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Thin-wall vacuum domain evolution
Volume 120B, number 1,2,3
PHYSICS LETTERS
6 January 1983
THIN-WALL VACUUM DOMAIN EVOLUTION V.A. BEREZIN, V.A. KUZMIN and I.I. TKACHEV Institute for Nuclear Research of the Academy of Sciences o f the USSR, 60-th October Anniversary Prospect 7a, Moscow 117312, USSR Received 29 July 1982 Revised manuscript received 22 September 1982
In the framework of general relativity the equation of motion of a thin-wall vacuum shell is derived at arbitrary values of the vacuum parameters inside and outside the shell. We obtain that the velocity of the walls of the true vacuum bubble at its expansion does not tend to the velocity of light. We find that vacuum shells could exist now in the universe, which do not contradict to observational cosmology. The types of black holes created are investigated. Restrictions on the mass of the black holes formed from remnants of the false vacuum and on the fraction of the vacuum energy pumped into such black holes are obtained.
Vacuum phase transitions [1] changing the symmetry group o f an effective interaction o f elementary particles have become recently the object o f rather intensive studies especially in the framework o f GUTs since in view of the very great magnitude o f the unification scale M X ~ 1015 GeV these phenomena could affect rather significantly the evolution o f the early universe. The decay of a metastable state of the universe proceeds via creation o f bubbles o f a correct vacuum in the interiors o f the old ones [ 2 - 5 ] . Until the whole universe will become covered by the new phase the process o f the formation o f new bubbles, their expansion and collisions will proceed continuously. Configurations o f surfaces separating the phases, influence essentially the s p a c e - t i m e geometry just after the phase transition. In particular the phase transition could result in the creation o f black holes and wormholes in the universe [6]. In the general case the problem o f formation and development o f such random surfaces is extremely complicated. The purpose o f the present paper is the strict investigation o f an idealized case o f the motion of a single bubble possessing the one vacuum, in the medium occupied by the other phase. We would especially emphasize that in the general consideration we do not make any difference between correct and false 0 0 3 1 - 9 1 6 3 / 8 3 / 0 0 0 0 - 0 0 0 0 / $ 03.00 © 1983 North-Holland
vacua. Indeed, at the early beginning of the phase transition there are isolated bubbles o f the new vacuum drifting in the space occupied by the old phase. At the moment o f the completion o f the phase transition the picture becomes just opposite: in the space with the new phase there are already drifting islands o f remnants o f the old phase, which in turn could be considered, approximately, as single isolated bubbles. These latter bubbles were studied in ref. [6]. However in these papers the assumption was made that just from the beginning the bubble walls are moving with the velocity o f light, i.e. in fact the surface energ y - m o m e n t u m density o f the wall was not taken into account. We do not make such an assumption and this may be very important in a number o f cases where the energy density of the wall could dominate, for instance, in SUSY GUTs or in studies of the domain structure o f the universe [ 7 - 9 ] (in particular, in the case o f S U 3 X SU 2 X U 1 and o f S U 4 × U 1 domains [8]). In general the configuration o f domain boundaries is arbitrary. The plane domain walls were investigated in ref. [9] where it was shown that the possible existence o f such walls is in contradiction now with the observational data due to the large magnitude o f the surface energy density. However, if the abundance o f one phase is smaller than that o f another (the corre91
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sponding volume fraction is given by percolation theory) then the domain boundaries form closed surfaces and the problem of the subsequent evolution of such a configuration could be solved by considering the motion of a single domain. As we shall see there could exist spherical domains which do not contradict to observational cosmology and do not form black holes. The investigation of the evolution of a vacuum bubble is also of importance in connection with the magnetic monopole problem [10] and with the problem of the formation of large-scale structure in the universe. We study the bubble motion in real time. We use the thin-waU approximation based on the investigation of thin shells in general relativity, a method developed by Israel [ 11 ]. The thin-wall approximation could turn out to be a crude one in a number of cases [12] but on the other hand the only essential quantities one needs to know in this approximation are the inner and outer metrics and the surface density of the energy-momentum tensor on the shell. Therefore our result is quite suitable for any shell, independently of the used model of quantum field theory. From the Einstein equations GuY = -(87r/M21)Tu v written for a singular spherically symmetrical shell with vacuum, we derive the following equation determining in the most general case the form of the threedimensional hypersurface separating the four-dimensional manifold onto two regions with different vacua* 1
S = (1/47r)M21[K22],
S = const.
(la,b)
Here K22 is the discontinuity of the K22 component of the outer curvature tensor of the shell; S - S00 for a time-like (TL) and S = S11 for a space-like (SL) hypersurface, respectively, where E
Si i = lira f e~O
dn Ti]
--e
6 January 1983
value of the surface energy density of the plane wall *2 Here we investigate the time-like hypersurface. Let us write the interval on the hypersurface in the form dl 2 = dr 2 -/92(r) d ~ 2 .
(2)
Calculating K 2 for spherically symmetric vacuum metric of the most general form ds 2 = f(r) dt 2 - dr 2 If(r) - r 2 d~22 , f ( r ) = 1-(~Tre/M21)r 2 - 2m/M21r + e2/M21r2 , (3) we obtain the equations describing the TL-shell: oi n [t~2 + fin(pp) ] 1/2 _ Oout [t52 + lout(/9)] 1/2
= (4~S/M21)/9,
(4)
where f i n and fout are the coefficients in the metric of the inner and the outer regions of the bubble, respectively, and o = +1 if the radii of two-dimensional spheres are growing in the direction of an outgoing normal and o = - 1 in the opposite case. Only the invariant quantities/9@) and ~(r) are present in eq. (4), therefore it holds true for the whole region of the maximal analytical continuation of the metric (3). We shall be interested here only in the motion of neutral shells (though just charged shells remain magnetic monopoles after their evaporation [6]). The equation of motion for such a shell takes the form ~2 =B2p2 _ 1 + (m//9)[1/M21 + ( g o u t - gin )/67rS2]
+ m2/16rr2S2/94 ,
(5a)
where B -1 - 3 S X [(gout + gin + 67rS2/M21)2 - 4goutgin] -1/2 . (5b) The Schwarzschild parameter m is the integral of motion and is determined by the initial values of/9 and
is the surface density of the energy-momentum tensor on the shell and n is the coordinate in the direction of an outgoing normal to the bubble. The quantity S for the vacuum bubble does not depend upon its radius, therefore S simply coincides with the
m = ~-Tr(gin - g o u t ) P 3
.1 Eq. (la) holds in the non-vacuum case too. In this case eq. ( l b ) is changed and determines the dependence o f S u p o n the coordinates [13].
*2 See, for example ref. [2] for a calculation of S for the plane wall. In particular, for a potential of the form 1 . 3 V = ~h (~o2 - ~p~)2 one obtains S = 4gx/h~oo.
92
+ 47rp2Soin[b 2 + 1 - (87rgin/3M21)/92] 1/2 - (8~r2S2/M21)p3 .
(6)
Volume 120B, number 1,2,3
PHYSICS LETTERS
For what follows it is appropriate to introduce the variables ~ [(~out --
exterior region of the bubble, during its expansion the velocity o f the shell tends not to the velocity o f light ,4 but to the value
~in)/67rS2]M21'
r~ - (m/8n2 S2 p 3 )M21 .
6 January 1983
drout/dtou t (7)
Only the values ~ < 0 correspond to the remnants o f the old phase in the new one. There may correspond to the domain structure any value o f ~. One may wonder what values of ~ are typical for phase transitions o f GUTs? Taking l g i n -- gout l ~M4x, $2/M21 (Mx/MI,1)2M 4 we get I~1 ~ (gpl/Mx) 2 >> I. In some special cases (for example in SUSY GUTs where the vacuum energy is equal to zero) the contribution o f the wall energy is essential and then ~ ~ 1. Let us discuss now the results o f our analysis o f the equation of motion of the shell. Consider first of all the shells with m = 0. Bubbles with m --- 0 describe at all values of I~1 ~> 1 *3 one and the same physical situation. It is just these bubbles which appear spontaneously as new vacuum bubbles, created during the phase transition as a result o f the decay of a metastable state. At - 1 < ~ < 1 the spontaneous materialization o f bubbles with m = 0 is forbidden (in such a case one cannot match the old and the new metrics at the SL hypersurface [13]). Such bubbles could however arise, like shells with m :~ 0 after the phase transition, when there is some environment, and such domains could exist in our universe till now without conflicting with observational data, due to m = 0, despite the enormous value of the wall energy. All bubbles with m = 0 have a point o f rest and expand infinitely. The quantity B -1 coincides for such bubbles with the value o f the radius o f the shell at the moment of its rest, which in turn coincides (at least in the case g i n = 0) with the radius o f the bubble at the moment o f its materialization [2,3]. Formula (5b) was obtained for the limiting case Mpl = oo in ref. [2] and for gin = 0 in ref. [3]. Notice that the equation of motion as rewritten in the coordinates o f the internal region of the bubble coincides in the case of m = 0 and g i n = 0 with the equation r2n - t2n = B - 2 obtained in refs. [2,3]. At arbitrary values o f g i n , the equation of motion is more cumbersome so we shall not give it here. Notice only that in the coordinates of the *3 For m = 0 the transformation ~ ~ - ~ corresponds to the interchange of the interior and the exterior regions.
~ [l/R(t)][1 - (87r/3M21)~out/B2] 1/2, P~=
(s)
R being the scale factor, p = rR. In GUTs with M X ~ M p l this value differs only slightly from the velocity of light. This, however, may not be so if the contribution of the wall energy is essential, which, for example, is the case in SUSY GUTs. Moreover drout/dtou t 0 when ~ -~ 1, i.e. the coordinate volume of such a bubble does not increase. The knowledge o f the asymptotic velocity of the shell is rather important in the estimates o f number densities of monopoles [ 15 ], etc. It was obtained in ref. [6] that phase transitions could result in the creation o f both black holes and wormholes. The main assumption made, consists in that the shell moves from the very beginning with the velocity o f light. As we have shown the velocity o f the real shell does not even tend to the velocity of light. Nevertheless the creation o f wormholes is possible in our case too, though our criteria differ from those of ref. [6]. Let us consider now the shells with m 4: 0. We did not intend to study the problem o f the creation o f such a shell. Assume that it was created somehow. In the case m > M c =M31/3(8rrlgout) 1/2 there are no horizons in the outer metric [16]. A shell with such a mass is expanding infinitely. At m < M c the outer geometry has two horizons: at p = Pc a cosmological event horizon is situated while the black hole event horizon is at p = PH < Pc [16]. In the latter case two possibilities may be realized: the trajectory o f the shell intersects the R+ region o f the outer metric (Oou t = +1) or it intersects the R _ region (Oou t = - 1 ) (see fig. 1). Any shell going through the R region forms the wormhole. Using the variables (~, r?) we arrive at the following conditions for the determination o f the signs o f Oout, *4 In refs. [5,14] the asymptotic value of the velocity of the shell does not equate to that of the velocity of light too, which reflects the thermal properties of a medium. Our statement that the velocity of the shell does not tend to the velocity of light is not connected with the properties of the medium but follows from the Hubble expansion of space. 93
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PHYSICS LETTERS
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/
C Fig. 1. Path of the shells on the Penrose diagrams of the Schwarzschild-de Sitter geometry. A space-like hypersurface represented by the dashed line is shown schematically in fig. 3.
tory 2 in fig. 1). However, independent of whether the shell has a rest point or not it may be shown that the mass of any shell intersecting the R+ region is bounded from above in the case ~ < 1. Indeed, from the conditions p > P9 > 2m/Mp1 and condition (9) we obtain for the shell shown in fig. 3a
Oin and b" : A: Oout ~ 0 B:
Oin~0
ifr/+~-
1 ~0,
ifr?+~+l~0,
c: b':eo
m < m c / ( 1 - ~)1/2,
if r/<> ¼([9(1 + ~)2 + (16/3rr)(gin/S2)M21] 1/2 -
(1
+ ~)}.
Fig. 3. Schematic view of the shell intersecting the R+ region (3a), and wormholes (3b), (3c) corresponding to the (~, rl) regions shown in fig. 2.
(9)
The whole set of corresponding conditions is shown in fig. 2. Profiles of the space are shown in the cases when the parameters of the shell lie under the line A, in between the lines A and B and below the line in figs. 3a, 3b and 3c, respectively. The mass spectrum of the shells of all types may in principle begin just from zero. Consider shells intersecting the R+ region. In R+ at ~ < 1 we have always p" < 0. Therefore such a shell forms a black hole if it has the rest point (trajectory 1 in fig. 1) but it may not form a black hole in the opposite case (trajec-
m c ~M21/8rrS.
(10)
In the case m > rnc(1 - ~)-1/2 the shells belong to the categories 3b or 3c but not to 3a. These shells form wormholes. The mass of the wormhole 3b is bounded from above if ~ < - 1 so that m < m,.(-1 _ ~)-1/2. The shell with rn > m c ( - 1 - ~)-l/l/I form a wormhole of type 3c. For ~ > 1 the formation of wormholes is not possible. Any shell has in its motion either a point of rest or a point when ~" = 0. In partic-
~/r2
~y8
fJ---f
l
0,4
Fig. 2. Various types o f shells on the (~, r/) plane.
94
I
~_
~
e~yx
Fig. 4. Dependence of the bubble mass u p o n shell size at rest m o m e n t in the case ~ o u t = &in = 0 (curve 1). Curve 2 represents the dependence o f the bubble mass on size where the resultant force vanishes for shells which do not possess rest points. The solid parts of the curves correspond to black holes while the other parts correspond to wormholes.
Volume 120B, number 1,2,3
PHYSICS LETTERS
ular, in the case of the pure Schwarzschild outer metric (gout = 0) all shells running across the R+ region of the metric have a point of rest. The dependence of the Schwarzschild mass of the bubble upon the radius at the rest point and on the null-force radius are shown in fig. 4 (curves 1 and 2 respectively). For definiteness we draw them for the case gout = gin = 0. Let us take the value of z/at p equal to the shell radius at the rest point. If the point (~, ~/) lies now below the curve C then the shell collapses while in the opposite case it should expand. The following criterion may be formulated for the shells having a rest point, to learn the fate of a shell: if at the rest moment the shell radius obeys the inequality
P
(11)
then a collapsing black hole is formed while in the opposite case a wormhole is created. We have thus considered the problem of the motion of a single shell created somehow, and have obtained the exact solution in the thin-wall approximation. Now we should like to speculate about the origin o f black holes and wormholes. For the creation of the wormhole we know the only mechanism, namely, for its creation it is necessary that the piece of old vacuum with mass m is separated out of the old phase simultaneously as a whole, as a result of the simultaneous production of the whole region o f new vacuum surrounding this remnant. Black holes may of course be created in this process too. The probability of the formation of a long living configuration is very small. We estimate it in the quasi-classical approximation as [131 Pm ~ P0 eXp [_43.-lr(m/Mpl )2] '
(12)
where P0 is the probability o f the creation of an empty bubble of the new vacuum, and equality takes place when S = 0 and the bubble is created at its own gravitational radius * s. However, the formation of the remnant of old vacuum with given mass (or domain)
+5 From (12) it follows for example that the probability of "spontaneous" materialization of a black hole, living longer than r ~ 10-37s, is less than 10-l°°.
6 January 1983
with the shell intersecting the R+ region of the Schwarzschild metric (and thus in particular the creation of black holes) is possible as a result of collisions of expanding bubbles of the new vacuum after the boundary conditions at spatial infinity have set in. A wormhole could not be formed in this way. Thus the mean mass of the shell, mH, is apparently restricted by the value m c ~ (Mp1/Mx)2MpI when the vacuum energy dominates the surface energy and by the value m c ~ M41/S ~ (MpI/Mx)3Mp1 in the opposite case for shells intersecting the R+ region, which follows from (10), though the question of the mass spectrum and of the number density of the created black holes requires further detailed investigation. However, what concerns the black holes resulting from GUT phase transitions as remnants of the former phase, we may put the following conclusions just now. For these phase transitions the number density of created black holes n H is proportional to that of the topologically created magnetic monopoles n H = f . n M ( f ~ 10) [17] due to similarities of the creation processes. We thus arrive at the inequality n H < f X 10-25s, s being the entropy density. We have exploited here the known bounds on the number density of the superheavy monopole. Requiring energy conservation we get mHn H = Pgv, Pgv being the fraction of the vacuum energy pumped into black holes. Then
p < (mHf/g~ v) 10-25s < lO-24(MpI/MX )3-4 . Considering the remnants of the old vacuum we assumed in fact that starting from some moment the old false vacuum does not percolate. It was shown however in ref. [18] that percolation, along with the suppression of the production of magnetic monopoles, could take place only at very fine tuning of the coupling constants of a theory. Nevertheless only if we do not live inside a single bubble [12], we have to suppose that percolation of the new true phase has taken place. The method we used is appropriate not only for the investigation of the motion of vacuum bubbles but also in studies o f bubbles created by means of thermodynamical fluctuations. It is appropriate also when there is no thermodynamical equilibrium in a system. If both ~in, ~out and S are functions of temperature then the equation of motion of the bubble with m = 0 can be obtained from eq. (5) by substitu95
Volume 120B, number 1,2,3
PHYSICS LETTERS
tion o f the corresponding functions instead o f the constants g~in, gout and S. We h o p e to discuss the m o r e c o m p l i c a t e d situations in subsequent papers. We are grateful to V.A. Matveev and M.E. Shaposhnikov for interest in the w o r k and useful discussions.
References [1] D.A. Kirzhnits, Pis'ma Zh. Eksp. Teor. Fiz. 45 (1972) 745; D.A. Kixzhnits and A.D. Linde, Phys. Lett. 42B (1972) 471; S. Weinberg, Phys. Rev. D9 (1974) 3357; L. Dolan and R. Jaekiw, Phys. Rev. D9 (1974) 3320. [2] S. Coleman, Phys. Rev. D15 (1977) 2929. [3] S. Coleman and F. De Luccia, Phys. Rev. D21 (1980) 3305. [4] A.D. Linde, Phys. Lett. 70B (1977) 306. [5] A.D. Linde, Lebedev Phys. Inst. preprints 265,266 (1981). [6] K. Sato, M. Sasaki, M. Kodama and K. Maeda, Prog. Theor. Phys. 65 (1981) 1443; K. Maeda, K. Sato, M. Sasaki and H. Kodama, Phys. Lett. 108B (1982) 98.
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[7] R.W. Brown and F.W. Stecker, Phys. Rev. Lett. 43 (1979) 315; V.A. Kuzmin, M.E. Shaposhnikov and I.I. Tkachev, Phys. Lett. 105B (1981) 167. [8] V.A. Kuzmin, M.E. Shaposhnikov and I.I. Tkachev, Phys. Lett. 102B (1981) 397; Z. Phys. C12 (1982) 83. [9] Ya.B. Zeldovich, I.Yu. Kobzarev and L.B. Okun, Zh. Eksp. Teor. Fiz. 67 (1974) 3. [10] Ya.B. Zeldovich and M.Yu. Khlopov, Phys. Lett. 79B (1978) 239. [11] W. Israel, Nuovo Cimento 44B (1966) 1; 48B (1967) 463. [12] A.D. Linde, Phys. Lett. 108B (1982) 389. [ 13] V.A. Berezin, V.A. Kuzmin and I.l. Tkachev, to be published. [14] P.J. Steinhardt, Phys. Rev. D25 (1982) 2074. [15] A.H. Guth and S.-H.H. Tye, Phys. Rev. Lett. 44 (1980) 631; M.B. Einborn and K. Sato, Nucl. Phys. B180 (1981) 385. [16] G.W. Gibbons and S.W. Hawking, Phys. Rev. DI5 (1977) 2738. [171 T.W.B. Kibble, J. Phys. A9 (1976) 1387; K. Sato, KUNS 599 (1981). [18] A. Guth and E. Weinberg, Phys. Rev. D23 (1981) 876; preprint MIT CTP 950 (1982).
PHYSICS LETTERS
6 January 1983
THIN-WALL VACUUM DOMAIN EVOLUTION V.A. BEREZIN, V.A. KUZMIN and I.I. TKACHEV Institute for Nuclear Research of the Academy of Sciences o f the USSR, 60-th October Anniversary Prospect 7a, Moscow 117312, USSR Received 29 July 1982 Revised manuscript received 22 September 1982
In the framework of general relativity the equation of motion of a thin-wall vacuum shell is derived at arbitrary values of the vacuum parameters inside and outside the shell. We obtain that the velocity of the walls of the true vacuum bubble at its expansion does not tend to the velocity of light. We find that vacuum shells could exist now in the universe, which do not contradict to observational cosmology. The types of black holes created are investigated. Restrictions on the mass of the black holes formed from remnants of the false vacuum and on the fraction of the vacuum energy pumped into such black holes are obtained.
Vacuum phase transitions [1] changing the symmetry group o f an effective interaction o f elementary particles have become recently the object o f rather intensive studies especially in the framework o f GUTs since in view of the very great magnitude o f the unification scale M X ~ 1015 GeV these phenomena could affect rather significantly the evolution o f the early universe. The decay of a metastable state of the universe proceeds via creation o f bubbles o f a correct vacuum in the interiors o f the old ones [ 2 - 5 ] . Until the whole universe will become covered by the new phase the process o f the formation o f new bubbles, their expansion and collisions will proceed continuously. Configurations o f surfaces separating the phases, influence essentially the s p a c e - t i m e geometry just after the phase transition. In particular the phase transition could result in the creation o f black holes and wormholes in the universe [6]. In the general case the problem o f formation and development o f such random surfaces is extremely complicated. The purpose o f the present paper is the strict investigation o f an idealized case o f the motion of a single bubble possessing the one vacuum, in the medium occupied by the other phase. We would especially emphasize that in the general consideration we do not make any difference between correct and false 0 0 3 1 - 9 1 6 3 / 8 3 / 0 0 0 0 - 0 0 0 0 / $ 03.00 © 1983 North-Holland
vacua. Indeed, at the early beginning of the phase transition there are isolated bubbles o f the new vacuum drifting in the space occupied by the old phase. At the moment o f the completion o f the phase transition the picture becomes just opposite: in the space with the new phase there are already drifting islands o f remnants o f the old phase, which in turn could be considered, approximately, as single isolated bubbles. These latter bubbles were studied in ref. [6]. However in these papers the assumption was made that just from the beginning the bubble walls are moving with the velocity o f light, i.e. in fact the surface energ y - m o m e n t u m density o f the wall was not taken into account. We do not make such an assumption and this may be very important in a number o f cases where the energy density of the wall could dominate, for instance, in SUSY GUTs or in studies of the domain structure o f the universe [ 7 - 9 ] (in particular, in the case o f S U 3 X SU 2 X U 1 and o f S U 4 × U 1 domains [8]). In general the configuration o f domain boundaries is arbitrary. The plane domain walls were investigated in ref. [9] where it was shown that the possible existence o f such walls is in contradiction now with the observational data due to the large magnitude o f the surface energy density. However, if the abundance o f one phase is smaller than that o f another (the corre91
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sponding volume fraction is given by percolation theory) then the domain boundaries form closed surfaces and the problem of the subsequent evolution of such a configuration could be solved by considering the motion of a single domain. As we shall see there could exist spherical domains which do not contradict to observational cosmology and do not form black holes. The investigation of the evolution of a vacuum bubble is also of importance in connection with the magnetic monopole problem [10] and with the problem of the formation of large-scale structure in the universe. We study the bubble motion in real time. We use the thin-waU approximation based on the investigation of thin shells in general relativity, a method developed by Israel [ 11 ]. The thin-wall approximation could turn out to be a crude one in a number of cases [12] but on the other hand the only essential quantities one needs to know in this approximation are the inner and outer metrics and the surface density of the energy-momentum tensor on the shell. Therefore our result is quite suitable for any shell, independently of the used model of quantum field theory. From the Einstein equations GuY = -(87r/M21)Tu v written for a singular spherically symmetrical shell with vacuum, we derive the following equation determining in the most general case the form of the threedimensional hypersurface separating the four-dimensional manifold onto two regions with different vacua* 1
S = (1/47r)M21[K22],
S = const.
(la,b)
Here K22 is the discontinuity of the K22 component of the outer curvature tensor of the shell; S - S00 for a time-like (TL) and S = S11 for a space-like (SL) hypersurface, respectively, where E
Si i = lira f e~O
dn Ti]
--e
6 January 1983
value of the surface energy density of the plane wall *2 Here we investigate the time-like hypersurface. Let us write the interval on the hypersurface in the form dl 2 = dr 2 -/92(r) d ~ 2 .
(2)
Calculating K 2 for spherically symmetric vacuum metric of the most general form ds 2 = f(r) dt 2 - dr 2 If(r) - r 2 d~22 , f ( r ) = 1-(~Tre/M21)r 2 - 2m/M21r + e2/M21r2 , (3) we obtain the equations describing the TL-shell: oi n [t~2 + fin(pp) ] 1/2 _ Oout [t52 + lout(/9)] 1/2
= (4~S/M21)/9,
(4)
where f i n and fout are the coefficients in the metric of the inner and the outer regions of the bubble, respectively, and o = +1 if the radii of two-dimensional spheres are growing in the direction of an outgoing normal and o = - 1 in the opposite case. Only the invariant quantities/9@) and ~(r) are present in eq. (4), therefore it holds true for the whole region of the maximal analytical continuation of the metric (3). We shall be interested here only in the motion of neutral shells (though just charged shells remain magnetic monopoles after their evaporation [6]). The equation of motion for such a shell takes the form ~2 =B2p2 _ 1 + (m//9)[1/M21 + ( g o u t - gin )/67rS2]
+ m2/16rr2S2/94 ,
(5a)
where B -1 - 3 S X [(gout + gin + 67rS2/M21)2 - 4goutgin] -1/2 . (5b) The Schwarzschild parameter m is the integral of motion and is determined by the initial values of/9 and
is the surface density of the energy-momentum tensor on the shell and n is the coordinate in the direction of an outgoing normal to the bubble. The quantity S for the vacuum bubble does not depend upon its radius, therefore S simply coincides with the
m = ~-Tr(gin - g o u t ) P 3
.1 Eq. (la) holds in the non-vacuum case too. In this case eq. ( l b ) is changed and determines the dependence o f S u p o n the coordinates [13].
*2 See, for example ref. [2] for a calculation of S for the plane wall. In particular, for a potential of the form 1 . 3 V = ~h (~o2 - ~p~)2 one obtains S = 4gx/h~oo.
92
+ 47rp2Soin[b 2 + 1 - (87rgin/3M21)/92] 1/2 - (8~r2S2/M21)p3 .
(6)
Volume 120B, number 1,2,3
PHYSICS LETTERS
For what follows it is appropriate to introduce the variables ~ [(~out --
exterior region of the bubble, during its expansion the velocity o f the shell tends not to the velocity o f light ,4 but to the value
~in)/67rS2]M21'
r~ - (m/8n2 S2 p 3 )M21 .
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drout/dtou t (7)
Only the values ~ < 0 correspond to the remnants o f the old phase in the new one. There may correspond to the domain structure any value o f ~. One may wonder what values of ~ are typical for phase transitions o f GUTs? Taking l g i n -- gout l ~M4x, $2/M21 (Mx/MI,1)2M 4 we get I~1 ~ (gpl/Mx) 2 >> I. In some special cases (for example in SUSY GUTs where the vacuum energy is equal to zero) the contribution o f the wall energy is essential and then ~ ~ 1. Let us discuss now the results o f our analysis o f the equation of motion of the shell. Consider first of all the shells with m = 0. Bubbles with m --- 0 describe at all values of I~1 ~> 1 *3 one and the same physical situation. It is just these bubbles which appear spontaneously as new vacuum bubbles, created during the phase transition as a result o f the decay of a metastable state. At - 1 < ~ < 1 the spontaneous materialization o f bubbles with m = 0 is forbidden (in such a case one cannot match the old and the new metrics at the SL hypersurface [13]). Such bubbles could however arise, like shells with m :~ 0 after the phase transition, when there is some environment, and such domains could exist in our universe till now without conflicting with observational data, due to m = 0, despite the enormous value of the wall energy. All bubbles with m = 0 have a point o f rest and expand infinitely. The quantity B -1 coincides for such bubbles with the value o f the radius o f the shell at the moment of its rest, which in turn coincides (at least in the case g i n = 0) with the radius o f the bubble at the moment o f its materialization [2,3]. Formula (5b) was obtained for the limiting case Mpl = oo in ref. [2] and for gin = 0 in ref. [3]. Notice that the equation of motion as rewritten in the coordinates o f the internal region of the bubble coincides in the case of m = 0 and g i n = 0 with the equation r2n - t2n = B - 2 obtained in refs. [2,3]. At arbitrary values o f g i n , the equation of motion is more cumbersome so we shall not give it here. Notice only that in the coordinates of the *3 For m = 0 the transformation ~ ~ - ~ corresponds to the interchange of the interior and the exterior regions.
~ [l/R(t)][1 - (87r/3M21)~out/B2] 1/2, P~=
(s)
R being the scale factor, p = rR. In GUTs with M X ~ M p l this value differs only slightly from the velocity of light. This, however, may not be so if the contribution of the wall energy is essential, which, for example, is the case in SUSY GUTs. Moreover drout/dtou t 0 when ~ -~ 1, i.e. the coordinate volume of such a bubble does not increase. The knowledge o f the asymptotic velocity of the shell is rather important in the estimates o f number densities of monopoles [ 15 ], etc. It was obtained in ref. [6] that phase transitions could result in the creation o f both black holes and wormholes. The main assumption made, consists in that the shell moves from the very beginning with the velocity o f light. As we have shown the velocity o f the real shell does not even tend to the velocity of light. Nevertheless the creation o f wormholes is possible in our case too, though our criteria differ from those of ref. [6]. Let us consider now the shells with m 4: 0. We did not intend to study the problem o f the creation o f such a shell. Assume that it was created somehow. In the case m > M c =M31/3(8rrlgout) 1/2 there are no horizons in the outer metric [16]. A shell with such a mass is expanding infinitely. At m < M c the outer geometry has two horizons: at p = Pc a cosmological event horizon is situated while the black hole event horizon is at p = PH < Pc [16]. In the latter case two possibilities may be realized: the trajectory o f the shell intersects the R+ region o f the outer metric (Oou t = +1) or it intersects the R _ region (Oou t = - 1 ) (see fig. 1). Any shell going through the R region forms the wormhole. Using the variables (~, r?) we arrive at the following conditions for the determination o f the signs o f Oout, *4 In refs. [5,14] the asymptotic value of the velocity of the shell does not equate to that of the velocity of light too, which reflects the thermal properties of a medium. Our statement that the velocity of the shell does not tend to the velocity of light is not connected with the properties of the medium but follows from the Hubble expansion of space. 93
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/
C Fig. 1. Path of the shells on the Penrose diagrams of the Schwarzschild-de Sitter geometry. A space-like hypersurface represented by the dashed line is shown schematically in fig. 3.
tory 2 in fig. 1). However, independent of whether the shell has a rest point or not it may be shown that the mass of any shell intersecting the R+ region is bounded from above in the case ~ < 1. Indeed, from the conditions p > P9 > 2m/Mp1 and condition (9) we obtain for the shell shown in fig. 3a
Oin and b" : A: Oout ~ 0 B:
Oin~0
ifr/+~-
1 ~0,
ifr?+~+l~0,
c: b':eo
m < m c / ( 1 - ~)1/2,
if r/<> ¼([9(1 + ~)2 + (16/3rr)(gin/S2)M21] 1/2 -
(1
+ ~)}.
Fig. 3. Schematic view of the shell intersecting the R+ region (3a), and wormholes (3b), (3c) corresponding to the (~, rl) regions shown in fig. 2.
(9)
The whole set of corresponding conditions is shown in fig. 2. Profiles of the space are shown in the cases when the parameters of the shell lie under the line A, in between the lines A and B and below the line in figs. 3a, 3b and 3c, respectively. The mass spectrum of the shells of all types may in principle begin just from zero. Consider shells intersecting the R+ region. In R+ at ~ < 1 we have always p" < 0. Therefore such a shell forms a black hole if it has the rest point (trajectory 1 in fig. 1) but it may not form a black hole in the opposite case (trajec-
m c ~M21/8rrS.
(10)
In the case m > rnc(1 - ~)-1/2 the shells belong to the categories 3b or 3c but not to 3a. These shells form wormholes. The mass of the wormhole 3b is bounded from above if ~ < - 1 so that m < m,.(-1 _ ~)-1/2. The shell with rn > m c ( - 1 - ~)-l/l/I form a wormhole of type 3c. For ~ > 1 the formation of wormholes is not possible. Any shell has in its motion either a point of rest or a point when ~" = 0. In partic-
~/r2
~y8
fJ---f
l
0,4
Fig. 2. Various types o f shells on the (~, r/) plane.
94
I
~_
~
e~yx
Fig. 4. Dependence of the bubble mass u p o n shell size at rest m o m e n t in the case ~ o u t = &in = 0 (curve 1). Curve 2 represents the dependence o f the bubble mass on size where the resultant force vanishes for shells which do not possess rest points. The solid parts of the curves correspond to black holes while the other parts correspond to wormholes.
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ular, in the case of the pure Schwarzschild outer metric (gout = 0) all shells running across the R+ region of the metric have a point of rest. The dependence of the Schwarzschild mass of the bubble upon the radius at the rest point and on the null-force radius are shown in fig. 4 (curves 1 and 2 respectively). For definiteness we draw them for the case gout = gin = 0. Let us take the value of z/at p equal to the shell radius at the rest point. If the point (~, ~/) lies now below the curve C then the shell collapses while in the opposite case it should expand. The following criterion may be formulated for the shells having a rest point, to learn the fate of a shell: if at the rest moment the shell radius obeys the inequality
P
(11)
then a collapsing black hole is formed while in the opposite case a wormhole is created. We have thus considered the problem of the motion of a single shell created somehow, and have obtained the exact solution in the thin-wall approximation. Now we should like to speculate about the origin o f black holes and wormholes. For the creation of the wormhole we know the only mechanism, namely, for its creation it is necessary that the piece of old vacuum with mass m is separated out of the old phase simultaneously as a whole, as a result of the simultaneous production of the whole region o f new vacuum surrounding this remnant. Black holes may of course be created in this process too. The probability of the formation of a long living configuration is very small. We estimate it in the quasi-classical approximation as [131 Pm ~ P0 eXp [_43.-lr(m/Mpl )2] '
(12)
where P0 is the probability o f the creation of an empty bubble of the new vacuum, and equality takes place when S = 0 and the bubble is created at its own gravitational radius * s. However, the formation of the remnant of old vacuum with given mass (or domain)
+5 From (12) it follows for example that the probability of "spontaneous" materialization of a black hole, living longer than r ~ 10-37s, is less than 10-l°°.
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with the shell intersecting the R+ region of the Schwarzschild metric (and thus in particular the creation of black holes) is possible as a result of collisions of expanding bubbles of the new vacuum after the boundary conditions at spatial infinity have set in. A wormhole could not be formed in this way. Thus the mean mass of the shell, mH, is apparently restricted by the value m c ~ (Mp1/Mx)2MpI when the vacuum energy dominates the surface energy and by the value m c ~ M41/S ~ (MpI/Mx)3Mp1 in the opposite case for shells intersecting the R+ region, which follows from (10), though the question of the mass spectrum and of the number density of the created black holes requires further detailed investigation. However, what concerns the black holes resulting from GUT phase transitions as remnants of the former phase, we may put the following conclusions just now. For these phase transitions the number density of created black holes n H is proportional to that of the topologically created magnetic monopoles n H = f . n M ( f ~ 10) [17] due to similarities of the creation processes. We thus arrive at the inequality n H < f X 10-25s, s being the entropy density. We have exploited here the known bounds on the number density of the superheavy monopole. Requiring energy conservation we get mHn H = Pgv, Pgv being the fraction of the vacuum energy pumped into black holes. Then
p < (mHf/g~ v) 10-25s < lO-24(MpI/MX )3-4 . Considering the remnants of the old vacuum we assumed in fact that starting from some moment the old false vacuum does not percolate. It was shown however in ref. [18] that percolation, along with the suppression of the production of magnetic monopoles, could take place only at very fine tuning of the coupling constants of a theory. Nevertheless only if we do not live inside a single bubble [12], we have to suppose that percolation of the new true phase has taken place. The method we used is appropriate not only for the investigation of the motion of vacuum bubbles but also in studies o f bubbles created by means of thermodynamical fluctuations. It is appropriate also when there is no thermodynamical equilibrium in a system. If both ~in, ~out and S are functions of temperature then the equation of motion of the bubble with m = 0 can be obtained from eq. (5) by substitu95
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tion o f the corresponding functions instead o f the constants g~in, gout and S. We h o p e to discuss the m o r e c o m p l i c a t e d situations in subsequent papers. We are grateful to V.A. Matveev and M.E. Shaposhnikov for interest in the w o r k and useful discussions.
References [1] D.A. Kirzhnits, Pis'ma Zh. Eksp. Teor. Fiz. 45 (1972) 745; D.A. Kixzhnits and A.D. Linde, Phys. Lett. 42B (1972) 471; S. Weinberg, Phys. Rev. D9 (1974) 3357; L. Dolan and R. Jaekiw, Phys. Rev. D9 (1974) 3320. [2] S. Coleman, Phys. Rev. D15 (1977) 2929. [3] S. Coleman and F. De Luccia, Phys. Rev. D21 (1980) 3305. [4] A.D. Linde, Phys. Lett. 70B (1977) 306. [5] A.D. Linde, Lebedev Phys. Inst. preprints 265,266 (1981). [6] K. Sato, M. Sasaki, M. Kodama and K. Maeda, Prog. Theor. Phys. 65 (1981) 1443; K. Maeda, K. Sato, M. Sasaki and H. Kodama, Phys. Lett. 108B (1982) 98.
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[7] R.W. Brown and F.W. Stecker, Phys. Rev. Lett. 43 (1979) 315; V.A. Kuzmin, M.E. Shaposhnikov and I.I. Tkachev, Phys. Lett. 105B (1981) 167. [8] V.A. Kuzmin, M.E. Shaposhnikov and I.I. Tkachev, Phys. Lett. 102B (1981) 397; Z. Phys. C12 (1982) 83. [9] Ya.B. Zeldovich, I.Yu. Kobzarev and L.B. Okun, Zh. Eksp. Teor. Fiz. 67 (1974) 3. [10] Ya.B. Zeldovich and M.Yu. Khlopov, Phys. Lett. 79B (1978) 239. [11] W. Israel, Nuovo Cimento 44B (1966) 1; 48B (1967) 463. [12] A.D. Linde, Phys. Lett. 108B (1982) 389. [ 13] V.A. Berezin, V.A. Kuzmin and I.l. Tkachev, to be published. [14] P.J. Steinhardt, Phys. Rev. D25 (1982) 2074. [15] A.H. Guth and S.-H.H. Tye, Phys. Rev. Lett. 44 (1980) 631; M.B. Einborn and K. Sato, Nucl. Phys. B180 (1981) 385. [16] G.W. Gibbons and S.W. Hawking, Phys. Rev. DI5 (1977) 2738. [171 T.W.B. Kibble, J. Phys. A9 (1976) 1387; K. Sato, KUNS 599 (1981). [18] A. Guth and E. Weinberg, Phys. Rev. D23 (1981) 876; preprint MIT CTP 950 (1982).