:EXACT ~0 s MAGNETIC-MONOPOLE SOLUTIONS ETC. 367. For this solution one may calculate the integral E =- 4=fV'~T0~ ' x, where To o is taken from the ...
L1'~TTERE AL NUOVO CIMENTO
VOL. 39, X. 15
14 A p r i l e
1984
Exact SO~ Magnetie-Monopole Solutions in the Expanding Universe. V. ~*..~,IELNIKOV a n d V. K. SIIIGOLEV USSR
State Committee/or
Sta.~,dard,s - _]Ioscow
(rieevulo 1'1 D i e e m b r e 1983) I'ACS. 98.80. - Cosmology.
The influence of g r a v i t a t i o n on classical magnetie-nionopole solutions of a nonA b e l i a n g a u g e t h e o r y (NGT) was i n v e s t i g a t e d in (i). L a t e r c o n s i d e r a b l e nutnbel" of l)ubli c a i i o n s a p p e a r e d in this field (el. ref. if.i), w h e r e t h e p r o b l e m was c o n s i d e r e d in t h e f r a m e s of t h e unified E i n s t e i n - Y o u n g - M i l l s - H i g g s model). A t t h e s a m e t i m e t h e e x i s t e n c e of e x a c t r e g u l a r NGT-soltttions of t h e m o n o p o l e t y p e in e x t e r n a l g r a v i t a t i o n a l fields is still an open q u e s t i o n , l h o u g h t h i s a p p r o a c h is a necessary s t e p to u n i f i c a t i o n of a g r a v i t a t i o n a l t h e o r y w i t h NGT (s). H e r e we p r e s e n t e x a c t m a g n e t i e - m o n o p o l e solntions of NGT in .'l. g r a v i t a t i o n a l backg r o u n d , n a m e l y in t h e S O a m o d e l of ele,c t r o w e a k interact.ions in the e x p a n d i n g universe. The F r i e d m a n - R o b e r t s o n - ~ V a l k e r m e t r i c is u s e d : (1)
d.s2 = - - (lt ~ ' - a~(t) {
dr2 1 ~--]~"r~ :- 1"2(d0Z :
sin ~ 0 d~ 2} ,
k := (i, .!: 1.
The La~rangitul is taken in the forni (2)
7, V(m) = 7 (~o '>- - / < ~ ) 2 ,
where
are Young-Mills and t t i g g s isotrilfiets. (1) F. A. BaI.q and 12.. J. I{I;SSEL: Phys. Rev. D, 11, 2692 (1975). (:) Y. M. Clio arid P. FREt'D: Phys. Rec. D, [2, 1588 (1975). (~) 3I. Y. \\-~xc,: Phys. Rep. D, 12, 3096 (1975). (t) P. VAN NIEI.:WENHL'IZEN, D. WII.KIN.'~ON"and 3[. J. PERRY: Phys. Rev. D, :13, 778 (1976). (~) 51. I)vyl.' and J. ~IADOlll.:: Phys. ReL,. D, [8, 2788 (1978). (';) V. N. POXOM.kI~EV and A. A. TSEITLIN: Wtzcl. Phys. U S S R , 29, 539 (1979). (:) M. K..~..qlTv~: Phys. Lc#l. 1~, ]03, 353 (1981). (~) 1(. P. ~TANUKOVICHand V. N. 3[ELNIKOV: Hydrodynamicz, Fields and Constants in the TDeory o] Gracilalio~ (Moscow, :Em~l'goatomizdat, 1983), in Russian.
364
)~XACT
SO 3 MAGNETIC-MONOPOLE
SOLUTIONS
365
ETC.
E q u a t i o n s of nmtions, corresponding to (2) are [ ~ ( v / - - - : g G ~'~ -t- ex/----g(cp -i- ~ r
(3)
[
~ ( V - - - ~ ~B~) - ( ~ / 2 ) r
(~-
= O,
P~)~ :, o.
U s i n g t h e substitution of ref. (9) q~ = H(r, t) n / e r ,
W o = J(r, t) n / e r ,
W"2 = [ K ( r , t) - - 1] m / e ,
W1 = O,
Wa = - - ( K ( r , t) - - 1) (sin O)l/e ,
where the u n i t vectors are n
= (sin 0 cos ~, sin 0 sin ~, cos 0) ,
1
,= (cos 0 cos ~, cos 0 sin q?, - - sin O),
m = (-- sin ~, cos ~, O), from (3) for the m e t r i c (1), we obtain the following system of equations:
r(1 - - kr2)t
--
~r
(1
--
kr2)tr 2
__
0r~r]
a ~t
a 3
~,t]
2K"H--
. .2e22. a 2. H .I H2
(4)
r ( 1 - - kr2)t-~, r
( 1 - - kr2)tr~ ~r
re(1--kre)t~r
(l--kr2)t
~r]
- - 2K2 J = O,
r~
~i
-~] -K[K2-
(erF)2 ] = 0 ,
~
= O,
1 ~- ae(H 2-J'-,)] = 0 ,
0 OK K ~ct (aJ) -f- 2 a J O--t- = O .
F o r a n y a(t) t h e system (4) has such solution of a dyon type, singular at r K ~ 0,
It = erF,
0:
j -- a - l ( t ) [ q ( 1 _ kr2)~ L Ccr],
where q and Co are constants. In t h e flat-space static case (k = 0, a = const) e x a c t reglrlar solutions of (4) are known for t h e t h e o r y of v a n i s h i n g self-interaction of t h e IIiggs field (~ --~ 0) (1o). W e shall find the e x t e n t i o n of these solutions for a n o n s t a t i c case and for o t h e r
(*) W. ~'~I'ECLENBURGand D. O'BRIE.w: Phys. Rein D, 18, 1327 (1978). (,o) M. 1~. I)RASAD and C. SOMMERFIELD: Phys. Re~,. Let[., 3 5 , 760 (1975).
366
.MELI(IKOV a n d
v..',L
eosnlolo~ieal models
v. K. SIII(~OLEV
ill ttle f o r m K = C X ( r , t) s i n h -1 I C Y ( r , t ) ] ,
(5)
C,X i r , t) e o t g h I C Y ( r , t)] - - Z ( r , t ) ,
Q
C -- coast,
u s i n g H - ~ a - l Q c o s h f l . J == a - l Q s i n h f i , fl is a n a r b i t r a r y constant. t h e t i m e c o - o r d i n a t e a s dt = a(~)d~] a n d t h e r a d i a l c o - o r d i n ' ~ t e a s
(6)
r=
,~J..(Z) -
We
transfornl
sinz,
l,'-- I,
"/,
k -- O,
siuh Z,
1,' - -
- 1.
Then d . ~ - - a ~ - O / ) [ . - - d ~ l e .... dz'-' I ~.(dO-~ : sineO&~.~)] ~ n d e q s . (4) in 1he, l ' r a s a d - S o m m e r f i e l d
limit
(7)
Z~[G .... i) ".: G(6,/a
(8)
~(/~"--
~- k ) ] -
(11)
KR
where G ~./r - -From if ~'i :/- 0, ~
_.-
=- a l l ,
/r := a J ,
(') -=- e / ~ Z ,
0,
O, { G ~ - - R 0-) : : 0 ,
~.~(K"--J~)--.K(K2--1
(10)
2K"-(I =-. O ,
k]~) .... 2.K2R = (~-xt t~)' =
(9)
F/
( 2 - ~ 0)(10) h'an.~form t o
-- 2R1"s == O, ( ' ) -= ~/~1 a n d
the consequence
of (6) is u s e d :
l,.
| h e a n a l y s i s of eqs. (7)-(9) a n d (11) it f o l h ) w s t h a t . if ~/ := O, t h e n )'l := 0 a n d , l h e n 12 : : 0. So, b e l o w ~ve c o n s i d e r t h e s e t w o e a s e s .
A ) (g -- 0, i.e. a :- a o . we c o m e l o l h e s y s t e m ~K"--
bo~I.
Taking
K ( K " - - - 1 + Q2) = o ,
K -
K ( Z ) , Q .... Q(Z) f l ' o m e q u a t i o n s (7)-(9),
~ ( Q " -~ 1,Q) -
2 K 2 Q == O ,
w h i c h is a d i f f e r e n t i a l c o n s e q u e n c e of the, f i r s t - o r d e r s y s t e m
{
(~2)
# . f l u = - - I((.? ,
, ~'~ Z~,Q - - r ? : - 1 - - K z .
S o l n l i o n s of (12) a r e f u u e t i o n ~ , d e f i n e d b y (5) w i t h (n)
[ (13)
X
--
~e~(Z) ,
]7
Z,
Z
~.
::
CO~Z,
k; =: I ,
l ,
It--
[ eosh Z, (n) V. K. ~mnom.:v: l:c. Vu.~ov, Fisira, 11, ,q9 (198~
k ....
0
1 .
367
:EXACT ~ 0 s MAGNETIC-MONOPOLE SOLUTIONS ETC.
F o r this solution one m a y calculate the integral E = 4 = f V ' ~ T 0 ~ '~x, where Too is t a k e n from t h e general form of the Young-Mills-IIiggs e n e r g y - m o m e n t u m tensor, defined from (2} as (14)
T~ - : - - g~P(2t~/--~) 8L/Sg ~ =
I ( G ~ . G ~ " - - l b~G,, G"r-:- ~t ~ q ~ q ~
l
)
In a static case E is the t o t a l proper energy of the gauge field. In t h e system of units c = 1 it is e q u a l to t h e p r o p e r dyon mass or, w h e n fl = 0, to t h e m a g n e t i c - m o n o p o l e mass (J~)
2)_rdro. = (cosh"-//l~)[Q(1-
(l 5)
-
K2)/~.llJx%.-.
This v a l u e is tinite for a n y possible t y p e of a cosmological background (k = 0, •
1).
B) L e t g +- 0, J -- 0. T h e n (7)-(11) becomes
I ~ ( K .... K ) - - - K ( K " - - - 1 + Q") = 0 , t ~.[Q'--"Q, -!- Q(ii/a -!- k)] - - 2KeQ =: O.
(16)
W e consider only the case it = -- ka, Ulfiverse (r,) and our nonsingular describes t h e self-consistent account t h e Higgs field. Using the m e t h o d of ref. (9) from
Yl K rr--
(17)
which includes t h e F r i e d m a n r a d i a t i o n - d o m i n a t e d model (s), when a - - - a o c o s h ~ / , k . . . . . 1, which of t h e spontaneous s y m m e t r y - b r e a k i n g effect of (16), we get J~( K 2 -
1 -t- Qz) -- 0 ,
Y'Q, r r - - 2K2Q = 0 . T h e solutions of (17) are again (5) w i t h X = Y, Z = 1. Besides, the following conditions m u s t be fullfilled: (18)
~(xl-i- x2)Y,., Y,~: = y 2 ,
where xx, a = (Y. =~ .~)/2.
T h e solution of equations (18) m a y be represented as
]7, .... = O,
xt[ xa
(19)
(20)
Y(x,, x2) =: e x p [ - -
/,=,g,~.~-(l,~,
f daf~Z.2(fi)d[~ + / ( x l ) -" g(xn) ] , xl+x2
+g,=,)I
~;2@d~ =~'
(*t) YA. B. ZEL'DOVICI[ a n d I, ]). NOVIKOV: T h e Strwcturc a n d Evolution o! the Universe (Moscow, 1975), in R u s s i a n .
368
V.N.
M E L N I K O V &rid v . K. S H I G O L E V
I n t e g r a t i n g (20), we obtain
(2])
g(x2) j
''
where C~, Ce, and C a are arbit.rarv constants. get t h e solutions
[g(x~)J
"
F i n a l l y , s u b s t i t u t i n g (21) into (19), we cos~]~ cosz,
(22)
X:
](=
Y~ = ~ - l n q ~
Z
1
:b1~ =
k ~ 1;
~ / e - x ~-
cZ
coshr]~ coshz,
k :--
1.
F o r the fiat-space model (k = 0) the solution (22) was o b t a i n e d earlier in ref. (9). An i n t e r e s t i n g feature of (22) is t h a t only the solution w i t h Y [ has no singularity on a lightcone and m a y lead to the finite i n t e g r a l E. F o r this solution ToO according to (14) has the form T o = - - ( 4 . ~ e ~ a~)-: { R 2
-T-
K,~
-Y
,
(K 2 - -
, ~2
9
,
2
L c,7
2
8;~
A t last, for a special choice of i n t e g r a t i o n constants in 121) e q u a t i o n s (19) lead to a n o t h e r class of solutions, which are nonsingular on a light-cone for a n y k: x = ]: = ? ; = e x p {-'- ( - / , ' ) ~ v } ~ ( z ) ,
z = ].
F o r k = 0 this solution is trivial (X = Y = 7~,Z = 1) and was o b t a i n e d in ref.(9).
b y S o c i e t k I t a l i a n a di F i s i c a Propriet~ letteraria riservata Direttore responsabile:
REIffATO ANGELO
I~ICCI
Stampato in Bologna dalla Tipografia Compositori coi t i p i d e l l a T i p o g r a f i a 1V[onograf Q u c s t o fascicolo 6 s t a t o l i c e n z i a t o d a i t o r c h i il 10-IV-1984
Questo periodico 6 iscritto all'Unione Stampa Periodica Italiana
