IL NUOVO CIMENTO VOL. 102 B, N. 5 Novembre 1988

The Stability of ,,Q Balls, in Rosen's Bimetric Theory of Gravitation.

C. WOLF

Department of Physics, North Adams State College - North Adams, MA 01247 Institut fi~r Theoretische Physik - D-74 Tubingen, BRD

(ricevuto fl 18 Agosto 1987)

Summary. -- The approximate solution of a S02 theory of two scalar fields in the bimetric theory of gravitation is discussed. Coleman's suggestion that such Q matter can be stabilized by an internal rotation in S02 space leads to stability criteria for the Q matter before the breaking of the S02 global symmetry. The two distinct masses lead to two different criteria of stability for the Q ball. The problem of the cosmological constant is left as an unanswered and pressing question in the bimetric theory.

PACS 04.90 - Other topics in relativity and gravitation.

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

Recently, Coleman has suggested that Q matter or matter composed of S02 matter fields can be stabilized against collapse by an internal rotation in S02 space in much the same way that a planet is stabilized by an effective potential term due to the existence of a conserved angular momentum (1). It is intriguing to inquire just what the fate would be of such Q balls if they were to collapse gravitationally and whether there is a conserved charge preventing them from succumbing to black holes. In general relativity, this is an interesting question, but in bimetric theory no such black-hole solutions occur. Certainly a conserved Q charge would deter collapse to a black hole but perhaps not prevent it. If such an event happened, could the newly formed Q ball black hole transcend the conservation of Q number as suggested by Wheeler and Beckenstein for baryon

Q ) S. COLEMAN: Lecture on Q balls, M.I.T., November 21, 1984.

3 2 - Il Nuovo Cimento B. 441

442 c. WOLF

number (2.~). It seems if the S02 theory is not gauged there would be no way of sensing the Q charge and thus the black hole might be a sink to Q number and in a certain sense a source of CP or C violation in the universe. The possibility that such Q balls might form prior to the GUT phase transition suggests that they might in some way be very strong catalysts to any inhomogeneties that might develop at or near the end of the period of inflation(4). Coleman has demon- strated that such clumps of matter execute various stable modes of oscillation which might provide a mechanism to form galactic structures on a very primitive level. If indeed the false vacuum is left outside the Q balls to suffer a global phase transition, energy might be pumped into the Q balls and an expansion might insue. Certainly a <(plumb-putting>> type of universe might develop with Q balls present and voids of true vacuum between. Such a picture might be an alternative to the formation of strings in the early universe as suggested by Vilenkin (5). In what follows, I do not develop the evolution of the S02 matter fields with cosmological expansion but rather examine the static solution for a pair of S02 matter fields in the bimetric theory. Such an analysis leads to two different domains of stability depending on the mass we assume is relevant to stability. It is also suggested that the bimetric theory has evaded experimental tests to date for the masses mentioned above only differ under extreme conditions of density or compactness of the object under investigation Q). The fact that there has not been any overwhelming experimental evidence for the existence of a black hole predicted by general relativity allows the student of the early universe to at least view the bimetric with some degree of plausibility.

2. - Q Ba l l s in the bimetr ic theory .

Consider the following matter Lagrangian:

(1) ~ = [ ~ 1 ~ ~lg" + ~"~2a~2g'2 U(~I'~2)] ~/L'g '

where

A2/ 2

(2) j. A. WHEELER: Proceedings of Sixteenth Solvay Conference (Brussels, 1974). (3) j . BECKENSTEIN: Phys. Rev. D, 10, 2903 (1972). (4) V. P, NAIR: Phys. Rev. D., 27, 2856 (1983). (5) A. VILENKIN and A. E. EVERETT: Phys. Rev. Lett., 48, 1867 (1982). (5) N. ROSEN: Ann. Phys. (N.Y.), 84, 455 (1974).

THE STABILITY OF <<Q BALLS- IN ROSEN'S BIMETRIC THEORY ETC.

Invariance under 71 = 71 - ~72, ~2 = ~2 + $91 implies

32PJ

leading to

(3) Q--fff(-92~T91~)drdod?,

443

where ~1, ~2 = 0 at r - - R , and 91, 92 regular at r = 0 . Equation (3) represents the conserved charge mentioned above. Now let us consider the isotropic metric in the bimetric theory:

(ds) 2 = exp [2~](c dtfi - exp [2~](dr) 2 - r 2 exp [2z](d02 + sin 2 0(d~)2),

c t = x 4 r = x 1 O = x 2, 9 = x 3,

(4)

from eq. (1) we have

+ = [~4 91 ~4 91 exp [ - 2~] + a191 ~1 91 (-- exp [ - 2~]) 4f L 2 2

~4 ~92 ~4 ~2 exp [ - 2~-] a1 ~2 ~1 ~2 ( - exp [ - 2~b]) +

2 2

o r

-- U(91, ~2) 1 ~ ' +

(5) Q = f f f [ - p 2 ~4 ~1 exp [ - 2~] + 91 a4 ~2 exp [ - 2~-]] r 2.

�9 sin 0 exp [-~ + ~ + 2 z] dr dO d~.

Now consider the internal rotation in S02 space giving the solution

(6) ~l(r, t) = ~(r) cos ~t , 92(r, t) = ~(r) sin ~ot.

Substi tuting eq. (6) into eq. (5) yields

(7) R

Q _- 47:~ f exp [ - ~ + ~ + 2z] r2(9(r)) 2 dr C J o

for the conserved charge. Now the Lagrangian for the S02 mat te r fields is, af ter substi tuting eq. (6) into eq. (1),

Ig44 j gll(9#)z U(9)] r 2 exp [~- + ~ + 2z] sin 0. (8) ~M = [ - - ~ C 2 (9(r)) ~ -~ 2

444 c. WOLF

V a r y i n g eq. (8) wi th r e s p e c t to 9@) y ie lds

(9) _exp [ - 2~] 2 9@)

C 2 U ) r 2 exp [~ + ~ + 2Z] +

+ ~ r (r~ exp [~ - - r + 2Z] 9,r)

for t he equa t ion to d e t e r m i n e ~(r). N o w the e n e r g y m o m e n t u m c o m p o n e n t s c o r r e s p o n d i n g to eq. (1) a re

1 . . . . . . + g ' ~ ' " - -~ ~'%(g~fl~1,~91,~ "~- g~2,~92,,~) + tLU( 9~ , ~2) T ,~- g, 91,a ~1,,~ 72,~ ~2,'~

= 0

(10) T44 _ exp [ - 2~-] 2 exp [ - 2~](9, ~)2 2c 2 (9(r))2 -4 2 -{- U(~I, ~2),

(11) T~ - - exp [ - 2~] J (9@))2 exp [ - 2~b](~,,.) 2 2c 2 - 2 F- U(91, ~2),

(12) T~ = T~ - exp [ - 2r (~ r) 2 exp [-- 2~] ~o2(9(r)) ~ ~ U(91,92). 2 ' 2c 2

F o r the me t r i c , eq. (4), the field equa t ions of the b ime t r i c t h e o r y a re (7)

1 87:G kT~v (13) N,~ - - ~ N g ~ - c4 ~ ,

o r

w h e r e k = (g/y)l/2, C = 1, G = g rav i t a t i ona l cons tan t ,

g44 = exp [2~], gu = - exp [24], g22 = - r2 exp [2z], g3a = - r ~ sin s 0 exp [2Z],

~'44 = 1, ~'11 = - 1, Y22 = - r2, ~'a~ = - ~ sin20-

T h e 4-4, 1-1, (2-2, 3-3) equa t ions r e a d

(14) - 9" 2 , - r V = - 8 r ~ k (exp [ - 2~] c o 2 ( ~ ( r ) ) 2 - - U(~l, ~2)),

(~) N. ROSEN: Gen. Rel . Grav., 4, 435 (1973).

THE STABILITY OF ((Q BALLS- IN ROSEN~S BIMETRIC THEORY ETC. 445

(15) _ ,~,,_ 2 , r '~ - sinh (2X - 2~b) = - 8r:Gk ( - exp [ - 2~] (~,~)2 _ U(?I, 72)),

(16) - Z " - 2 , 1

r Z - -~ sinh (2~ - 2Z) = - 8,-:Gk(- U(~ , ~2)),

where k = exp [~ + ~ + 2X]. An exact solution would entail solving eq. (9), eqs. (14), (15) and (16) subject to the constraint equation (7) with the BC,

--~ = z = 0 at r = 0, ~ = 0 at r = R, and the metric at r = R is to be matched with

(16a) (dsf i=exp[-~M-](dx4)2-exp[~-~---](dr2 + r~(dO)2 +r2sineod~2).

The reason for the two masses in the above metric was discussed by Rosen (~).

He showed that only for a point particle or an extended object such that P = 0 could the above masses M, M be the same (M = M). This follows from the fact

tha t only for a point chargeless particle can the analogue of the Di Donder condition (kg"~)/~--0 be consistently imposed so as not to contradict the field equations, this condition then leads to the equality of the masses. Also for an extended object with P -- 0 the equality of the masses follows from the solution for ~-, ~, Z. In our case, since nei ther of these conditions holds, we must assume the masses different (M:/: M).

Le t us construct an approximate solution by set t ing exp[~] = e x p [ ~ ] = -~ exp [z] = 1 in eq. (9), this implies

(17) r 2 p" + 2r~' + (r 2 ~2 + Air 2) ~(r) - r2A2 ~3 = O.

In eq. (17), if the ~ - t e r m were absent, a series solution exists. Le t us t ry ~ a~r' to approximate our solution. ':~

Subst i tut ing into eq. (17) we have

r(2al) + r2(2a2 + 4a2 + a0 J + A1 a0 - A2 a0 3) = 0 , . . .

a l -- 0, a2 = - 6 (~ + Ai - A2 a~),

also, since ~( r )= 0 at r = R , ao + a2R 2= 0, keeping up to quadratic te rms

a0 (18) a2 - R2.

446

Thus

(18a)

C. W O L F

6) a~ A2~ + A I - ~ - ~ �9

I f AI >> 6/R 2, o~2/c 2, then a~ = Ai/A2 which is jus t the condition for the global

minimum of eq. (2). Our solution for ~ is

(19) p(r) = ao 1 - ,

the t rue vacuum at r = 0 and the false vacuum at r = R. F ro m eq. (7) we have

(20) Q = 4~co ~ J r 2 a~ 1 - d r , 0

where exp [g] -- exp [~] = exp [z] -- 1 to approximate eq. (20).

Equat ion (20) yields

8 a 2 / 8 \ a A1

H e re ~o = -BQ/R 3, where B- = 105/32~ao(1 - r2/R2). Using eq. (19) for p ( r )= ao (1 - r2/R~) in eq. (14) we have

2--, [ ( _r2~2_.~(oj(r)2_A1~2 ] (21) ~ " + r ~ =8rd7 ~o ~ao 2 1 R2 ] -~2]J'

af te r using e x p [ - 2 ~ ] = 1 on the right-hand side.

Equat ion (21) has the solution

(22) ~(R)- ~ j r L~ ao 1 Re ] -~ ~(r)e--~2]jdr which, if matched to the exter ior solution (~)

( dS)2 = exp [- ~ - ] ( dx4)2 - exp E ~-~-] ( dr2 + r2( dO )2 + r2 sin2 0( dp )2) ,

at r = R yields

R I r2~ 2 -~( Al~2]dr (23) M=8r: f r 2 r ~(r) 2 - o -

for the mass corresponding to X 4.

C=I ,

THE STABILITY OF ,,Q BALLS- IN ROSEN'S BIMETRIC THEORY ETC. 447

Now using eq. (15) and eq. (16) we have after multiplying eq. (16) by 2 and adding to eq. (15)

S" + 2 S ' = 87~Gk(- exp [ - 2~](#,r) 2 - 3U(91,02)) r

yielding

(24) S : ~-~ f r2(exp [ - 2~b](9,r) 2 + 3U(9)) dr ,

for S = ~b + 2Z, ~b = Z at r -- R to match the exterior solution [eq. (16a)], we have

R

= 8~G f ~I~=R 3R 0J r2(exp [ - 2~](?'~)2 + 3U(~))dr

leading to

(25) ~ = _~5 f r2(exp [_ 2~b](9, ~)2 + 3U(9)) dr

upon matching the interior solution with eq. (16a). Equation (23) yields a value of

(26) M = ~ + C~ R 3

for the mass M if we suppress the constant term in the potential as giving rise to an effective cosmological constant in the bimetric theory. In the above equation

C, = 8~a~ Q 2~28 2~A~ 2 C2 =---7-- (O. 05). 105 ' -~2

- - i

Equation (25) [(M)] would give a different value for M and we would have to keep the o} term in a~, eq. (18a), to get the full dependence on the conserved charge Q. We have also neglected the term k = exp [~ + ~ + 2x] and set it equal to 1 in eqs. (14), (15), (16) as a first approximation. According to the result for M (eq. (26)) the energy (mass) has 1 turning point.

Fig.

M(R)

m-

R

1. - x 4 mass (M(R)) of Q ball vs. radius of Q ball.

448 c. WOLF

This Q ball is stable against collapse and also will be stable against radial pulsation. Equation (25) also will produce a curve, which has a region of absolute stability. This is inferred from eq. (18a) and its dependence on a~ for large ~2 and the structure of U(~) from eq. (2) along with the definition of Q from eq. (20).

~(R

i i

R

Fig. 2. - x I mass (M(R)) of Q ball vs. radius of Q ball.

Thus we conclude just on the basis of our crude model calculation that the bimetric Q ball will be stable according to the X 4 mass and stable according to X 1 mass. In normal general relativity the ADM mass is extracted from the X 1 component of the metric. It is also well known that, if M = M, the bimetric theory outside matter agrees with the isotropic Schwarzchild solution to the post-Newtonian approximation. If experiment ever suggests the possible existence of two different gravitational masses, it would certainly lend more support to Rosen's theory and help us to reshape our entire picture of space- time. A last issue that deserves attention is the fact that we have not dealt the cosmological constant for r > R in the above analysis which is a representation of the vacuum energy for r > R. This would give rise to additional terms in eq. (16a) as in general relativity with the De Sitter term. We have assumed here that this does not couple to the mass term as a product directly in eq. (16a) but would contribute an additional term in the exponent for r > R as well as an additive term in eqs. (23), (25) for the expression for the masses which could be matched at r = R. The interesting question is: Does a cosmic no-hair theorem exist for De Sitter space in the bimetric theory(8), as has been proved for GR under restrictive conditions for De Sitter space? It is also of interest to ask in closing if there are compactified theories from other than 11 or 6 d general relativity that yield Rosen-like theories with the possibility of two distinct gravitational

masses.

I would like to thank the Physics Department at Williams College for the use of their facilities where this work was carried out.

(~) J. D. BARROW: Phys. Lett. B, 183, 285 (1987).

THE STABILITY OF ,,Q BALLS. IN ROSEN'S BIMETRIC THEORY ETC. 449

�9 R I A S S U N T 0

Si esamina la soluzione approssimata di una teoria SOz di due campi scalari nella teoria bimetrica della gravitazione. L'ipotesi di Coleman che tale mater ia Q possa essere stabilizzata da una rotazione interna hello spazio SOe porta a criteri di stabilit~ per la mater ia Q prima della ro t tura della s immetria globale SOz. Le due masse distinte portano a due diversi criteri di stabili ta per la sfera Q. I1 problema della costante cosmologica r imane un problema irrisolto e pressante nella teoria bimetrica.

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