IL NUOV0 CIMENT0 VOL. 91A, N. 2 21 Gennaio 1986

New Solutions of Einstein-Yang-Mills Field Equations (').

S. SLNZINKAYO

Universit~ du Burundi, D~partme~t de Physique, Faculhi des Sciences B.P. 2700 - Bujumbura, Burundi

(ricevuto il 28 Ottobrc 1985)

Summary . --- Using the Corrigan-Fair l ie- ' t Hooft-Wilczek ansatz, we find now solutions of the Einstein-Yang-Mills field equations. These solutions depend on the Lorentz invariants v = k~ x ~, k being a constant and uni- form four-vector.

PACS. 11.10. - Field theory.

1 . - I n t r o d u c t i o n .

I n a r ecen t w o r k (1), we h a v e shown h o w some k n o w n cosmolog ica l so lu t ions

of o r d i n a r y E i n s t e i n f ie ld e q u a t i o n s m a y be g e n e r a t e d b y a Yang-Mi l l s gauge

field. I n such an a p p r o a c h , s p a c e - t i m e c o n f o r m a l g r o u p a n d some of i ts m a x i m a l

s u b g r o u p s were exp lo i t ed . The Yang-Mi l l s fields s t u d i e d in t h a t w o r k (1) were

a s s u m e d to be o b t a i n e d v i a t h e C o r r i g a n - F a i r l i e - ' t H o o f t - W i l c z e k (CFHV~')

a n s a t z (2). L e t us r eca l l t h a t some of t h e s e so lu t ions of t h e E i n s t e i n - Y a n g -

Mills f ield e q u a t i o n s wexe i n t e r p r e t e d as h a v i n g a p h y s i c a l m e a n i n g on ly in

t h e c o n t e x t of s t r o n g g r a v i t y (3). I n p a r t i c u l a r , th i s was t h e case for t h e E i n s t e i n

mic roun ive r se .

(*) To speed up publication, the author of this paper has agreed to not receive the proofs for correction. (i) S. $INZINKAYO and J. DEMAR:ET: Gen. Rel. Grav., 17, 187 (1985). (a) E. CORRIGAN and D. B. FAIl,LIE: Phys. I~ev. B, 67, 69 (1979); G. 'T HOOFa': un- published (1977); F. WILCZ};K: in Quark Confinement and Tield Theory, edited by D. S~uMP and D. W]~INGARTX~" (J. Wiley, .-%'ew York, 1~. Y., 1977). (3) G. SIVARA.~I and K. P. SI.~'HA: Phys. ~ep., 51, 112 (1979).

174

NJ~W SOLUTIONS OF :EINST]!IIN-YANG-MILL8 :FI:ELD EQUATIO:NS 175

We want now to derive new solutions of the Einstein-Y~ng-Mil]s field equations by assuming tha t the scalar field used in the CFtt-W ansatz depends

on the Lorentz invariants u = x ~ or v z k . x = k~x ~, k being a constant and uniform four-vector. These two variables were recently used in the s tudy of

the solutions of r Einstein field equations coupled to a massless scalar field (*) on the one hand, or the determination of the vacuum solutions (~) of massless scalar, Yang-Mills and Einstein field equations on the other hand.

We determine, in sect. 2 of this note, all the solutions of the Einstein- Yang-~Vlills field equations which depend on the variable v. In sect. 3, we give an ordinary differential equation which has to be satisfied by the eonformal factor, if the space-time dependence is defined by u = x% Some solutions of

such an equation are given. Lastly, sect. 4 is devoted to the concluding remarks.

2. - Scalar f ield depending on the var iable v.

Let us consider a massless scalar field ~v(x), which is ~ssume4 to be ~ solution of the wave field equation

(1) D~(x) + ~ ( x ) = 0

on the ]Vfinkowski space. Then, it is well known tha t the C F H W ansatz may be

used to generate SU2 gauge potentiaIs (2)

(2) At,(x) = i a , ~ q J ( x ) e ~ ( x ) '

solutions of the Yang-l~Iills field equations

~ F ~ , ~- e [ A ~ F~ ~] = 0 (3)

with

(4) F ~ ~ ~ A ~ - - ~A~, + e[Az, A~] ,

where e is the gaugo coupling constant. The energy-momentum tensor asso- ciated with this Yang-Mills field is given by (6)

(5) e~O~(x) DqD 2 ~ O , qD + ~ , ~q~ -- O~,qD~q~

where we use for the metric tensor the signature ( A - - - - - ) .

(4) J. B~CK~Rs, S. S~ZINKAYO and J. DISMAleST: Phys. ~ev. D, 30, 1846 (1984). (5) S. SI~ZI~KAYO ~nd J. D:EMARET: ~e~t. ~uovo Cimento, 39, 239 (1984). (8) A. ACTOR: l~ev. Mod. Phys., 51, 461 (1979).

176 s. SI~ZlXKAYO

Let us now replace the )/Iinkowskian background by a conformally flat space-time

(6) guy(x) = 92(x)~u~

and assume tha t ~ is the same scalar field as in the ansatz (2). Consequently in the new space-time, we have to replace eq. (3) by

(7)

where

(s)

~ [ ( - - g)+ F~*] + e(-- g)+[A~, F.~] ---- 0 ,

Fs'~(x) = g~d ' (x)g '~(x)F~,~(x) .

In addition, the new energy-momentum tensor ~akes the form

(9 ) - Y ~ - 2 Y ~ 0~,~ (x) = ~ (x)O.~ (x) ,

where Ou~(x) is given by eq. (5). We want now to search for solutions of the Einstein field equations, coupled

with the Yang-~Sills field, namely

(10) R~,,(x) - - �89 + Ag~, ,(x) = - - z O ~ ( x ) .

I f the scalar field in eq. (6) depends on the variable v = kax ~, then the cor- responding Ricei tensor and scalar curvature are, respectively, given by (5)

2k~ k~ k2 ~u~ R ~ ( x ) -- 9~ [gq~"-- 2(~') ~] + - ~ - - [ ~ " + (9')~] (11)

and

(12) R - - 6 k " 9 " 93 '

where the derivatives are taken with respect to the variable v. The energy- momentum tensor (5) takes now the form (5)

(13) v~ 2k2

e 0~ (x) ~- -~- {k2~?,~[~ 2 - (~o') 2] -- 2k~,k,[q~cf"-- 2(~')2]}.

Using eqs. (5) and (9) on the one hand, (11) and (12)on the other hand, we find t ha t (10) is fulfilled providing t ha t (1)

2~A (14) e 2 =

3

N E W SOLUTIONS OF ~INST:EIN-YA:NG-MILLS FIELD :EQ~_IATIONS 177

x traceless) and (we t ake b = 1 and use the fac t t h a t A = R/4 since 0~ ( ) is the scalar field ~ sa t i s fy ing the fol lowing o rd ina ry differential equa t ion :

(15) F " - - m 2 ~ 3 : 0 ,

where

A -k k 2 (16) m ~ - - - -

2k 2

Le t us dis t inguish different cases cor responding to the l ightl ike ( k 2 = 0) or nou-l ight l ike (k2V= 0) cha rac te r of k.

a) I f k s = 0, t h e n the va lue of m is unde te rmined . This is easi ly seen

f r o m eq. (16) and the fac t t h a t A = R/4 implies t h a t A = 3k 2. I n this case, f r o m eq. (15) we deduce b y a s imple in t eg ra t ion t h a t

m 2 ~04 • A ~ (17) ~'~ - - 2 '

• A4/2 being a r b i t r a r y cons tan ts . Consequent ly , we have , for nonzero A 4,

the fol lowing solut ions (7) (note t h a t we have the same differential equa t ion as in this reference, bu t wi th different a r g u m e n t ) :

(is)

(19 )

and

i) for + A ~

ii) for - - A 4

A Vl--enz ~ = • -- cnz '

A Vll-~- cn2 ~ = • ~ - c n ~

A (20) q - - x / ~ e n d '

where

( 2 1 ) Z

and cn Z is the J a c o b i a n elliptic funct ion . I n the pa r t i cu la r case in which the in tegra t ion consSant A 4 is zero, t h e n we

(~) Tm HULEIItIL: Int. J. Theor..Phys., 24, 571 (1985).

178

have

(22)

with m~ = m~/2.

(23)

F r o m eq. (22), we deduce i m m e d i a t e l y t h a t

+ 1 ~P-- • ~

8 . S I N Z I N K A ~ Y O

d being ~n a r b i t r a r y cons t an t of in tegra t ion . Le t us no te t h a t a s imple solut ion is easi ly o b t a i n e d in the pa r t i cu la r case

in which m = 0 in eq. (16). In fact , in this case, we have , as a solut ion of eq. (15) :

(24) cf =- clv • e8,

where el and c2 are a r b i t r a r y cons tan t s .

b) I f k 2 # 0 (k is t imel ike or spaeelike) we ob ta in f r o m eq. (16) w i th

(25) A ---- 3k 2 t h a t m 2 ---- 2 .

Consequent ly , in this case, we recover solut ions (18)-(20) and (23)-(24) b u t wi th condi t ion (25). P u t t i n g these solut ions in eqs. (2) and (6), t h e n we ob ta in solut ions of the coupled E i n s t e i m Y a n g - ~ i l l s field equat ions . I n fac t , when

the fou r -vec to r k is no t l ightl ike, ~hen the scalar field (23) is associa ted wi th the two de Si t ter space- t ime metr ics (5).

3. - S c a l a r f i e ld d e p e n d i n g o n t h e v a r i a b l e u.

Let us n o w consider the case in which the scalar field ~(x) in eqs. (2) and (6) depends on the var iab le u = x 2. On the one hand , we deduce f r o m eq. (5) t h a t (5)

(26) e 2 Y M 0.~ (x) = - - 2 ~ [ ~ " - - 2(~')~][x~v.~-- 4Gx~] ,

where the der iva t iones are now t aken wi th respect to the new var iable u = x 2.

On the o ther hand, the Ricci t ensor and the scalar c u r v a t u r e take , respect ively, the fo rm

(27) 4 12T' (p,

R ~ = ~ [ ~ " - - 2 ( T ' ) 2] [2x,,x~ + uW~ ] + 7 - [ ~ + ~] n,,

]flEW SOLUTIONS OF :EINSTEI~-YA~,G-~MILLS F I E L D :EQWATIONS 179

and

(28) 24 R = ~ , [u~'~+ 2 r

T-

Using eqs. (26)-(28) and tak ing into account relat ion (9), i t is easy to check tha t the Einsteiu-Yang-Mills equat ions (7) and (10) are satisfied, providing t ha t

e 2 (29) u _

and

(3o) A~ 3 - 6(u?" + 2q') = 0 .

Equa t ion (30) admi t s in par t icu lar the two de Si t ter cosmological models, bu t in this case condit ion (29) is not necessary (s).

Le t us note tha t , if we require the cosmological constant to be zero, then we have the simple condition

(31) u~o" + 2 ~ 0 ' : O.

By integrat ion, we immed ia t e ly obtain the solution

(32) q~ - - ~- e ~ ,

where cl and e2 are a rb i t r a ry constants. I f we pu t re lat ion (32) in eq. (6), we get a curved space- t ime with a vanishing scalar curvature . The par t icular case in which c2 ~ 0 is associated with a flat space-t ime.

4. - Concluding remarks.

Using the gaeobian elliptic functions (18)-(20) in the CFIt 'W ansatz (2), we obta in new solutions of the coupled Einstein-Yang-Mills field equations. I n fact , the space- t imes obta ined b y pu t t ing the scalar field (18)-(20) in eq. (6) have a constant scalar curvature . Le t us note t h a t this scalar curva ture is positive, negat ive or zero according to the fact t h a t the four-vector k is t imelike, spaeelike or lightlike.

In the last case, condition (14) is not necessary, since Y~ 0,v (x) = 0 f rom eq. (13). I n fact , the Einste in equat ions are reduced to

2k, k~ (33) R~,(x) - - - - [~0~0"-- (2~0') ~] = 0 . q2

180 s. SI~ZlXKAYO

There fo re , we m u s t r equ i r e t h a t

(3~) q ~ " - - 2(~'): = 0

a n d t h e so lu t ion is

b (35) q) - - k ~ x ~ '-- n ' k " = O.

This sca la r f ichl g ives a so lu t i on of Yaug-Mi l l s f ie ld e q u a t i o n s in f ia t space-

t i m e , d i scussed b y OI~ a n d T) :n (B).

L a s t l y , le t us r eca l l t h a t t h e so lu t ions d i scus sed here h a v e a lahysical

m e a n i n g on ly in t h e c o n t e x t of s t r o n g g r a v i t y , s ince t h e cond i t i ons (14) a n d

(29) a re i n c o m p a t i b l e w i t h t h e o b s e r v e d cosmolog ica l c o n s t a n t of t h e o r d i n a r y

g e n e r a l r e l a t i v i t y .

The a u t h o r w o u l d l ike to t h a n k Prof . A]3])vs SAI,AM for h o s p i t a l i t y a t t h e

I n t e r n a t i o n a l Cent re for T h e o r e t i c a l P h y s i c s , I t a l y , where th i s w o r k was com-

p l e t ed .

(8) C. II. Ou and R. Tx~n: P h y s . Le t t . B , 87, 83 (1979).

@ RIASSUNTO (')

Usando l 'ausatz di Corrigau-Fair l ie- ' t Hooft-Wilczek, si t rovano nuove soluzioni allc equazioni 4i campo di Einstein-Yang-Mills. Queste soluzioni dipendono dagli invar iant i di Lorentz v = k a x ~, 4ove k b u n quadrivet tore costantc e uniforme.

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