Stability of self-gravitating magnetic monopoles

18 Pages·0.34 MB·English

by  Guillermo Arreaga

Stability of self-gravitating magnetic monopoles
Guillermo Arreaga∗
Departamento de F´ısica, Centro de Investigaci´on y de Estudios Avanzados del IPN
Apdo. Postal 14-740,07000 M´exico, DF, MEXICO
Inyong Cho†
Department of Physics, Emory University, Atlanta, Georgia 30322-2430, USA
Jemal Guven‡
Instituto de Ciencias Nucleares, Universidad Nacional Aut´onoma de M´exico
Apdo. Postal 70-543, 04510 M´exico, DF, MEXICO
The stability of a spherically symmetric self-gravitating magnetic monopole is examined in the
thin wall approximation: modeling the interior false vacuum as a region of de Sitter space; the
0
exterior as an asymptotically flat region of the Reissner-Nordstr¨om geometry; and the boundary
0
separating the two as a charged domain wall. There remains only to determine how the wall gets
0
embeddedinthesetwogeometries. Inthisapproximation,theratiok ofthefalsevacuumtosurface
2
energy densities is a measure of the symmetry breaking scale η. Solutions are characterized by

n this ratio, the charge on the wall Q, and the value of the conserved total energy M. We find that
a for each fixed k and Q up to some critical value, there exists a unique globally static solution,
J with M ≃ Q3/2; any stable radial excitation has M bounded above by Q, the value assumed in
5 an extremal Reissner-Nordstr¨om geometry and these are the only solutions with M < Q. As M
2
is raised above Q a black hole forms in the exterior: (i) for low Q or k, the wall is crushed; (ii)

for higher values, it oscillates inside the black hole. If the mass is not too high these ‘collapsing’
1
solutions co-exist with an inflating bounce; (iii) for k, Q or M outside the above regimes, there is
v
8 a unique inflating solution. In case (i) the course of the bounce lies within a single asymptotically
7 flat region (AFR) and it resembles closely the bounce exhibited by a false vacuum bubble (with
0 Q=0). Incases(ii)and(iii) thecourseofthebouncespanstwoconsecutiveAFRs. However,from
1 the point of view of either region it resembles a monotonic false vacuumbubble.
0
0 I. INTRODUCTION for an inflationary universe [5].
0
They discovered that the essential radial dynamics of
/
c thescalarfieldiscapturedextremelyaccuratelybyasim-
q Several years ago, Linde and Vilenkin pointed out the ple one dimensional mechanical caricature of the field
r- possibility that the core of a localized topological defect configuration. Thisledthemtoconsiderthedynamicsof
g could inflate under appropriate conditions in a process a spherically symmetric region of false vacuum which is
v: that was aptly dubbed topological inflation [1,2]. See separatedbyadomainwallfromaninfinite truevacuum
i also [3]. exterior (with zero energy density).
X
A common characteristic of such defects is some non- The energy momentum tensor of false vacuum is in-
r linear scalar field (the Higgs field) forced up in its core herently homogeneous and isotropic. The Birkhoff theo-
a
intoaconstantexcitedfalsevacuumstate,fallingthrough remthenconstrainsthespacetimeitoccupiestocoincide
a transition layer to the true vacuum value remote from with some patch of de Sitter space; the exterior is sim-
the core. ply the Schwarzschild geometry truncated at the false
Without further refinement, a configuration of this vacuum boundary. In this model, it becomes clear that
type will always collapse, with or without gravity. The collapsedoes notnecessarilyspell the demise ofthe false
dynamics of this basic configuration, a so called false vacuum interior. When gravity is taken into account,
vacuum bubble, was studied in detail in the eighties by there are (at least) two spherically symmetric configura-
various groups, perhaps most comprehensively by Blau, tions associated with each sufficiently low value of con-
Guendelman and Guth (BGG) in Ref. [4] where refer- served Arnowitt-Deser-Misner (ADM) mass M. While
ences to the earlier literature are provided. Their work the smaller of the two, no matter what its initial radius,
was motivated, like Linde and Vilenkin’s, by the possi- will always collapse to a vanishing radius, the other will
bility that a false vacuum bubble could provide a seed inflate; but unexpectedly from an euclidean perspective,
∗Electronic address: garreaga@fis.cinvestav.mx
†Electronic address: [email protected]
‡Electronic address: [email protected]
1
not at the expense of the exterior. This peculiar state of 1. Fixη: foreachnon-vanishingvalueofQuptosome
affairs is possible because the expansionoccurs behind a critical value Q0 there exists a unique stable and
minimal surface (a wormhole) in the Schwarzschild ge- globally static configuration with a fixed core ra-
ometry connecting the false vacuum coreto the exterior. dius (a monopole) with mass M0 ∼ Q(Q/Q0)1/2.
While false vacuum is destroyed by the motion of the Stable radial oscillations of this configuration ex-
boundary, it is created exponentially faster in the inte- ist for all M < Q above M0. These are the only
rior. The wormhole itself, however,alwayscollapses into solutions with M <Q.
a black hole.
Now raise M above Q but below some value M+:
Field theories admitting static configurations when
gravity is turned on were first constructed in the eight-
2. (i) For non-vanishing values of Q up to some value
ies. Self-gravitating global monopoles were considered
Q+ lower than Q0, this radially oscillating solu-
byBarriolaandVilenkin[6](withadditionalinsightpro-
tion falls through an event horizon and is termi-
vided in Ref. [7]). Gauge monopoles were considered by
natedbythecollapseoftheexteriorintoaReissner-
Gibbons [8], with the subsequent numerical solution of
Nordstro¨mblackhole;(ii)forhigher(butbounded)
the static Einstein-Yang-Mills-Higgs equations by Ortiz
values of Q radial oscillations lie within the inner
[9] and Breitenlohner, Forgacs and Maison [10]. How-
horizon and get isolated by the collapse of the ex-
ever, as Gibbons himself realized the balance of forces is
terior(amonopoleinsideablack-hole);aninflating
notalwayspossiblewhengravityisacting. AsLindeand
bounceco-existswiththesesolutions;(iii)ifQorM
Vilenkin were later to show, if the symmetry-breaking
lies outside these two regimes, but M lying above
scaleisincreasedtowardsthePlanckscale,atsomepoint
some minimum value, there is a unique inflating
the interior radius will exceed the corresponding cosmo-
bounce.
logical horizon, triggering the inflation of the interior.
A bound exists beyond which static configurations nec- 3. The bounce occuring in these three regimescan be
essarily become unstable. This process was studied by characterized roughly as follows: (i) the course of
Sakai et al. [11,12]by numerically solving the dynamical the bounce lies within a single asymptotically flat
Einstein-HiggsandEinstein-Yang-MillsHiggsequations. region (AFR) and it resembles closely the bounce
The inflating core was not unlike an inflating false vac- exhibited by a false vacuum bubble (with Q = 0);
uumbubble. InRef.[13]twoofusshowedhowthemodel (ii),(iii) the course of the bounce spans two con-
of a false vacuum bubble could be adapted to imitate secutive AFRs. However, from the point of view
thedynamicsofaself-gravitatingglobalmonopoleunder of either region it resembles a monotonic false vac-
these extreme conditions. Technically this was simple, uumbubble; Inallcases,theexpansiontakesplace
involvingthesubstitutionoftheBarriola-Vilenkingeom- behind an event horizon. These configurations are
etry describing the asymptotics of a globalmonopole for analogues in this model of the topologically inflat-
theexteriorSchwarzschildgeometryoftheformer. Itwas ingsolutionsobservednumericallybySakai[12]for
possible to capture, remarkably faithfully, the essential large η.
underlying physics of topological inflation found earlier
bySakaietal. Theonsetoftopologicalinflationwasvery 4. All collapsing and monotonic solutions are ruled
clearly indicated at 8πη2 =1 (in natural units). out as either unphysical or inconsistent with
Thedetailsoftopologicalinflationinagaugemonopole asymptotically flat boundary conditions.
are very different. Here, the only long range field is the
magnetic Coulomb field and the total energy is finite. Aspects of the model have been examined before. In-
However, the source resides on the core boundary which deed in the sixties, it received its first incarnation in
suggeststhe same mechanicalcaricature: de Sitter space Dirac’s proposal (without spin!) of a model of the elec-
inside, a domain wall, but with the Reissner-Nordstro¨m tronasaclosedchargedconductingmembranesurround-
geometryoutside. The structureofthis geometryis very ing a vacuum interior [14].
different from Schwarzschild. If the conserved charge to Tachizawa, Maeda and Torii focused on the stability
mass ratio of the configuration Q/M ≤ 1, the analyi- of the monopole from the point of view of catastrophe
callycontinuedgeometrypossesseshorizons,otherwiseit theory [15], modeling the monopole core and exterior
contains a naked singularity. In this paper, we examine as we do but without an intermediate surface layer, a
the above model in detail. Though simple in principle, model originally proposed by Lee, Nair and Weinberg,
a thorough analysis of the three-dimensional parameter in Ref. [16]. In this limit, the core radius exceeds the
space(η,Q,M)iscomplicatedinpractice. Wewillfocus cosmological horizon when η ∼ 0.33, signaling inflation.
on the identification of the regimes of parameter space However, without the domain wall to transmit energy
admitting solutions which are either static, collapsing or fromthefalsevacuum,alldynamicalpossibilitiesarenot
inflating within the core and we will examine the fate of faithfullyrepresented. Morerecently,Alberghi,Loweand
these solutions as the relevant boundaries in parameter Trodden [17] considered the model within the context of
space are crossed. the Anti- de Sitter space/conformal field theory corre-
Our results can be summarized as follows: spondence. For this purpose they catalogued accurately
2
thepossibletrajectoriesofthechargedfalsevacuumbub- Reissner-Nordstro¨m geometry described by the line ele-
ble. However, they did not consider how these trajecto- ment
ries depend on the values of Q, M or η and they did not 1
consider the parameter regime M <Q corresponding to ds2 =−AMdTM2 + A dR2+R2dΩ2 , (2.4)
M
stable configurations.
where
The paper is organized as follows. In Sec. II we intro-
duce the model. In Sec. III, we determine all possible 2M Q2
A =1− + . (2.5)
trajectories of the wall radius consistent with set values M R R2
ofQ,η andM. InSec.IVwedescribebrieflytheinterior
Here M is the conserved ADM mass which represents
andexteriorspacetimes and how the routingof trajecto-
the combined material and gravitational binding energy
riesisdeterminedineach. InSecs.V-IX,weidentifyall of the configuration. M must be positive.1
physically interesting solutions and compare our results
In this model, we attempt to capture the dynamics of
with earlier work. Finally, in Sec. X we conclude with a
the bubble wall in a single variable, the radius r of the
few brief comments.
core boundary or wall. Following Ref. [4], it is straight-
forwardto cast the Einstein equations at the wall in the
form
II. THE MODEL
β −β =4πσr ≡κr, (2.6)
D M
The configuration possesses a core in which the mag- where we define β = ± r˙2+A , and the over-
D,M D,M
nitude of the Higgs field approximates its false vacuum dot represents a derivative wpith respect to proper time.
value, φ = 0; in the core region, the potential energy of Equation (2.6) can be exploited to express both β and
D
theHiggsfielddominatesthegradientenergyintheHiggs β as functions of the wall radius:
M
and gauge fields. We model this core by a spherically
1
symmetric region of false vacuum, and for the Mexican β = [−(1∓k2)z4+2mz−q2], (2.7)
D,M 2kz3
sombrero potential
where we have rescaled variables as follows:
V(φ)= λ(φ2−η2)2, (2.1) κ/H =k, HM =m, H2Q2 =q2, Hr=z .
4
(2.8)
thisenergydensityisgivenbyV(φ=0)= λη4. Thecor-
4 Now Eq. (2.6) can be recast as
responding spacetime is then described by the de Sitter
line element z˙2+U(z)=−1, (2.9)
1 wheretheoverdotrepresentsaderivativewithrespectto
ds2 =−ADdTS2+ A dR2+R2dΩ2 , (2.2) propertimerescaledbyH. ThepotentialU(z)appearing
D
here can be expressed in either of two equivalent forms
where 2m q2
U(z)=−β2 −z2 =−β2 − + . (2.10)
A =1−H2R2 . (2.3) D M z z2
D
The Einstein equations determine the local geometry in
The Hubble parameter H appearing here is given by the neighborhood of the wall. The sign of the functions
H2 = 83πV(0)= 2π3λη4. βD,M encodes the boundary conditions required to con-
We will suppose that there is a charge Q localized on struct the complete global geometry.
the boundary of this core. The energy in the neighbor- Finally, we note that, in terms of the symmetry-
hood of the core is dominated by field gradients. This breaking scale η, the ratio k is given by
boundary layer can be modeled as a relativistic domain
24π
wallwith a surfaceenergy density (tension) σ ∼η3,[18]. k= sη. (2.11)
r λ
Outside the core, the energy density in the massive
fieldsfallsoffexponentiallyfastsothat,toagoodapprox- Here, we have exploited the fact that ρ∼η4 and σ ∼η3
imation,theenergyinmatterisdominatedbytheasymp- with constants of proportionality λ and s of order unity.
toticmagneticCoulombfield. Thesphericallysymmetric For a GUT symmetry-breaking scale, η ∼ 1016GeV,
exteriorspacetimecanthenbemodeledasaregionofthe k ∼10−3. For Planck scale η, k ∼1.
1The charge Q appearing here is related to the magnetic
charge of the monopole g by Q2 = g2 where g= 4π and e is
4π e
thegauge coupling strength.
3
III. ALL LOCAL SOLUTIONS leftanditsrightrespectively. Wenotealsothatz
−
(and never z+) is alwaysthe absolute maximum of
The potential U given by Eq.(2.10) is parametrized U. Lowering m through Mcrit at a fixed values of
by three positive dimensionless parameters characteriz- Q and k we find that z0 and z+ (not z−) coalesce
ing the mass, the charge, and the symmetry breaking when m=Mcrit. The value Mcrit increases mono-
scale m, q and k respectively. We will consider sections tonically from zero (as a function of both Q and
ofconstantk andofconstantQofthisthreedimensional k).2
space. Because both m and q have η folded into their
This completes the discussion of the topological
definition, when we varyk, it is appropriate to undo the
form of the potential, as characterized by its crit-
‘natural’scalingdepending on η one exploits for calcula-
ical points. This topology is not, however, always
tional purposes.
relevantphysically. Thiswillbe the caseifthewell
In general, the potential is always negative. In addi-
is not accessible physically.
tion, U →−∞ at z =0 and as z →∞ so that it always
possesses at least one maximum. To discuss the qualita-
The domains of z which are physically accessible
tive dependence of the potential on the values of M, Q
in the potential are determined by the mass shell
andk,itis usefulto identify the followingboundarieson
conditionEq.(2.9). Tolocatethese domainsweex-
the parameter space:
amine where the critical points of the potential lie
with respect to the fixed ‘energy’ −1. Again we
1. The location of the extremal exterior Reissner-
fix Q and k. These conditions will identify three
Nordstro¨m geometry, M : M =Q
hor
values of M.
If M > Q the complete Reissner-Nordstro¨m ge-
ometry possesses an (outer) event horizon at R+
and an (inner) Cauchy horizon at R− <R+ which 3. The upper limit on monotonic motion MM:
are given by the two positive solutions of A = 0
M
where AM is given by Eq.(2.5). When M = Q, If M is below some value MM the absolute maxi-
thetwohorizonspossessthesameradius(thisdoes mum of the potential will lie below the value −1.
not mean that they coalesce). If, however, M <Q All values of r are then accessible and all candi-
therearenohorizonsandthe correspondingspace- date physicaltrajectoriesnecessarily monotonic —
time possesses a naked singularity at R = 0. This either expanding from zero radius or collapsing to
criterion is independent of k. it.3 If M > MM there are no monotonic trajec-
tories. For each such M there are always at least
This boundary will play an important role in de-
two trajectories, each with a single turning point,
termining the limit of stability of a self-gravitating
one bounded and another unbounded. When we
object.
refer to them below we will describe the trajectory
WereferthereadertotheM−QandM−kparam-
initially at rest at the turning points: the former
eter planes representedin Fig. 1 and Fig. 2 respec-
collapsesfromafinitemaximumtozeroradius;the
tively. As a visual aid, the former is reproduced
latter expands from a minimum to infinite radius.
zoomed-inasFig.3andzoomed-outasFig.4. The
corresponding potential in different regions of pa- Whether the trajectories we have described trans-
rameter space is plotted in Fig. 5. late into configurations which are compatible with
theboundaryconditionsisaquestionwhichwead-
2. Thelowerboundonthemassprovidingapotential
dress in the following section. The Einstein equa-
with a well, M :
crit tions, as we will see, do admit spurious solutions
SupposenowthatwefixQandk. Considerthede- which do not correspond to the isolated lump of
pendence of the potential on M. Below some fixed energy we are interested in.
value M , U possesses a single maximum; there
crit
is no well. Above M , U possesses a well: with The value M like M increases monotonically
crit M crit
minimum z0 (say), and maxima z− and z+ on its from zero as a function of both Q and k.
2Weremark that thetechnicaldetails enteringthedetermi-
nation of boundaries such as M on the parameter plane
crit
have been discussed elsewhere by two of the authors in the
context of global monopoles and will be omitted here. See
Ref. [13].
3WhenM =MM,U(z−)=−1andanunstableequilibrium
withthewallpoisedprecariouslyatz− is,ofcourse,possible.
4
4. The limits of oscillatory motion, M0 and M+: pointisrepresentedbythetrajectoryindicatedR=0on
thelefthandsideofthespacetimediagram. Thediagonal
The analogue in our model of a radially deformed
running from the upper right to the lowerleft represents
monopole corresponds to an oscillatory trajectory.
the cosmologicalhorizon of this point.
Thesearetheonlytrajectorieswhichshouldsurvive
Thecoreinteriorisrepresentedbythespacetimeregion
when gravity is turned off. When is such motion
to the left of the trajectory on this diagram. It is clear
possible?
that turning points of the motion of the wall must occur
To accomodate a stable oscillating trajectory in within the static regions I and III with R < H−1 where
the potential, the well must be accessible on shell, the Killing vector ∂ is timelike, and ∂ spacelike. In
U(z0) < −1, and confine the motion on the right, particular, oscillatorTySsolutions are necesRsarily confined
U(z+) > −1. Clearly these conditions will not be to these regions (one should not rule out, a priori, an
realized for every specification of Q and k. When oscillatingcoreboundaryinregionIwithaninflatingin-
they are they will limit M to values within a fi- terior). A globally static core must, however, lie in the
nite band [M0,M+]. The values M0 at which left hand quadrant III. Any trajectory which crosses the
U(z0) = −1 and M+ at which U(z+) = −1 are horizon necessarily inflates inside. See Fig. 7.
indicated in Fig. 1). These two boundaries in the The nature of the Reissner-Nordstro¨m spacetime de-
three dimensional parameter space coalesce on the
pends crucially on the charge to mass ratio, Q/M. If
boundaryMcritalongsomecriticalcurveM∗where M <Q there are no horizons in the Reissner-Nordstro¨m
they terminate. For fixed k, we denote the limit-
geometry and a Penrose conformal diagram of its maxi-
ing value of the charge on M∗ by Q∗(k). We have malanalyticextensionconsistsofasingleasymptotically
plotted Q∗ as a function of k in Fig. 6. Note that flatgloballystaticspacetimewithanakedtimelikesingu-
Q∗ decreases monotonically to zero as k → ∞. In larity at R=0. See Fig.7. In our analysis,the Reissner-
particular,the relationshipQ=Q∗(k) is invertible Nordstro¨m geometry will always be truncated at some
for the corresponding limiting value of k at fixed finite radius within which it is replaced by a patch of de
Q, k =k∗(Q). Sitter space. If this radius does not fall to zero, the sin-
Even without consulting spacetime diagrams, it is gularity does not appear in the physical spacetime and
already possible to conclude the following: thereis nophysicaljustificationtolimitourselvesto val-
ues of M exceeding Q as one does in vacuum.
For a given k there exists a unique ‘static’ trajec-
The maximal analytic extension of the Reissner-
tory for each Q up a limiting value, Q (k); and
∗ Nordstro¨m geometry when M >Q is represented on the
that for a given Q, there is a corresponding limit-
Penrose-Carter(PC) diagram, Fig. 8. See Ref. [19] and
ing value of k, k (Q). As we will see when we ex-
∗ also Ref. [20] for a recent pedagogical discussion. This
aminethecorrespondingspacetimes,notall‘static’
consists of an infinite tower of identical connected uni-
trajectoriescorrespondtostaticspacetimessothat
verses. The singularities at R = 0 are not visible at
the physical limiting values will be lower.
infinity in this geometry. Within the regions R < R
−
There exists, at best, a finite spectrum [M0,M+] and R > R+, the Killing vector ∂TM is timelike: both
bounded below by M0, of stable oscillations about of these regions are static. In the inter horizon region,
any static configuration. R− <R<R+, ∂TM becomes spacelikeand ∂R generates
temporal evolution. The spacetime in this region is dy-
Finally, we comment that the boundary structure on namicalno matter how one cares to look atit. As in the
parameter space is captured completely by either of the interiordeSitterspace,anyturningpointsofthemotion
two sections we have considered. The M − Q section must occur in the static regions. This will be useful to
contracts continuously towards the unique fixed point, remember when locating turning points in spacetime.
M =0, Q=0 as k is raised. Its topological structure is We remark that the Cauchy horizon is unstable [21].
unchanged. Under a small generic perturbation in the metric, it has
In the following section, we will consider the embed- been shown to collapse into a Schwarzschild type space-
dingofthewalltrajectoriesinboththe interiordeSitter like singularity limiting motion towards the future. A
and the exterior Reissner-Nordstro¨mspacetime. consequenceis thatthe exoticpossibilitiesevokedbythe
Reissner-Nordstro¨m tower are irrelevant. The life span
of the physicalsystem is limited to just one floor on this
IV. EMBEDDING OF THE WALL TRAJECTORY
tower.
IN SPACETIME
The exterior is representedby the spacetime regionto
the right of the trajectory on this spacetime diagram.
In the present context, de Sitter space is represented The topology of a regular spatial slice is R3 with a disk
most conveniently by a Gibbons-Hawking diagram. For removed.
details, in the present context the reader is referred to It can be shown that the fugacities β are propor-
D,M
Ref. [4]. In this diagram the center is placed at the tional to the derivative of the corresponding coordinate
(north) pole of a round sphere. The evolution of this time with respect to proper time,
5
β =∓A t˙ . (4.1) S, QSI,andQSIIonFig.1. Theoscillatorymotioncom-
D,M D,M D,M
patible with each of these regions is different.
In the case of de Sitter space and the M > Q Reissner- S: There exist stable oscillating trajectories with both
Nordstro¨m geometry, the right hand side of Eq.(4.1) in static interiorand exterior: in the interior,β >0 along
D
turn relates these two functions to the course of the po- the trajectory so that it lies in region III of a Gibbons-
larangleθD,M subtendedbythetrajectoryaboutafixed Hawking diagram — the interior does not inflate; the
pointinthecorrespondingspacetimediagramwhichper- exterior Reissner-Nordstro¨m geometry with M < Q is
mits the routing of the trajectory about this point to be globally static, The trajectory is indicated O in Fig. 7.
determined. For β one has
D QSI,II: Stable oscillating trajectories would also appear
to be admitted in these neighboring regions of param-
β ∝−θ˙ , (4.2)
D D eter space. However, whereas the interior geometry in
both is essentially identical to that of an S trajectory,
We note that β > (<)0 indicates clockwise (counter-
D the exterior geometry necessarily contains a black hole.
clockwise) motion about the origin.
If a genuine static trajectory with M > Q exists, it
Unlike the de Sitter geometry, the Penrose-Carter di-
agram for Reissner-Norsdtro¨m geometry with M > Q, mustdosoalongthatsectionofthe boundaryM0 where
possessesneitherpreferredorigin,noruniquecorrespond- M0 > Q, which occurs within QSII. Because r is con-
stant, it mustlie entirely within one ofthe static regions
ing polar angle on the spacetime diagram. We consider
the routingofthemotionaboutthebifurcationpointsof with R < R− or R > R+. Outside this domain, R is a
timelike coordinate and a constant value of R defines an
R+ and R−. We find
impossible spacelike trajectory.
βM ∝∓θ˙± . (4.3) If r < R−, the static trajectory lies within a black
hole. Only if r > R+ (with no horizon) is the exterior
where θ± are the corresponding angles. spacetimegeometrygloballystatic,sothatwe canspeak
Theinterpretatinonofβ isdifferentonthe Reissner- of a genuinely static solution. There are, however, no
M
Norsdtro¨m spacetime with M < Q. Due to the absence solutions of this form: within QSII the turning points
of horizons, βM necessarily possesses a fixed sign. It is rmin and rmax of oscillatory motion both lie below R−.
easily checked that βM must be positive for an isolated Infact, the possibilityrmin,rmax >R+,while consistent
monopole with an infinite exterior. A negative β in with the spacetime geometry,neveroccurs. Stable static
M
this case corresponds to a finite exterior geometry with monopoles (and stable radial oscillations about them),
a naked singularity. appear always to configure themselves so that M <Q.
We are now in a position to describe the wall motion For a given Q there exists an upper bound on η ad-
in spacetime which corresponds to any given set of pa- mitting such a solution (as for a given η there exists an
rameters. upper bound on Q), determined by the crossing of M0
and M =Q, strictly below the ‘naive’ bound on the ‘os-
cillatory’ regime discussed in Sec. III. The existence of
V. LIMIT OF STABLE OSCILLATORY MOTION the limit on η was predicted within the simplified zero
surface tension model by Tachizawa,Maeda and Toriiin
We begin with a discussion of trajectories which cor- Ref. [15]. The existence of this limit was also noted by
respondto the intuitive notionofamonopole asa stable Sakai in [12]. This value of η signals the onset of topo-
compactlumpofenergy. Aswehaveseen,suchsolutions logical inflation.
must lie in the ‘oscillatory’ regime of parameter space Consider, now, the fate of an oscillating trajectory as
admitting bounded radial motion, with mass bounded M is raised through Mhor from some initial value in S
below by M0 and above by M+. The boundary M = Q maintaining Q and k constant. Because Mhor > M0 in
provides a natural partition of this region. Indeed, we this regime, the trajectory must undergo finite oscilla-
note that for low values of Q, M0 < Q, with equality tion in r (there are no static trajectories). The surplus
alongQ=Q0(k). ThisvalueisstrictlylowerthanQ∗(k). M providesthe wall with radialkinetic energy. As Mhor
TheboundaryM+,onthe otherhand,liesstrictlyabove is crossed, two horizons with initially equal radii appear
Q except along Q = Q+(k) where the two touch (with intheexteriorgeometry. Wheretheturningpointsofthe
a common tangent). These two features are clearly indi- oscillatorymotion,rmin andrmax say,liewithrespectto
cated on the zoom-in of the M −Q parameter plane. In the horizons at R+ and R− will depend on the values of
Fig.6weplotQ0 andQ+ asfunctionsofk. Theyclearly Q and k.
converge as k becomes large. Q0, Q+ and Q∗ partition If Q < Q+(k), QSI is entered with rmin < R− and
the oscillatory regime into three regions which we label rmax >R+; whereasif Q lies between Q+(k) and Q0(k),
4WenotethatthereisalsoaregionwithinQSIIcorrespond-
6
QSII is entered with both rmin and rmax less than R−.4 Q<Q0(k),sothatstablestaticsolutionsexist,thisvalue
When Q=Q+, rmax coincides with the right maximum isM0 [24];(ii)ifQ≥Q0(k)andtheredonot,aboundis
of the potential and rmax =R− =R+. providedbyQ. Thelaterboundwillbesharpenedbelow.
The exterior spacetimes which correspond to ‘oscilla- For a constant k, the mass of a static solution M0 ≃
tory’ trajectories OI in QSI and OII in QSII are illus- Q(Q/Q0)1/2 which has the same functional form as the
trated in Figs.8 and 9, respectively. Minkowski space limit.
The ‘oscillatory’ trajectory O interpolates between
I
a maximum in region I and a minimum in region V.
Its apparent subsequent oscillatory course up through VII. INFLATING BOUNCES WITH M >Q
the Penrose-Carter tower is an analytical accident with-
out any observable consequences. The physical solution
In Sec. III, bounce solutions were identified in the pa-
clearlydoesnotoscillatecomingasitwilltotheunpleas-
rameter regime bounded below by M . Such solutions
ant end described in the previous section as it crosses M
coexist with the quasi-static solutions we have described
the Cauchy horizon. The exterior geometry collapses in
in each of QSI and QSII.
a black hole.
Weagaindiscardthecollapsingsolutionasanunphys-
The trajectory O oscillate within region V. The ex-
II ical closed universe with a naked singularity. However,
terior geometry again collapses in a black hole isolating
theexpandingbouncetrajectoriesareconsistentwiththe
the monopole inside. The gauge monopole analogues of
boundary conditions.
bothsolutionswereobservednumericallybySakaiin[12].
The regime admitting bounces partitions naturally
TheirzerotensionanaloguewasidentifiedinRef.[15]by
into three regions: QSI (as before), B (which contains
Tachizawa,Maeda and Torii.
QSII) indicated on Fig. 1, and B′ indicated on Fig. 3.
Finally, we note that the Penrose singularity theorem
The interiorspacetimeofanexpandingbounce clearly
placesnoclassicalobstructionontheformationofanyof
inflates. The trajectories are embedded on the Reissner-
the solutions we have described from nonsingular initial
Nordstro¨mspacetimeasB onFig.8forQSI;BonFig.9
conditions, be they static or black hole. [22,23] I
for B; and B′ on Fig. 10. for B′.
The expansion in all cases occurs behind a throat
geometry which subsequently collapses into a Reissner-
VI. LOWER BOUND ON THE MONOPOLE
Nordstro¨mblackhole(inthesamewayasitdoesoutside
MASS
theoscillatorycounterpartsdiscussedinSec.V)Thisex-
pansiondoes notoccuratthe expense ofthe exteriorge-
TheoscillatorysolutioninSdescribedaboveistheonly ometrybut(withrespecttoareasonableslicingofspace-
solution of the Einstein equations satisfying the bound- time) does getcut offfromthe exteriorby the formation
aryconditionswhichcorrespondstoanisolatedmonopole of a black hole.
in the parameter regime M < Q. Technically, this is Qualitatively,thebounceoccuringinQSIisverysimi-
because β < 0 along the remaining trajectories, be larto the false vacuum bubble bounces describedin Ref.
M
they monotonic, collapsing or expanding bounces. This [4]. Note that the Penrose singularity theorem places
is just as well: while gravity might be sufficiently strong an obstruction to its formation from non singular initial
to provoke the collapse of a charged object, one would conditions. Accessible or not classically, this trajectory
not expect this to happen if the charge exceeds M; nor is of interest because of the possibility of tunneling into
would one expect gravity to promote the explosion of a it from its bounded counterpart, [23].
monopole. A negative β corresponds, in the regime The bounce occuring in B is very different, contract-
M
M < Q, to an exterior which is a finite region of the ing from infinity in one asymptotically flat region of the
Reissner-Nordstro¨m geometry with an unphysical naked Reissner-Nordstro¨mspacetime to a minimum on the left
singularity at the antipode. The spatial geometry is a hand side of the Penrose-Carter tower before expanding
closedthreespherewhichisinconsistentwiththeasymp- to the corresponding asymptotically flat region on the
totically flat boundary conditions that we associatewith next floor of the tower. Clearly, the full bounce is not a
an isolated monopole. Because it contains a naked sin- physically realizable configuration. The physically rele-
gularity we consider it unphysical. vantleg ofanybounce is its expansionfroma stationary
An immediate corollary of the above observation is minimum. Bounces correspond either to the thermody-
the existence of a lower bound on the mass of a phys- namicalorquantummechanicalmaterializationofacon-
ically realistic configuration, static or otherwise: (i) if figuration.
ing to values of Q and k within the range [Q0(k),Q∗(k)]
which cannot be considered as excitations of any S static
configuration.
7
Interestingly, there is a narrow window in the neigh- some critical mass M , both collapsing and expanding
cr
borhoodofthisstationaryinitialconfigurationwherethe bounce motion occur as we described in our introduc-
Penrosetheoremdoesnotpresentanyobstructiontothe tion. For masses above M all motion is monotonic:
cr
classicalassemblyoftheBbouncefromnon-singularini- the core expands from a singular zero radius behind the
tial conditions. These are the analogues of topologically Schwarzschild horizon and like the bounce described in
inflating gauge monopoles. the introduction is connected to the asymptotically flat
In his numerical simulations of the dynamics of gauge regionbyathroat. Thoughthethroatcollapses,thecore
monopoles,Sakaialsoobservedinflatingmonopoles(cor- expands forever with an inflating interior. The reader is
responding to our bounces) to co-exist with collapsing referred to [4] for details. Both the expanding bounce
monoples (corresponding our O ). and the monotonic solution violate the Penrose theorem
I
Bounces occuring in the narrow regime of parameter along their course [22].
space indicated B′ occur in a convexpotential. Whereas This limit shouldbe consistentwithsolutions lyingon
theasymptoticsofsuchbouncesareidenticaltothosefor theM-axisontheM−Qsection. Atfirstsight,however,
B,theirminimumoccursnowontherightofthePenrose- the limitQ→0ofourmodelappearsto contradicttheir
Cartertower. IncontrasttoBbounces,the Penrosethe- findings. Specifically, there does not appear to be any
oremimpliesthatitsformationisunphysicalonthecom- analogofM atQ=0 —the monotonictrajectorieswe
cr
plete expanding leg. On its contracting leg, there is no find do not even exist in this regime. To resolve this ap-
asymptoticallyflatspatialslicecontainingthetrajectory. parentcontradiction,notethat, asQ→0inthis regime,
We must conclude that such trajectories are unphysical. the left maximum of the potential U occurs at ever de-
creasing radius (z → 0) while, simultaneously, the well
−
depth becomes infinitely deep, U(z0) → −∞. We also
VIII. ALL MONOTONIC TRAJECTORIES ARE note that, as Q → 0, the inner horizon of the Reissner-
UNPHYSICAL Nordstro¨m geometry appoaches zero, R → 0, while
−
the outer horizon at R = R+ becomes the Schwarzchild
horizon. The bounce trajectory occuring in B thus ap-
We have already discounted monotonic trajectories
proaches arbitrarily close to r = 0, the Penrose window
withM <Q. TheboundaryM partitionstheremain-
crit we discussed in Sec. VII closes and the trajectory on its
der of this regime. In the bounded regime M < M
M crit
expanding leg becomes indistinguishable from a mono-
the potential is convex and both β and β possess
D M
tonically growing false vacuum bubble. It is clear that
definite signs. We have indicated the trajectory by M
on Fig. 11. In the remaining unbounded region with weshouldidentifyMcr withM+ atQ=0,notwithMM.
M <M , the effective potential develops a well and We also note that below M+, in QSI the a quasi-
crit M
oscillatorytrajectoryapproachesz =0arbitrarilyclosely
both β and β change sign in the course of their evo-
D M
and become indistinguishable from a collapsing bounce.
lution. Apart from this single dynamical detail, motion
Theexpandingbounce,aswecommentedearlierdoesnot
is qualitatively identical in both regimes.
suffer any signifacant local change.
Is this motion physical from a classical point of view?
The partofthe trajectorylyingwithin r <R necessar-
−
ily contains a nakedsingularityin its exterior;moreover,
theinteriorinitiallycontainsathree-sphere’sworthofde X. CONCLUDING REMARKS
Sitter space. The trajectory is clearly unphysical in this
regime. Infact,thePenrosetheoremforbidstheassembly
We have examined in some detail the dynamics of a
of such a trajectory by classical means [23]. We dismiss
charged false vacuum bubble within the thin wall ap-
this solution as unphysical. It would appear that there
proximation. We claimthat, with the identificationofQ
are no physical monotonic trajectories in this model.
withthemagneticchargeg (relatedtotheelectriccharge
If we take in account the elimination as unphysical of
by g =4π/e) the model mimics the radialdynamics of a
all possible trajectories in both M and B′, the lower
sphericallysymmetricmagneticmonopole. Inparticular,
bound on the mass of any asymptotically flat configura-
themodelprovidesavaluableguidetounderstandingthe
tionis raised. As QincreasesaboveQ0,the lowerbound physics which underlies both the onset of instability of a
follows the line M = Q, then M = M , and finally
crit static monopole as well as the conditions which need to
M =M .
M be met to produce a topologically inflating object.
It would appear that inflation does not necessarily re-
quire dialling up the symmetry-breaking scale η; an in-
IX. FALSE VACUUM BUBBLE LIMIT
flatingsolutiononlyrequiresthattheADMmassbesuffi-
cientlylarge,whichis possibleinprinciple forarbitrarily
WearefinallyinapositiontoexaminethelimitQ→0, low values of η or Q. In this respect, the monopole we
where the model had better reproduce the “false vac- consider differs from the ‘global’ monopole discussed in
uum bubble” investigated by Blau, Guendelman and [13]whereinflationisonlypossiblewhenηisraisedabove
Guth and others. Briefly, for each value of M below the Planck scale. However,it should also be pointed out
8
that in a field theory of monopoles the mass is itself a [1] A. Linde, Phys.Lett. B327, 208 (1994).
function of η. It is not clear if the high mass and low η [2] A. Vilenkin,Phys. Rev.Lett. 72, 3137 (1994).
inflating solutions we find can be realized in practice. [3] E. Guendelman and A. Rabinowitz, Phys. Rev. D44,
There are a few interesting extensions of this work: 3152 (1991).
We note that, for everymonopole which collapses into [4] S. K. Blau, E. I. Guendelman and A. H. Guth, Phys.
a black hole in the parameter regimes QSI and QSII, Rev. D35, 1747 (1987).
there will be a corresponding expanding bounce con- [5] Alan Guth, The Inflationary Universe (Addison-Wesley,
figuration with identical values of the conserved mass Reading, MA.1997).
[6] M. Barriola and A. Vilenkin, Phys. Rev. Lett. 63, 341
and charge. On semiclassical grounds, one would an-
(1989).
ticipate a finite amplitude for tunneling from the former
[7] D. Harari and C. Lousto, Phys.Rev. D42, 2626 (1990).
to the latter. The construction of the instanton medi-
[8] G. W. Gibbons, in Proceedings of the XII Autum School
ating this passage should provide a valuable exercise in
on the Physical Universe, Lisbon, 1990, edited by J.D.
semi-classical quantum gravity.
Barrowetal.,LectureNotesinPhysicsVol.383(Springer
We haveconsideredadescriptionofasphericallysym-
Verlag, Berlin, 1991).
metricfieldtheoreticalmonopoleinwhichitscorebound- [9] M. E. Ortiz, Phys. Rev. D45, R2586 (1992).
ary is modeled as a relativistic membrane. How robust
[10] P.Breitenlohner, P.Forgacs andD.Maison, Nucl.Phys.
is this description when spherical symmetry is relaxed? B383, 357 (1992); ibid.B442, 126 (1995).
In the seventies it was shown that, in Dirac’s extensi-
[11] N. Sakai, H. A. Shinkai, T. Tachizawa, and K. Maeda,
ble model of the electron, the static charged membrane Phys. Rev.D53, 655 (1996); ibid. D54, 2981 (1996).
is unstable to non-radial deformations [25]. The origin [12] N. Sakai, Phys.Rev. D54, 1548 (1996).
of this instability is similar to that which triggers fission [13] I. Cho and J. Guven,Phys. Rev. D58, 63502 (1998).
of the atomic nucleus (the boundary conditions differ). [14] P. A. M. Dirac, Proc. R. Soc. A 268, 57 (1962).
On the other hand, in Ref. [26] Goldhaber argued that [15] T. Tachizawa, K. Maeda and T. Torii, Phys. Rev. D51,
a global monopole suffers from a cylindrical string-like 4054 (1995).
instability. Superficially, this would appear to be analo- [16] K. Lee, V. P. Nair and E. Weinberg, Phys. Rev. D45,
goustothespike(zeroarea)instabilityofaNambu-Goto 2751 (1992).
membrane. However, it is likely that higher curvature [17] G. L. Alberghi, D. A. Lowe and M. Trodden, JHEP
(rigidity)correctionsto the Nambu-Gotoactionmustbe 9902:020 (1999).
included to model the field theory when spherical sym- [18] For a rigorous justification of this approximation in the
metry is relaxed. Such additions would tend to moder- context of a domain wall, see B. Carter and R.Gregory,
Phys. Rev.D51, 5839 (1995).
ate (or eliminate) the instabilities of Nambu-Goto mem-
[19] B. Carter, Phys.Lett. 21, 423 (1966).
branes. Itwouldbeinterestingtoexplorethemembrane-
[20] P. K. Townsend, Black Holes, Lecture Notes, gr-
topological defect correspondence in greater detail.
qc/9707012.
[21] See, for example, W. Israel, in ‘Black Holes, Classical
and Quantum’ (Mazatl´an, Mexico, 1998).
ACKNOWLEDGMENTS
[22] E. Farhiand A.H.Guth,Phys.Lett. B183, 149(1987).
[23] E. Farhi,A.Guthand J.Guven,Nucl.Phys.B339, 417
Thanks to Gilberto Tavares for technical assistence. (1990).
G.A.wassupportedbyaCONACyTgraduatefellowship. [24] ThisisconsistentwiththegeneralresultsofD.Sudarsky
I.C. was supported in part by the Institute of Cosmol- and R. Wald, Phys. Rev. D47, 5209 (1993); ibid. D46,
ogy at Tufts University. The work of J.G. has received 1453 (1992).
support from DGAPA at UNAM, CONACYT proyect [25] P. Hasenfratz and J. Kuti, Phys. Rep. 40, 75 (1978); P.
32307E and a CONACyT-NSF collaboration. Gnadig, Z. Kunszt, P. Hasenfratz, and J. Kuti, Annals
of Phys.116, 380 (1978).
[26] A. Goldhaber, Phys.Rev.Lett. 63, 2158 (1989).
9
FIG.1. M−Qsection ofparameterspaceforη=0.5indicatingthefollowing boundaries: (i)thelowerboundonthemass
providing the potential U(z) with a well, Mcrit; (ii) the upper limit on monotonic motion MM; (iii) the limits of oscillatory
motion, M0 and M+. M+ and M0 terminate at M∗ on Mcrit. The extremal exterior Reissner-Nordstr¨om geometry occurs at
M =Q and is indicated M . The parameter regimes S, QSI, QSII, B, and M are indicated. For details see thetext.
hor
10