IL NUOVO CIMENTO VOL. 106 A, N. 7 Luglio 1993
Structureless Magnetic Monopoles in an Expanding Spacetime (*).
J. R. MORRIS
Department of Chemistry-Physics-Astronomy, Indiana University Northwest 3]~00 Broadway, Gary, IN ~6~08
(ricevuto il 22 Dicembre 1992; approvato il 12 Febbraio 1993)
Summary. -- A model consisting of an SO(3) gauge field interacting with an isovector Higgs field in the background geometry of a Robertson-Walker spacetime is examined. The gauge field and field tensors are parametrized by the Higgs field directionality and a gauge field structure function. Two simple, but exact, solutions of the field equations are obtained which represent structureless magnetic mono- poles in an expanding spacetime. There is an SO(3)-symmetric solution with a ,,color,) magnetic field, along with a broken-symmetric solution corresponding to the point particle limit of a 't Hooft-Polyakov monopole. The former solution represents a generalization of a type of solution previously studied within the context of a Minkowski spacetime. Within the context of an expanding spacetime each solution possesses a magnetic field which, on a spherical comoving surface, depends upon the cosmic scale factor a(t), but not the curvature constant k.
PACS 14.80.Hv - Magnetic monopoles.
It is well known that non-Abelian gauge theories admit magnetic-monopole solutions . The equations of motions are usually of sufficient complexity, however, that complete analytical solutions describing such field configurations are difficult to obtain. It can therefore be useful to consider mathematically simplified limiting cases in order to gain some physical insight regarding certain aspects of these non- perturbative solutions. Examples of such limiting cases include, for example, structureless magnetic-monopole solutions in Minkowski spacetime [2-5].
Within a cosmological context, however, magnetic-monopole solutions are expected to be physically realized within an expanding spacetime: The model presented here is that of an S0(3) Yang-Mills theory wherein the gauge field is coupled to an isovector Higgs field, and a background geometry descibed by a Robertson-Walker spacetime is assumed. Simple, but exact, solutions of the field equations are found which signal the presence of a structureless point magnetic-
(*) The author of this paper has agreed to not receive the proofs for correction.
956 J . R. MORRIS
monopole field. There is an SO(3)-symmetric solution with a radial ,,color, magnetic field, which corresponds to a generalization of a type of point magnetic-monopole solution previously studied[2-5], along with a broken-symmetric solution corre- sponding to the structureless limit of a 't Hooft-Polyakov monopole in an expanding spacetime. The magnetic field possesses an explicit time dependence on a spherical surface with constant comoving radial coordinate, but there is no induced electric field on this surface. This is seen to be in accord with Faraday's law in the expanding spacetime. It is expected that the essential features outside the core of a structured monopole will coincide, at least approximately, with those associated with the structureless solutions.
The model being considered contains an S0(3) gauge field coupled to a scalar isovector Higgs field . The Lagrangian is
(2) D r
is the gauge covariant derivative, V, and
=V, ,+g& x
is the ordinary spacetime covariant derivative,
(8) = a,,A - a,,r + x ,r
is the SO(3) gauge field tensor. The equations of motion are given by
8V D~D'~dp + -~ =0,
D rG ~v- g(D~d~ x dp) = O.
(6) = r
where ~ is an arbitrary unit vector, we can introduce a gauge field structure function /~(r, t) by writing [7, 8]
(7) D r ~b = ~a~ r + flCa~ ~.
Using (2), we also have
(8) D~b = ~a~r + Ca~ - gr x A~).
Taking the outer product of (7) and (8) with ~ and comparing expressions yields
(9) & = (~ x O,~) + a~, a t = ~.A~.
We can now use (3) and (9) to split the field tensor into two orthogonal parts ,
(10) G~ = ~r , , + H~,
STRUCTURELESS MAGNETIC MONOPOLES IN AN EXPANDING SPACETIME 957
fl -1 (11) F,~ = - - -~ [~. (a~ a~)] +f~,
(12) f,~ = a~ a~ - a~ a~,
1 [a~,fl(~, x a,,h) - a~fl(~, x a~,~,)] - f l (a~a~ft - a~a~ft). (13) H,~ = ~-
If the potential V(r possesses a global minimum at r = z > 0, then the original S0(3) symmetry is spontaneously broken down to a U(1) gauge symmetry that can be identified with the electromagnetic gauge symmetry. The electromagnetic-field tensor can be conventionally defined by F,~ = ~'G~, where F~ is given by (11).
Consider now the case where a, = 0 and
(14) n = (sinxcos~b, sinzsin~, cosx),
where ~b and X are assumed to be arbitrary functions of position and time. Then the triplet of equations represented by (4) is given by
aV (15) V~0~r - (0~ Za~Z + sin 2 za~ba~)fl2r + ~- = 0,
(16) V~ (flee a'z) - flr sin Z cos Z X, ~bX~b = 0,
(17) V, (flr sin Z X~) + flr cos Z X, X a'~ = 0.
Similarly (5) is represented by
(18) V~F "~ + ~M~a, fl = 0,
~ - lsinx[ge/~r + V,(X fix ~b - X~flX'~b)] - (19) flF X~X "
1 - cosz + a c,a z) - 2a a za,4 } = o,
1 ~ (20) f ls inzF~a~b + gflr x + ~V, (a fix X - X'flX~Z) +
+ ~sinzcoszX,~b(X fix ~ - X~flX'~b) = 0,
(21) M,v = sin z(a, Xav~ - a~xa~)
63 - l l Nuovo Cimento A
J. R, MORRIS
(25) r = r t), /~ =/~(r, t), a~ = 0.
By (6), (9), (24) and (25) it follows that
(26) , = r t)~, A~=[ ~(r't)-I ]
Now using (23)-(25) the equations of motion reduce to
1 - kr 2 ) O~fl (28) a~fl + 3Haofl -
( -Os inO0~ + ~0~0).
~V + + = 0,
+ a ~kra'fl +g2r 1) = O,
where the Hubble parameter is defined by
aoa (29) H - a
Notice that in the Minkowski limit (k ~ 0, a--* const) for time-independent functions r and fl(r), eqs. (27) and (28) coincide with the 't Hooft-Polyakov equations of motion in the temporal gauge.
Equations (27) and (28) admit two simple solution sets. To examine these solutions let the potential assume the standard form
= lz ( r 2)2, (30) V(r
so that for ~ > 0 the S0(3) symmetry is spontaneously broken down U(1). Then a simple broken-symmetric solution is obtained by noting that for ~ > 0 eqs. (27) and
is the field tensor defined by (11). Let us assume the background geometry to be that of a Robertson-Walker
spacetime described by the line element
1 - kr 2
where a(t) is the cosmic scale factor and k = O, + 1, or -1 for a flat, closed, or open universe, respectively. Let us choose
(24) ~ = 0, ~b = ~,
so that by (14) ~ = ~. Upon assuming spherical symmetry for the functions r and ~, we
STRUCTURELESS MAGNETIC MONOPOLES IN AN EXPANDING SPACETIME 959
(28) are solved by
(31) r = ~, fl = 0.
By (21)-(24) the .physical., or ((ordinary- radial component  of the magnetic field is given by
1 1 (32) B~=~'B- ~-=- :F0~=- - . vg~gvv ga 2r2
Note that on a constant r surface B~ is explicitly time-dependent, but from (21), (22), (24) and (31) there is no associated electric field. This is in agreement with Faraday's law in the expanding spacetime since the magnetic flux through a portion of constant r comoving surface S is
(33) 11 fs in0d0d? . S S S
Therefore the flux ~ through the comoving surface S is time-independent (due to the growth in proper area compensating for the increase in the proper radial distance from the origin and the hence the decrease in the -physical, magnetic field), so that by Faraday's law in the expanding spacetime there is no associated e.m.f, or electric field. (Another way to see that no electric field is induced is from the Maxwell equation 8~F~ + a~Fm + a ,F~ = 0, which upon choosing ~, v, 2 = 0, 0, reduces to 8oFov = 8o [gV~oogv~Br] = 0. ) The total magnetic flux through a spherical comoving surface is -4=/g, so that the enclosed magnetic charge is q i = -47~/g. Thus the solution given by (31) represents the structureless, point particle limit of a 't Hooft-Polyakov magnetic monopole in an expanding spacetime.
For the case z = 0 in (30), then (27) and (28) admit the trivial solution
(34) r = 0, fl = 0.
In this case the S0(3) symmetry remains unbroken and by (10)-(13) H,~ = 0 so that G,~ = ~F~. Therefore F~ is the S0(3) gauge-invariant field strength tensor. As in the previous case For =-sin0/g__so that there is a ,,physical, S0(3)gauge-invariant ,,magnetic--field component Br = -1 / (ga~r 2) giving rise to the associated ,,color, magnetic field
(35) ~ = ~ CBr _ ~c (c = 1, 2, 3), ga2r 2
which is a generalization of a type of previously studied gauge-symmetric point monopole field [2-5].
From the generalized 't Hooft-Polyakov ansatz (in the temporal gauge) given by (25)-(28) it is seen that a finite-energy monopole solution will generally have a core structure dependent upon the curvature (k), but the asymptotic solution given by (31) is valid for an arbitrary Robertson-Walker spacetime, regardless of the curvature or the particular form of the cosmic scale factor a(t). One therefore expects the salient features of the monopole outside the core (such as the magnetic field given by (32)) to be exhibited quite generally by the solution generated by (31).
960 J . R. MORRIS
 G. 'T HOOFT: Nuel. Phys. B, 79, 276 (1974); A. M. POLYAKOV: JETP Lett., 20, 194 (1974).  M. CARMELI: Phys. Lett. B, 68, 463 (1977).  J. R. MORRIS: Phys. Rev. D, 23, 556 (1981).  M. CARMELI and KH. HULEIHIL: Nuovo Cimento B, 67, 21 (1982).  See, for example, M. CARMELI, KH. HULEIHIL and E. LEIBOWITZ: Gauge Field:
Classification and Equations of Motion (World Scientific, Singapore, 1989).  See, for example, P. GODDARD and D. I. OLIVE: Rep. Prog. Phys., 41, 1357 (1978).  J. R. MORRIS: Nuovo Cimento A, 103, 1145 (1990).  Since ~b and ~ are scalar isovector, V, ~ = a, ~.  See, for example, S. WEINBERG: Gravitation and Cosmology: Principles and Applications of
the General Theory of Relativity (Wiley, New York, N.Y., 1972), Chapt. 4, sect. 8.