U M B E R 6 1 5 S E P T E M B E R 1 9 7 5

Gravitating 't Hooft monopoles*

Y. M. Cho and P. G. 0. Freund The Enrico Fermi Institute and the Department of Physics, The University of Chicago, Chicago, Illinois, 60637

(Received 12 May 1975)

The 't Hooft magnetic monopole solution of the Sq3) Higgs model generates, when coupled to gravity, a geometry which for large distances is of the Reissner-Nordstrom form corresponding to a (magnetic) charge I/e. The Higgs fields can contribute a cosmological term. In the absence of scalar fields the corresponding Wu-Yang solution of the SO(3) gauge theory still generates the Reissner-Nordstrom geometry.

Wu and Yangl have found classical solutions of the SO(3) isospin gauge theory. Similar solutions appear a l so in the Higgs-type model that couples a n isotr iplet of s c a l a r fields to the isospin gauge fields. There ' t Hooft has shown2 that they cor - respond t o Abelian magnetic monopoles (the cor - responding magnetic charge being conserved on topological grounds3). The question we want to answer h e r e is what happens when this sys tem i s fur ther coupled to gravitation.

F o r the monopole case, the SO(3) gauge fields approach for l a rge dis tances precisely one of the Wu-Yang solution^.^ By a singular gauge t rans- formation3 this Wu-Yang gauge field can be brought to point in a given, say the third, isospace direc- tion. In this gauge a l so the Higgs field points in the third isospace direction, s o that the theory behaves a t l a rge distances (where these consider- ations apply) a s if i t w e r e Abelian.3 A s f a r a s the space dependence i s concerned, in this "Abel- ian" gauge, a t l a rge distances2 the surviving Higgs field i s constant and the surviving isocomponent of the gauge field i s precisely equal to the vector potential corresponding to a usual Abelian Dirac magnetic monopole of magnetic charge l/e.3 If we now switch on gravity, the constant Higgs field will contribute a cosmological t e rm, whereas the magnetic monopole will have an energy-mu- mentum tensor of the s a m e form a s for a n electr ic charge with value l / e . One therefore must obtain fo r large dis tances a Reissner-Nordstrom geom- e t ry corresponding to a (magnetic) charge l / e . The energy-momentum tensor and the gravitational field being invariant under local SO(3)-gauge t rans - formations, the s a m e Reissner-Nordstrom geom- e t ry must appear fo r large distances in every gauge, in part icular in the original gauge in which a t l a rge dis tance the gauge fields were of the Wu- Yang form. We shal l check this by a d i rec t cal- culation below. Concerning the cosmological t e r m , it can be removed by hand, by adding an " i r re le - vant" constant to the original Higgs Lagrangian before coupling it to gravitation.

In the p rocess of treating this problem we shal l a l so find exact (i.e., valid a t a l l not only a t l a rge distances), albeit singular, solutions of the cou- pled gravitational-gauge-scalar sys tem which reduce to the corresponding known exact but sing- ular solutions of the non-Abelian Higgs theory without gravitation. The Reissner-Nordstram geometry without cosmological t e r m i s valid if one couples just the Wu-Yang solution to gravi ta- tion without introducing any s c a l a r Higgs fields.

F o r a s c a l a r isotr iplet of fields @" (a = 1 , 2 , 3 ) coupled to SO(3)-gauge fields and to gravitation, consider the general-relat ivis t ic Higgs Lagrangian

Here V,$' = a , , p a +i?f ; ,~$ @ C i s the gauge and generally covariant der ivat ive of the Higgs field, while a l l the other quantities have their usual meanings, and the met r ic has s ignature - + + + .

We a r e interested in spherically symmetr ic s tat ic solutions fo r which, in Schwarzschild coor- dinates, the s c a l a r and vector fields have the f o r m

( @ I , q2 , (ti3) =S( r ) ( s inOcos@, s i n e sin$, costi),

while the met r ic corresponds to the general Schwarzschild line element

(ds) ' = -A(v) (d t )'+ B( r ) (dr ) ' + ~ ~ ( d 8 ) ~

The s c a l a r and vector field equations take the fo rms (pr ime indicates derivat ive with respect to v )

12 - G R A V I T A T I N G ' t H O O F T M O N O P O L E S 1589

Independently of the gravitational field [ i .e . , of A ( r ) and B(Y)] they admit fo r l a rge dis tances (up to exponentially damped t e r m s ) the 't Hooft-Wu- Yang-type solution

1 y = - , s = p / a . e

(4)

Equations (4) even provide an exact solution a t a l l distances of the field equations (3) . These solutions, however, yield fields singular a t the origin which in the absence of gravitation cor - respond to infinite energy. They a r e therefore l ess interesting. All the following considerations a r e , fo r that mat te r , valid a t a l l dis tances with these qualifications. In addition to solution (4) our considerations a l so apply to a pure Wu-Yang field without any s c a l a r fields whatsoever.

With the solution (4), the gravitational field equations can be reduced to the fo rm

where p 2 4 ~ . (9.3) K=G/(e2/4a), A = - 8 n G V o , V o = - - -

These equations have the general solution

Here C and 2G1Mare integration constants. C can be determined, in the s tandard way, by requiring the only departure f rom asymptotic f la tness to originate in the induced cosmological t e r m (-A). This yields

M i s to be equated to the m a s s of the ' t Hooft mono- pole. We reca l l that fo r the physically interesting c a s e the met r ic takes the fo rm (6) only a t l a rge distances. The geometry [(6a) and (6b)] i s p re - cisely of the Reissner-Nordstrom f o r m (with cos- mological constant A ) . One can remove the in- duced cosmological t e r m by adding an explicit cosmological t e r m - A- to the Lagrangian (1). The geometry then becomes asymptotically flat. No induced cosmological t e r m appears in the ab- sence of s c a l a r fields (Wu-Yang case). Topological considerations ensure the quantization of the ' t Hooft magnetic charge already a t the c lass ica l level.3 In part icular for the s ingle monopole c a s e the charge i s not a r b i t r a r y but l / e even in the presence of gravitation.

T o conclude, we have found solutions of non- Abelian gauge theories coupled to gravitation4 and Higgs fields. The relevant solutions a r e of the ' t Hooft (or Wu-Yang) type for the vector and s c a l a r fields and of the Reissner-Nordstrom type for the gravitational field. The charge that de- termines the geometry i s the magnetic charge l / e , conserved for topological reasons.

*Work supported in part by the National Science Foun- dation, Contract No. MPS74-08833. 'T. T. Wu and C . N. Yang, in Propert ies of Matter Under

Umsual Conditions, edited by H. Mark and S. Fernbach (Interscience, New York, 1969), p. 349.

'G. 't Hooft, Nucl. Phys. x, 276 (1974).

3 ~ . Arafune, P. G . 0. Freund, and C . J. Goebel, J. Math. Phys. 16, 433 (1975).

4 ~ e also note that any solution of the Einstein-Maxwell system i s also a solution of the Einstein-Yang-Mills system for any group G in an Abelian gauge.