Gravitational Effects on and of Vacuum Decay

PHYSICAL REVIE% D VOLUME 21, NUMBER 12 15 J U NE 1980

Gravitational effects on and of vacuum decay

Sidney Co1eman* Stanford Linear Accelerator Center, Stanford University, Stanford, California 94305

Frank Be Luccia Institute for Advanced Study, Princeton, Sew Jersey 88548 (Received 4 March 1980)

It is possible for a classical field theory to have two stable homogeneous ground states, only one of which is an absolute energy minimum. In the quantum version of the theory, the ground state of higher energy is a false vacuum, rendered unstable by barrier penetration. There exists a well-established semiclassical theory of the decay of such false vacuums. In this paper, we extend this theory to include the effects of gravitation. Contrary to naive expectation, these are not always negligible, and may sometimes be of critical importance, especially in the late stages of the decay process.

I. INTRODUCTION There exist algorithms for computing the coeffi- cients A and B; indeed, in the limit of small ener- Consider the theory of a single scalar field de- gy-density difference between the two vacuums, fined by the action it is possible to compute B in closed form. Also, in this same limit, it is possible to give a closed- s = d'xI-,'(s„y)' —U(p) j, form description of the growth of the bubble after its quantum formation. %e will recapitulate this where Uis as shown in Fig. 1. That is to say, U analysis later in this paper. has two local minima, g„only one of which, P In this paper, we extend the theory of vacuum is an absolute minimum. The classical field the- decay to include the effects of gravitation. At ory defined by Eq. (1.1) possesses two stable ho- first glance, this seems a pointless exercise. In mogeneous equilibrium states, $ = Q, and Q = Q . any conceivable application, vacuum decay takes In the quantum version of the theory, though, only place on scales at which gravitational effects are the second of these corresponds to a truly stable utterly negligible. This is a valid point if we are state, a true vacuum. The first decays through talking about the formation of the bubble, but not barrier penetration; it is a false vacuum. This is if we are talking about its subsequent growth. The a prototypical ease; false vacuums occur in many energy released by the conversion of false vacuum field theories. In particular, they occur in some to true is proportional to the volume of the bubble; unified electroweak and grand unified theories, thus, so is the Schwarzschild radius associated and it is this that gives the theory of vacuum de- with this energy. Hence, as the bubble grows, the cay possible physical importance. For simplicity, Schwarzschild radius eventually becomes compar- though, we will restrict ourselves here to the the- able to the radius of the bubble. ory defined by Eq. (1.1); the extension of our This can easily be made quantitative. A sphere methods to more elaborate field theories is of radius A and energy density e has Schwarz- straightforward. schild radius 2Ge(4nA'/3), where G is Newton's The decay of the false vacuum is very much like constant. This is equal to A when the nucleation processes associated with first-or- -'~2. der phase transitions in statistical mechanics. ' A = (8~m/3) (1.3) The decay is initiated by the materialization of a For an e of (1 GeV)', the associated A is 0.8 km. bubble of true vacuum within the false vacuum. Of course, in a typical unified electroweak or This is a quantum tunneling event, and has a cer- grand unified field theory, the relevant energy den- tain probability of occurrence per unit time per sities are larger than this and the associated unit volume, I /V. Once the bubble materializes, lengths correspondingly smaller. We are dealing it expands with a speed asymptotically approaching here with phenomena which take place on scales that of converting false vacuum into true as light, neither subnuclear nor astronomical, but rather it grows. civic, or even domestic, scales far too smal1. to In the semiclassical (small P~ )limit, I'/'V admits neglect if we are interested in the cosmological an expansion of the form consequences of vacuum decay. Contrary to naive r/V=&e-s ~"I1+O(S)]. expectation, the inclusion of gravitation is not point-

3305 1980The American Physical Society 3306 SIDNEY COI. EMAN AND FRANK DE I, UCCIA 21

mation. In the two special cases described above, we explicitly compute the effects of gravitation on the decay coefficient J3of Eq. (1.2). As we have just argued, for any conceivable application, these effects are too small to worry about. Neverthe- less, when we discovered it was within our power to compute them, we were unable to resist the temptation to do so. We have made no attempt to study the effects of gravitation on the coefficient A. This computation would involve the evaluation FIG. 1. The potential U(p) for a theory with a false of a functional determinant; even if we had the V RCUUDl. courage to attempt such an evaluation, we would be frustrated by the nonrenormalizability of our the- less,' indeed, any description of vacuum decay that ory. In Sec. IV we study the growth of the bubble. neglects gravitation is seriously incomplete. Section V states our conclusions. We have tried to Not only does gravitation affect vacuum decay, write it in such a way that it will be intelligible to vacuum decay affects gravitation. In Eq. (1.1), a reader who has skipped the intervening sections. there is ro absolute zero of energy density; adding a constant to U has no effect on physics. This is not so when we include gravitation: II. OLD RESULTS SUMMARIZED

~ In this section, we recapitulate the known results d'&q:g [,'g s-„ys,y —U(y) —(16~a)-'Il], on vacuum decay in the theory defined by Eq. (1.1). The reader who wishes the that be- (1.4) arguments lie hind our assertions is referred to the original lit- where R is the curvature scalar. Here, adding a erature. ' constant to U is equivalent to adding a term pro- The Euclidean action is defined as minus the for- portional to g-g to the gravitational Lagrangian, mal analytic continuation of Eq. (1.1) to imaginary that is' to say, to introducing a cosmological con- time, stant. Thus, once the vacuum decays, gravita- d' tional theory changes; the cosmological constant [.'(s, e)"-U(e)], (2.1) inside the bubble is different from the one outside where the metric is the usual positive-definite the bubble. Hence, in our computations, we need one of Euclidean four-space. The Euclidean equa- an initial condition not needed in the absence of tion of motion is the Euler-Lagrange equation as- gravitation. We must specify the initial value of sociated with Ss. I et p be a solution of this equa- the cosmological constant; equivalently, we must tion such that (1) Q approaches the false vacuum, specify the absolute zero of energy density. Q„at Euclidean infinity, (2) Q is not a constant, The observation that the current experimental and (3) P has Euclidean action less than or equal value of the cosmological constant is zero gives to that of any other solution obeying (1) and (2). two cases special interest, (1) U(P, ) is zero. Then the coefficient J3in the vacuum decay amplii- w' This would be the appropriate case to study if e tude is given by were currently living in a, false vacuum whose — apocalyptic decay is yet to occur. (2) U(P ) is a= s,(y) s,(y, ) . (2.2) zero. This would be the case to study " appropriate P is called "the bounce. (The name has to do if we were living after the apocalypse, in the de- with the corresponding entity in particle mecha- bris of a false vacuum which decayed at some nics. ) early time in the history of the universe. Although For the theories at hand, it can be shown' that our methods a,re applicable to arbitrary initial the bounce is always O(4) symmetric, that is to value of the cosmological constant, we pay special say, p is a function only of p, the Euclidean dis- attention to these two cases. tance from an appropriately chosen center of co- The organization of the remainder of this paper ordinates. The Euclidean action then simplifies, is as follows: In Sec. II we summarize the theory of vacuum decay in the absence of gravitation. We S =2~' p'&p[-', (4')'+ (2.3) ~ U], emphasize the thin-wall approximation, the ap- 4p proximation that is valid in the limit of small en- as does the equation of motion ergy-density difference between true and false vacuum. In Sec. III we begin the extension to gra- dU 3, (2.4) vitation, again emphasizing the thin-wall approxi- p dP' 21 GRAVITATIONAL EFFECTS ON AND OF VACUUM DECAY 3307

where the prime denotes d/dp. thin-wall approximation, we will also justify our It is possible to obtain an explicit approximation ea,rlier neglect of the P' term in Eq. (2.4). Away for Q in the limit of small energy-density differ- from the wall, this term is negligible because P' ence between the two vacuums. I et us define e by is negligible; at the wall, it is negligible because p ls large, ~ = U(4, ) —U(0 ), (2.5) We will. now determine P by computing Bfrom and let us write U as Eq. (2.2) and demanding that it be stationary under variations of p. The region of integration breaks U($) = Uo(g) +0(&), (2.5) naturally into three parts: outside the wall, in- where Uo is a function chosen such that Uo(p) side the wall, and the wall itself. We divide B ac- = U, (Q,), and such that dUO/dQ vanishes at both Q, cordingly. Outside the wall, Q Q„. Hence, and . Q (2.13) The approximate Q obeys the equation Inside the wall, P = P . Hence, dUo 7) 2 (2. 71 4 hillside '= p & ~ (2.14) 2 Note that we have not only discarded the term in Within the wall, in the thin-wall approximation, Eq. (2.4) proportional to e, we have also discarded the term proportional to P'. We will justify this shortly. Equation (2.7) admits a first integral,

[-,'(y')' —U, ]' = O. (2.8) (2.15) Equation (2.9) gives us an integral expression for Its value is determined by the condition that P(~) ls sz (2.9) dt(2[U. (e) —U.(4.) ]]". (2.16) Thus, as p traverses the real line, Q goes mono- For future reference, we note that we can also tonically from P to . Equation (2.9) deter- P, write S, as mines P in terms of a single integration constant. We will choose this to be the point at which is p, P = 2 — the average of its two extreme values: S, dp [U,(A) U, (4,)]. (2.17)

'~' We can now compute dP [2(U —U (P,) ] = p —p . (2.10) "(4++0 ) /2 a= --,'~ p'c+ 2~'p'S, . (2.18) Thus, for example, if ' This is stationary at Uo = 8 A. (&P' —p.'/X) (2.11) p = 3S1/E . (2.19) then We have justified our approximation: p indeed becomes large when e becomes small. We now know tanh[-'V(p -P)]. (2.12) B=27w S~ /2e (2.20) All that remains is to determine P. We will do this on the assumption that p is large compared to We have used the bounce to compute a coefficient the length scale on which Q varies significantly. which enters into the probability for the quantum For the example of Eq. (2.11), this means that p p, materialization of a bubble of true vacuum within is much greater than one. This assumption will be the false vacuum. We can a,iso use the bounce to justified (for sufficiently small e) at the end of the describe the classical growth of the bubble after computation. its materialization. The surface t = 0 is the inter- If p is large, the bounce looks. like a ball'. of true section of Euclidean space (imaginary time) and vacuum, P = Q, embedded in a sea; of false vac- Minkowski space (real time). It can be shown uum, P = P„with a transition region ("the wall" ) that the value of P on this surface can be thought separating the two. The wall is small in thickness of a.s the configuration of the field at the moment compared to the radius of the ball; in our example, the bubble materializes. Also, at this moment, the its thickness is O(p, '). It is for this reason that time derivative of the field is zero. These initial- the approximation we are"describing is called "the value data, together with the classical field equa- thin-wall approximation. When we justify the . tions in Minkowski space, suffice to determine the 3308 SIDNEY COLEMAN AND FRANK. DE LUCCIA 21 growth of the bubble. the three-spheres through which they pass. We Of course, there is no need to go to the bother of choose our radial coordinate ( to measure dis- explicitly solving the classical field equations. All tance along these radial curves. Thus, the ele- we need to do is analytically continue the Euclidean ment of l.ength is of the form solution we already possess; that is to say, all we need to do is make the substitution (3.1)

where (dQ)' is the element of distance on a unit Thus, Euclidean O(4) invariance becomes Minkow- three-sphere and p gives the radius of curvature skian O(3, 1) invariance. In the thin-wall approxi- of each three-sphere. Note that rotational invari- mation, the bubble materializes at rest with radius ance has made its usual enormous simplification; P. As it grows, its surface traces out the hyper- ten unknown functions of four variables have been boloid p =p. Since p is typically a quantity of sub- reduced to one unknown function of one variable. nuclear magnitude, this means that from the view- Note also that we can redefine t' by the addition of point of macrophysics, almost immediately upon a constant without changing the form of the metric; its. materialization the bubble accelerates to essen- equivalently, we can begin measuring g from tially the speed of light and continues to grow in- wherever we choose. definitely at that speed. Given Eq. (3.1), it is a straightforward exercise in the manipulation of Christoffel symbols to com- III. INCLUSION OF GRAVITATION: MATERIALIZATION pute the Euclidean equations of motion. We will OF THE BUBBLE give only the results here. The scala. r field equation is In this section, we begin the extension of the 3P dU analysis of Sec. II to the theory described by Eq. ~, (3.2) (1.4), the theory of a scalar field interacting with gravity. where the prime denotes d/dt'. The Einstein equa- As before, we begin by constructing a bounce, a tion solution of the Euclidean field equations obeying ~ (3.3) appropriate boundary conditions. This is appar- LK KK ently a formidable task; we now have to keep track where K = BpG, becomes of not just a single scalar field but also of the ten = components of the metric tensor. However', things p" 1+ —,'ap'(-,'y" —U) . (3.4) are not as ba, d as they seem, for there is no rea.- The other Einstein equations are either identities son for gravitation to break the symmetries of the or trivial consequences of these equations. Fi- purely scalar problem. Thus it is reasonable to assume that, in the presence of gravity as in its nally, absence, the bounce is invariant under four-dimen- S = 2m' ' Q" + sional rotations. d( (p'(-, U) We emphasize that, in contrast to the case of a single scalar field, we have no theorem to back up (3.5) this assumption. We will shortly construct, in the thin-wall approximation, an invariant bounce, but In the thin-wall approximation, the construction this will still leave open the possibility that there of the bounce from these equations is astonishingly exist noninvariant bounces of lower Euclidean ac- simple. Equation (3.2) differs from its counter- tion. We do not think it likely that such objects part in the pure scalar case, Eq. (2.4), in only two exist, but we cannot prove they do not, and the respects. Firstly, the independent variable is reader should be warned that if they do exist, they called t' rather than p. This is a trivial change. dominate vacuum decay, and all our conclusions Secondly, the coefficient of the P' term involves a are wrong. factor of p'/p rather than one of 1/p. But this is We begin by constructing the most general rota- also a trivial change, since in the thin-wall approx- tionally invariant Euclidean metric. The orbits of imation we neglect this term anyway. (This is a. the rotation group are three-dimensional mani:- bit facile. Of course, because the term is different folds with the geometry of three-spheres. On each in form, the eventual self-consistent justification of these spheres, we introduce angular coordinates of the approximation must also be different in in the canonical way. We define a radial curve to form. We will deal with this problem when we be a curve of fixed angular coordinates. By rota- come to it.) Thus, in the thin-wall approximation, tional invariance, radial curves must be normal to we need only to copy Eq. (2.10), 21 GRA VITATIONAL EFFECTS ON AN D OF VACUUM DECAY u09 )

— 12' d4(2[U. U. (4,)]}"'=5 -5 (3.6) Binside "&@+++ )l2

Here $ is an integration constant, but, as we have explained, one with no convention-independent " meaning. (U(4 ) 'gl-l~P'U(4 ))"-I] Once we have P, we can solve Eq. (3.4) to find p. This is a first-order differential equation; to (3.13) specify its solution, we need one integration con- [As a consistency check, it is easy to verify that stant. We will choose this to be this reduces to Eq. (2.14) when g goes to zero. ] This an and continue our p -=p((). (3.7) is ugly expression, to investigation in full generality would quickly in- This does have a convention-independent meaning; volve us in a monstrous algebraic tangle. Thus it is the radius of curvature of the wall separating we now restrict our attention to the two cases of false vacuum from true. We do not need the ex- special interest identified in Sec. I. plicit expression for p for our immediate pur- The first special case is decay from a space of poses, so we will not pause now to construct it. positive energy density into a space of zero ener- Our next task is to find p. The computation is gy density. This is the case that is relevant if we patterned on that in Sec. II: First we compute B, are now in a postapocalyptic age. In this case the difference in action between the bounce and the U()t), ) =e, U(P ) =0. (3.14) false vacuum. Then we find p by demanding that Bbe stationary. It is then a trivial exercise to show that Bis sta- We first eliminate the second-derivative term tionary at from Eq. (3.5) by integration by parts; the surface term from the parts integration is harmless 4e + 3&S,' because we are only interested in the action difference between two solutions that agree at infinity. p0 (3.15) We thus obtain 1+ (p,/2A)' ' where p, = 3S,/e, the bubble radius in the absence ' — ' s, =s ss( ')-', s"+s)- ( "+ )). (3.8) of gravity, and A= (Ke/3) ', as in Eq. (1.3). At this point,

now use 4) to eliminate p'. We find BD e Eq. (3. (3.16) [1+(p,/2A)']' ' 3p Sa = 4w' d$ p'U-— (3.9) where B,=27))'S,'/2e', the decay coefficient in the K absence of gravity. These equations have some interesting proper- So far, we have made no approximations. We ties, but we will postpone discussing them until now evaluate B, from Eq. (3.9), in the thin-wall we write down the corresponding equations for the approximation. As before, we divide the integra- second special case. This is decay from a space tion region into three parts. Outside the mall, of zero energy density into a space of negative en- bounce and false vacuum are identical; thus, as be- density, the case that is relevant if we are fore, ergy now in a preapocalyptic age. In this case B-t.d. = o. (3.10) U(y„) =0, U(y ) =-~ . (3.17) In the wall, we can replace by and Uby p P, U, As before, trivial algebra shows that Thus, pa (3.18) = 4&'P ' — 1 — /2A)' B .)) 4 [UQ(4') UQ(4'+) ] (p, and —2mpS, (3.11) Inside the a constant. BQ by Eq. (2.17). wall, P is — /2A)']' ' (3.19} Hence, [1 (p, '~' These equations have been derived in the thin- d$ =dp(1 ——,'~p'U) (3.12) wall approximation. Before we discuss their im- and plications, we should discuss the reliability of the 3310 SIDNEY COLEMAN AND FRANK DE LUCCIA 21 approximation. In the absence of gravitation, the 4n E=-—ep +4nS~ thin-wall approximation was valid if p was large 3 compared to the characteristic range of variation of P; the significant quantity was P because 1/p (3.22) multiplied the neglected P' term in Eq. (2.4). In the presence of gravitation, 1/p is replaced by We see that this vanishes for the actual bubble, p p'/p [see Eq. (3.2) ]; thus it is this quantity that p p Thi s is as it should be. The energy of the must be small at the wall. world vanishes before the bubble materializes, By Eq. (3.4), and, whatever else barrier penetration may do, it does not violate the conservation of energy. — , =—,+ (-, y' —U). (3.20) We now see how to compute the effects of gra- p p vitation on the bubble radius, in the limit that The left-hand side of this equation is certainly these effects are small. All we have to do is to small if both terms on the right are small. The compute the effects of gravitation on the total en- first term is just (1/p)', as before. As for the ergy of the unperturbed bubble. If the gravitational second term, the qua. ntity in parentheses is ap- contribution is positive, the bubble radius wiQ have proximately constant over the wall, vanishes on to grow, so Eq. (3.22) will develop a small nega- one side of the wall, a,nd has magnitude e on the tive contribution and total energy will remain zero. other. Thus, it is certainly an overestimation to Qn the other hand, if the gravitational contribution replace it by c everywhere; this turns the second is negative, the bubble will have to shrink. Of term into (1/A)'. course, we already know that it is the former al- Thus, the thin-wall approximation is justified if ternative that prevails, and not just in the limit of both p and A are large compared to the character- small gravitational effects. However, the point of istic range of variation of P. This condition puts the computation is to understand why it prevails. no restraint on p,/A, the ratio that measures the There a,re two terms in the gravitational contri- importance of gravitation; thus, it is not senseless bution to the energy of the unperturbed bubble. to discuss our results for arbitrarily large values The first is the ordinary Newtonian potential ener- of this ratio. (Although it is not senseless, it is gy of the bubble. This is easily computed by in- useless; as we said in Sec. I, in any conceivable tegrating over all space the square of the gravi- application this ratio is negligible. We will pro- tational field, itself easily computed from Gauss's ceed anywa, y.) law. The answer is In the first special case, decay into the present (3.23) condition, we see that gravitation makes the materialization of the bubble more likely (diminishes Note that this is negative, as a gravitational po- B), and makes the ra.dius of the bubble at its mo- tential energy shouM be. The second term comes ment of materialization smaller (diminishes p). from the fact that the nonzero energy density in- In the second special case, decay from the pre- side the bubble distorts its geometry. Thus, there sent condition, things are just the other way is a correction to the volume of the bubble and thus around; gravitation makes the materialization of to the volume term in the bubble energy. We can the bubble less likely and its radius larger. In- determine this correction to the volume from Eq. deed, gravitation can totally quench vacuum decay; (3.12); the infinitesimal element of volume is at p, =2A or, equivalently, 4~p d$ = 4''dp (1 ——,'p'/A') + O(G') . (3.24) 3 2 E = (3.21) gKSj Note that this is smaller than the Euclidean for- the bubble radius becomes infinite and the decay mula; thus this geometrical correction reduces probability vanishes. For higher values of po or, the magnitude of the negative volume energy and equivalently, smaller values of e, our equations hence makes a positive contribution to the total admit of no sensible solution at all. Gravitation energy. Integration yields has stabilized the false vacuum. E = 2 epv/50A' (3.25) We believe we understand this surprising phe- ~ nomenon. Our explanation begins with a computa- in the small-G limit. The total gravitational cor- tion of the energy of a thin-walled bubble, in the rection is the sum of these two terms, absence of gravitation, at the time of its materiali- E „=vapo'/3A'. (3.26) zation. For the moment, we will give the bubble an arbitrary radius p, postponing use of our knowl- This is positive; hence, the bubble is larger in the edge that P is p,. The energy is the sum of a nega- presence of gravitation than in its absence. tive volume term and a positive surface term, We now understand what is happening. Vacuum 21 GRAVITATIONAL EFFECTS ON AND OF VACUUM DECAY decay proceeds by the materialization of a bubble. boloid with spacelike normal vector in Minkowski By energy conservation, this bubble always has space. [The overall minus sign has appeared be- energy zero, the sum of a negative volume term cause we adhere to the convention that a Minkow- and a positive surface term. In the absence of ski metric has signature (+ ———).j In this region, gravity, we can always make a zero-energy bubble @ is Q(p). In the thin-wall approximation, the bub- no matter how small c is; we just have to make the ble wall almays lies within this region at p =p. bubble large enough, and the volume/surface ra- Thus, this is all the manifold we need if we are tion will do the job. However, in the presence of only interested in studying the expansion of the gravity, the. negative energy density inside the . bubble as it appears from the outside. bubble distorts the geometry of space in such a However, if we wish to go inside the bubble, we way as to diminish the volume/surface ratio. Thus may encounter vanishing p. This is a pure coor- it is possible that, for sufficiently small e, no bub- dinate singularity analogous to reaching the light ble, no matter how big, will have energy zero. cone, the boundary of the spacelike region, in Min- What Eq. (3.18) is telling us is that this is indeed kowski space. We get beyond the singularity by the case. continuing to the timelike regi. on. We choose $ to We cannot exclude the possibility that decay be zero when p is zero, and continue to $ =i7, mith could proceed through nonspherical bubbles, al- v real. We thus obtain though we think this is unlikely; we guess thai such ds' = dr' — (t7)'(dAr)', (4.2) configurations would only worsen the volume/sur- p face ratio. Nor can me exclude the possibility that where dQ~ is the element of length for a. unit hy- decay proceeds through some nonsemiclassical perboloid with timelike normal in Minkowski space. process, one that does not involve bubble forma- 0'ne way of describing this equation is to say that tion at all, and whose probability vanishes more the interior of the bubble always contains a Ro- rapidly than exponentially in the small-A limit. bertson-Walker universe of open type. About this possibility me cannot even make gues- I et us nom apply this general prescription to ses. the specia, l cases we have analyzed in Sec. III. In the thin-wall approximation, no analytic continua- IV. INCLUSION OF GRAVITATION: GROWTH OF THE tion is needed for it is equal to outside the BUBBLE P; P, bubble and &f& inside the bubble. The metric out- In Sec. II, we explained how to obtain a descrip- side the bubble is obtained by solving Eq. (3.4): tion of the classical growth of the bubble after its P quantum materialization, in the absence of gravi- p"=1- V{y,). (4.3) tation. All we had to do was analytically continue the scalar field Q from Euclidean space to Min- Inside the bubble, it is obtained by solving the same kowskian space. Because of the enormous sym- equation with P, replaced by P . The two metrics metry of the bounce —O(4)-invariant in Euclidean are joined at the bubble wall, not at the same $, space, O(3, 1)-invariant in Minkowski space —for but at the same p, p =p. much of Minkowski space this continuation was In our first special case, decay into the present trivial. To be more precise, if we choose the cen- condition, U vanishes inside the bubble. Thus the ter of the bubble at its moment of materia, lization interior metric is p =$, ordinary Minkowski space. to be the center of coordinates, for all spacelike Outside the bubble, Eq. (4.3) becomes points the contribution was a mere reinterpretation p/2 1 p2/A2 (4.4) of p a.s Minkowskian spacelike separation rather than Euclidean radial distance. It mas only for The solution is timelike points that nontrivial continuation (to p =Asin($/A). (4 6) imaginary p) was needed. All of this carries over to the case in which We shall now shorn that this is ordinary de Sitter gravitation is present. The only difference is that space, written in slightly unconventional coordi- we have to continue the metric as mell as the sca- nates. 4 lar field, turning an O(4)-invariant Euclidean man- We begin by recapitulating the definition of de ifold into an O(3, 1)-invariant Minkowskian mani- Sitter space. Consider a five-dimensional Min- fold. kowski space with O(4, 1)-invariant metric Thus, a large part of the manifold is analogous to the spacelike region described above. Here, ds' = —(dw)'+ (dt)' —(dx)' ' — )' —(ds)' ds' = —dt —p (&)'(dQs)', (4.1) (dy . (4.6) where dA~ is the element of length on a unit hyper- In this space, consider the hyperboloid defined by 3312 SIDNEY COLEMAN AND FRANK DE LUCCIA

A =81 —t +x +y +Z (4 7) verse. We normally think of oscillating universes as necessarily closed, but this rule depends on the with A some positive number. This is a four-di- positivity of energy, very much violated here. mentional manifold with a Minkowskian metric; it The metric defined by Eq. (4.12) has singulari- is de Sitter space. Note that de Sitter space is as ties when T is an integral multiple of m&. We homogeneous as Minkowski space; any point in the shall now show that these singularities are spur- space can be transformed into any other an by ious, mere coordinate artifacts. O(4, 1) transformation. Consider a five-dimensional Minkowski space To put the metric of de Sitter space into our with O(3, 2)-invariant metric standard form, we must choose the location of the center of the bubble at its moment of materializa- ds' = (dw)'+ (dt)' —(dx)' —(dy)' —(dg)'. (4.13) tion. Since de Sitter space is homogeneous, we In this space, consider the hyperboloid defined by can without loss of generality choose this point to 2 be (A, 0, 0, 0, 0). The O(3, 1) group of the vacuum A 2 =so +PM —x 2 —J 2 —82 (4.14) decay problem is then the I orentz group acting on with A some positive number. This is a four-di- the last four coordinates. Thus we replace these mensional manifold with a Minkowski metric; in- by "angular" coordinates, as in Eq. (4.1): deed, by applying the same coordinate transfor- ds' = -(dw)' —(dp)' —p'(dQs)'. ($.8) mations we used in our analysis of de Sitter space, one can easily show that the metric is that defined Equation (4.7) then becomes by Eqs. (4.11) and (4.12). A' =re'+ p'. Although the hyperboloid is free of singularities, it is not totally without pathologies; it contains If we now define & by closed timelike curves, for example, circles in w = A cos($ /A), p = A sin(t /A), (4 9) the zo-t plane. These can easily be eliminated. The hyperboloid is homeomorphic by xS'. (That the metric falls into the desired form. R, is to say, once we have freely given x, y, and z, In this metric, p is bounded above by A. The zv and t must lie on a circle. Thus, it is not sim- geometrical reason for this is clear from Eq. ) ply connected, and we may replace it by its simply (4.7). A spacelilte slice of de Sitter space (say the connected covering space. [In Eq. (4.12), this hypersurface t = 0) is a hypersphere of radius A; corresponds to interpreting 7 as an ordinary real on a hypersphere, no circle has greater circum- va.riable rather than an angular variable. This ference than a great cir'cle. This also explains a ] covering has neither singularities nor closed curious feature of Eq. (3.15). , timelike curves. It is much like de Sitter space, except that its symmetry group is O(3, 2) rather pp ' (3.15) 1+ (p,/2A)' than O(4, 1); it is called anti —de Sitter space. ' Anti-de Sitter space is the universe inside the No matter how we choose is always less than p„p bubble. We shall now show that this universe is or equal to A. The reason for this is now obvious; dynamically unstable; even the tiny corrections the bubble cannot be bigger than this because a to the thin-wall approximation are sufficient to bigger bubble could not fit into the false vacuum. convert the coordinate singularity in Eq. (4.12) We now go to our second special case, the decay into a genuine singularity, to cause gravitational of the present vacuum. Here U vanishes outside collapse. the bubble, so it is the exterior metric that is or- For our discussion, we need the exact field dinary Minkowski space. Inside the bubble, Eq. equations in the timelike region. These are (4.3) becomes

' 3p ~ dU p" = 1+p'/A'. (4.10) Q+ —P+ =0 (4.15) p dP The solution is

p = A sinh($ /A) . (4.11) p' =1+ (-'4'+ &), (4.16) Since we are now inside the bubble, we will also need the continuation of this to the timelike region, where the dot indicates differentiation with respect the Robertson-Walker universe inside the bubble. to T. In general, initial-value data for this system By Eq. (4.2) this is consists of a point in the P-P plane at some value ds' = dv' —A' sin'(T/A) [dn~]'. (4.12) of p. (The associated value of ~ has no convention- independent meaning. ) In the special case of This is an open expanding-and-contracting uni- vanishing p, a nonsingular solution must have va- 21 GRAVITATIONAL EFFECTS ON AND OF VACUUM DECAY

'~' nishing P, jpst as, in Euclidean space, a rotation- p = ap cos(m Ap, - p p+ 8) . (4.23 ally invariant function must have vanishing gra- If we attempt to continue all the to the sec- dient at the origin of coordinates. way ond zero of predicted by Eq. (4.19), becomes For notational simplicity in what follows, w' e p P large and our approximations break down. How- choose the zero of P to be the true vacuum P =0. ever, whatever happens to a second zero is in- Also, we define P, evitable. From Eq. (4.16) and the assumed prop- d U erties of U, (4.1 t)

p )] p/A (4.24) Until now, we took the initial-value data at p = T = 0 to be P = P =0. This led to the solution Thus, once p is much less than A and diminishing, (4.18) it must continue to diminish all the way to zero, and must do so in a time of order and it quickly, p. Qf course, vanishing p does not imply singular p = A sin7/A. (4.19) behavior. Qn the contrary, we can always obtain a one-pa, rameter family of nonsingular solutions Now, vanishing P at p = 0 is an exact result; the by integrating the equations of motion backward bounce is rotationally invariant. But vanishing g from the second zero of using as final-value is just an approximation; as can be seen from the p, data any point on the line g = 0. If we continue this explicit formula for the bounce, at p =0, P is family of solutions into the region of validity of O(exp t-pp]), exponentially small but not zero. Eq. (4.23), then, . for any fixed they will define As long as remains exponentially small, we p, P a curve in the p-2I2 plane. Because we are inte- can neglect its effects on and continue to use Eq. p grating the equations of motion over a very short (4.19). Also, we can replace Eq. (4.15) by its lin- time interval, this curve can have only a negligible ear approximation dependence on the small parameter e, the energy- density difference between the two vacuums. Qn 0+—0+ p'y=o ~ (4.20) p the other hand, the angle in the P-P plane obtained from Eq. (4.23) at fixed p is a very rapidly vary- Of course, if in the course of time P grows large, ing function of e because A is proportional to e '~'. we can no longer make these approximations and Thus for general Eq. (4.23) does not continue must return to the exact equations. e, to a. nonsingular solution. This argument does not We begin by solving Eq. (4.20) for p/A«1. In eliminate the possibility that there might be spec- this region, p is approximately ~, anf Eq. (4.20) is ial values of e for which the singularity may be a Bessel equation. Thus, is simply an exponen- P avoided. However, we believe this possibility is tially small multiple of the nonsingular oscillatory of little interest; an instability that can be removed, solution of this equation and remains exponentially only by fine tuning the parameters of the theory is small throughout this region. going to return once we consider other corrections We next turn to the region p, » 1. Note that un- p to our approximation (e.g. the effects of a small der the conditions of the thin-wall approximation , initial matter density). This completes the argu- this overlaps the preceding region. In this region, ment for gravitational collapse of the bubble inter- Eq. (4.20) is the Newtonian equation of motion for ior. a weakly damped harmonic oscillator, with a slow- ly varying damping coefficient 3p/p. Thus, V. CONCLUSIONS cos(2r+ 2) exp — dr 22/22) (4.21) Although some of the results in the body of this ( paper hold in more general circumstances, we Ol will restrict ourselves here to the thin-wall approximation, the approximation that is valid in the =ap '~'cos(ps+ 9). (4.22) P limit of small energy-density difference between Here a is an exponentially small coefficient and true and false vacuum, and to the two cases of 8 is an angle independent of any of the parameters special interest identified in Sec. I. of the. theory, whose explicit value is of no inter- The first special case is decay from a space of est to us. We see that P remains exponentially vanishing cosmological constant, the case that small throughout the expansion and subsequent applies if we are currently living in a false vac- contraction of the universe, all the wa, y down to uum. In the absence of gravitation, vacuum decay the region where p/A is once again much less than proceeds through the quantum materialization of a one. In this region, we may rewrite Eq. (4.22) as bubble of true vacuum, separated by a thin wall SIDNEY COLEMAN AND F RANK DE LUC CIA 21

from the surrounding false vacuum. The bubble is collapse of the interior universe is on the order at rest at the moment of materialization, but it of the time A discussed in Sec. I, microseconds rapidly grows; its wall traces out a hyperboloid in or less. Minkowski space, asymptotic to the light cone. This is disheartening. The possibility that we If all we are interested in is vacuum decay as are living in a false vacuum has never been a seen from the outside, not a word of the preceding cheering one to contemplate. Vacuum decay is description needs to be changed in the presence of the ultimate ecological catastrophe, ' in a new va- gravitation. At least at first glance, this is sur- cuum there are new constants of nature, ' after va- prising because one would imagine that gravitation cuum decay, not only is life as we know it impos- would have some effect on the growth of the bub- sible, so is chemistry as we know it. However, ble. There are two ways of understanding why this one could always draw stoic comfort from the does not happen. The first is a mathematical way: possibility that perhaps in the course of time the The growth of the bubble in the absence of gravita- new vacuum would sustain, if not life as we know tion is O(3, 1) invariant; the inclusion of gravita- it, at least some structures capable of knowing tion does not spoil this invariance; the only O(3, 1)- joy. This possibility has now been eliminated. invariant hypersurfaces are hyperboloids with The second special case is decay into a space light cones as their asymptotes. The second is a of vanishing cosmological constant, the case that physical way: Quantum tunneling does not violate applies if we are now living in the debris of a the law of conservation of energy; thus the total false vacuum which decayed at some early cosmic energy of the expanding bubble is always identically epoch. This case presents us with less interesting zero, the negative energy of the interior being physics and with fewer occasions for rhetorical ex- canceled by the positive energy of the wall. Be- cess than the preceding one. It is now the interior cause the bubble is spherically symmetric, the of the bubble that is ordinary Minkowski space, gravitational field at the outer edge of the wall is and the inner edge of the wall that continues to determined exclusively by the total energy within. trace out a hyperboloid. The mathematical reason That is to say, it is zero, and neither accelerates for this is the same as before. The physical rea- nor retards the growth of the edge. son is even simpler than before: Within a spheri- Of course, gravitation affects the quantitative cally symmetric shell of energy there is no gra- features of vacuum decay. In any conceivable ap- vitational field. plication, these effects are totally negligible, but As before, the effects of gravitation are neglig- we have computed them anyway. In general, gra- ible in any conceivable application, but we have vitation makes the probability of vacuum decay computed them anyway. The sign is opposite to smaller; in the extreme case of very small en- that in the previous case; gravitation makes vac- ergy-density difference, it can even stabilize the uum decay more likely. false vacuum, preventing vacuum decay altogether. The space-time outside the bubble is now con- We believe we understand this. For the vacuum ventional de Sitter space. Of course, neither this to decay, it must be possible to build a bubble of space nor the Minkowski space inside is subject total energy zero. In the absence of gravitation, to cata.strophic gravitational collapse initiated by this is no problem, no matter how small the en- small perturbations. ergy-density difference; all one has to do is make Finally, we must comment on the problem of the the bubble big enough, and the volume/surface ra- cosmological constant in the context of spontaneous tio will do the job. In the presence of gravitation, symmetry breakdown. This problem was raised though, the negative energy density of the true some years ago' and we have little new to say vacuum distorts geometry within the bubble with about it, but our work here has brought it home to the result that, for a small enough energy density, us with new force. there is no bubble with a big enough volume/sur- Normally, when something is strictly zero, there . face ratio. is a reason for it. Vector-meson squared masses Within the bubble, the effects of gravitation are vanish because of gauge invariance, those of Dirac more dramatic. The geometry of space-time with- fields because of chiral symmetry. There is no- in the bubble is that of anti-de Sitter space, a thing to keep the squared masses of scalar me- space much like conventional de Sitter space ex- sons zero, but it is no disaster if they go negative, cept that its group of symmetries is O(3, 2) rather merely a sign that we are expanding about the than O(4, 1). Although this space-time is free of wrong ground state. singularities, it is unstable under small perturba- But there is no reason for the cosmological con- tions, and inevitably suffers gravitational collapse stant (equivalently, the absolute energy density of of the same sort as the end state of a contracting the vacuum) to vanish. Indeed, if it were not for Friedmann universe. The time required for the the irrefutable empirical evidence, one would ex- 21 GRAVITATIONAL EFFECTS ON AND OF VACUUM DECAY 83l5 pect it to be a typical microphysical number, the change, and change at a length scale much larger radius of the universe to be less than a kilometer. than the. Planck length. Even worse, zero energy density is the edge of disaster; even the slightest negativity would be ACKNOWLEDGMENTS enough to initiate catastrophic gravitational col- This work was supported in part by the Depart- lapse. ment of Energy under Contract No. DE-AC03- There is something we do not understand about V6SF00515 and Contract No. DE-AS02- V6ER02220 gravitation, and this something has nothing to do and by the National Science Foundation under with loops of virtual gravitons. There has to be Grant No. PHYV7-22864.

*Permanent address: Department of Physics, Harvard S. Coleman, V. Glaser, and A. Martin, Commun. Math. University, Cambridge, Mass. 02138. Phys. 58, 211 (1978). We follow here the treatment given in S. Coleman, Phys. Our treatment of de Sitter space closely follows that of Rev. 0 15, 2929 (1977); 16, 1248(E) (1977); and C. G. S. Hawking and G. Ellis, The Large Scale St~cture of Callan and S. Coleman, ibid. 16, 1762 (1977). These Space-Time (Cambridge University Press, New York, contain references to the earlier literature. 1973). A. Linde, Pis'ma Zh. Eksp. Teor. Fix. 19, 320 (1974) 5For more on anti —de Sitter space, see Hawking. and [JETP Lett. 19, 183 (1974}); M. Veltman, Phys. Rev. Ellis (Ref. 4). Lett. 34, 777 (1975).