Russian Physics Journal, Vol. 38, No. 2, 1995

PHYSICS OF ELEMENTARY PARTICLES AND FIELD THEORY

N O N T R I V I A L S O L U T I O N S O F T H E E I N S T E I N - Y A N G - M I L L S

E Q U A T I O N S F O R A S Y S T E M O F M A S S I V E M O N O P O L E S

L. M. Chechin UDC 530.12:531.51

Nontrivial solutions are obtained for the Einstein-Yang-Mills equations, as applied to a system of N slowly

moving massive monopoles with a linear accuracy relative to the interaction constant.

The idea of colored black holes proposed by Perry [1] stimulated two directions of study: 1) finding nontrivial solutions

of the self-consistent system of Einstein-Yang-Mills (EYM) equations (see, for instance, [2-4]), and 2) analyzing the motion of non-Abelian particles in external EYM fields (see, for instance, [5, 6]). Moreover, the specific representation of a chromogravitational system of bodies in models of quark stars [7] and diquark stars [8], in models of strange pulsars [9], in the theory of a hot universe [10], etc., makes it possible to relate these directions and to pose the problem of the motion of a system of N colored black holes in self-consistent EYM fields (in a similar formulation, this is the same problem that was

dealt with in synopsis [11]). An important part of this problem is finding the potentials of the Einstein-Yang-Mills equations generated by a

system of slowly moving sources. Here we will cite some nontrivial solutions of the EYM equations for a system of massive

monopoles in the first relativistic approximation, with a linear accuracy relative to the interaction constant. The general-covariant Yang-Mills equations, described in terms of potentials and supplemented by Lorentz gauging,

can in the adopted notatJ[on be written as

[]AL 0, = 2gstJ~ A'~ O, Ay~ + gjyX A/O,A~., - -

- - F~,, (0~ At" - - O" A~-~ + g~U. Ay, A~,) +

+ F~, (0~ Af" -- O" At~ + g~J~ Az~A~, ) -- 4~j~.

(1)

These equations are solwed using the method of approximations, all the quantities being expanded in series in powers of k =

v /c and the current being specified as well. In [12] it was shown that, satisfying the formula of Gell-Mann and Nishijima, a chromoelectric charge should be of

second order in X. Therefore, it follows from Wong's equations describing the motion of an isospin particle in external non- Abelian fields that a chromomagnetic charge q should be of the order of h. Consequently, for the problem at hand in the

general case the Yang-Mills potentials can be represented as series

= + + + . . .

A~ t A l_ ; = A ~ + 3 ° + A~ + . . . . (3) 2

t Now let us determine the form of the current tensor, keeping in mind the following: 1) potential A? as the principal 1

term must be static; 2) it must describe a field produced by a point distribution of colored charges. Thus

Astrophysics Institute, Kazakh Academy of Sciences. Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika,

No. 2, pp. 59-62, February, 1995. Original article submitted February 24, 1994.

1064-8887/95/3802-0151512.50 ©1995 Plenum Publishing Corporation 151

/ = = E ?,,o,, ( , , - a (4)

With regard to the timelike component, it is clear that

• i . i a a a a = ~ m (5) j~ = j~ "~, q~jdim (x - - t ) ,

• ~ 1 1

di

where ~,'~ is the space component of the 4th velocity of the a-th charge. 1

For a point distribution of massive monopoles, the lower order of the Riemann-Cristoffel symbols is proportional to

k 2. Therefore, the corresponding contributions to the potentials will be of third order. Thus, in the fundamental approximation from (1)-(4) we find the equations

A = - - 4 ~ tn'n ~ ~ ( x - - ~), a

a di dil l AA~= - - 4=Jn'n ~ q n,,, ~,o ~ ( x - - D.

= (7) a

Their solutions are:

di

q i ~inm ~ 1 a A a = - - ~ , , , , (8)

1 " I x - ~ l di

q a l l gt 1 ~ ~inm ~ 1 ~,0 7Ira.

A~ ~ Ix - -{ I ' (9)

a

The potentials obtained for g,~ = 0 describe the field produced by a system of monopoles of the Wu-Yang type. 1

Now let us calculate the potentials in the next (first relativistic) approximation. From (1) we get the equation of the

field

-- A,O,,A,a = J:~, I-- 1 - -

aA}a = -- 2e~'J" A'~, _ O, a:o=_ - g,':~ AI,_ O? A_~, = I~. ~''

(10)

(11)

Next, using (8) and (9) to calculate the self-action current of the Yang-Mills field f~ and substituting it into (10) and (11),

we arrive at the solutions of these equations:

a b

A : , qq ) __ ~ijK ~Klr g]nS : I I 7~s ~_

2 n 2~¢ a a b b

a b

qq dib( a ) Jr 4~ g-- eqr $] lr S KIs ~ t al la b l~r

(12)

and

152

a b

q q ,~ A I .~_ g $ t j x s g l r f f j s l o ~.z¢ - .... 1 1

2~ ~ - o ~' "~'~-- a , ~ l x - a l l ~ - q k l ~ - q / , ~ , '

a b

qq ao ~q., t, a (g

(13)

Similarly, on the basis of (12) and (13) and with the aid of the explicit form of the Riemann-Cristoffel symbols in

the Infeld method, it is easy to find the Yang-Mills potentials in the post-Newtonian approximation. However, we will not

give them here, because the results are extremely cumbersome.

Now let us find with the required accuracy the solutions of the Einstein equations with the corresponding right-hand

side. For the tensor densities they have the form

where

(14)

(15)

is the tensor of the Yang-Mills field. Using expansion (2) and the standard notation of the Infled method, from (14) and (15)

it is easy to obtain the equations of the gravitational field in the Newtonian approximation in harmonic coordinates

_ I hoo u = - - 2 = - ; (O,n A ! - - O, A*--~ "~ =too. (16) 2 2 ' 1 l 1 m* 2

By combining (8) and (16), we arrive at the explicit form of the field equations

A/Zoo = 4 ~ " ~ q ~'='J I q ~*'~' -- _~'~* 8(x. -- ~).

\ I x - q / , ~ \ I x - ~1 l , , . J

(17)

Solution of these yields the following simple correction to the zeroth component of the metric tensor:

a b

oo=- Z x

ab ab

\1~-- ~1/' an

(18)

In the first re, lativistic approximation the Einstein equations (14) can be written as

1 boo u = _ 4 ~ . f (OmA_~ --e ,A~)(O, .AI - O , AL)= too. 2 ' 2 I l I 3 (19)

Now, we introduce potentials (8) and (12) to obtain the Poisson equation with a source too.. Its solution is already quite

complicated: a

a b c

3 .~ ~ rZ a a , ; I x - al [ \I, - ~1 ),~m \ [ ~ _ ~l/,a,J

153

a b a c

t7 a C a

a b a c a c

+ 2 " l ~ - ~ l t q - . - ~ I / ' ~ " Ia-'--.l '~.' \l~-a!/'~'kt~-~l '~" a c a *

(20)

Expressions (8), (9), (12), (13), (18), and (20) are the desired nontrivial solutions of the Einstein-Yang-Mills equations for a system of N massive monopoles.

REFERENCES

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2. 3. 4.

5.

,

7.

8. 9.

10. 11.

12.

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