Gravitating skyrmions

Physics Letters B 297 (1992) 55-62 North-Holland

PHYSIC S LETTERS B

Gravitating skyrmions P. Bizon Institut Fdr Theoretische Physik, Universitiit Wien, A- I 090 Vienna, Austria and lnstitute of Physics, Jagellonian University, PL-30 059 Cracow, Poland

and T. C h m a j N. Copernicus Astronomical Center, Cracow, Poland

Received 10 June 1992

We study classical solutions of the SU (2) X SU (2) chiral theory (Skyrme model)interacting with gravity. We show that this model admits a surprisingly rich spectrum of asymptoticallyfiat static sphericallysymmetric solutions, both with and without horizons. The stability of these solutions is also examined and the status of no-hair conjecture is elucidated.

l.Introduction Nonlinear field theories coupled to gravity have received a lot of interest in the past few years. In particular, it was discovered that gravitational interaction may lead to genuinely nonperturbative phenomena like gravitationally bound configurations of nonabelian gauge fields in the Einstein-Yang-Mills (EYM) theory [ 1 ]. The study of black hole solutions in various models revealed the possibility of nonlinear hair on black holes (see for instance refs. [ 2 - 5 ] ) which put into question the validity of the unqualified nohair conjecture. More recently, several authors have analysed the effects of gravity on magnetic monopoles in the framework of EYM-Higgs theory [ 6-8 ]. With a few notable exceptions [ 9,10 ], this was the first attempt to understand the properties of gravitating topological solitons #1. To see which of these properties are generic it is useful to consider other models with topological solitons coupled to gravity. Having mainly this motivation in mind, in this letter we investigate the Skyrme model coupled to gravity. ~ Gravitational equilibria of non-topologicalsolitonslike boson stars or Q-baUswere studied, for example,in refs. [ 11,12].

Some aspects of the Einstein-Skyrme (ES) model were studied previously in the literature. Luckok and Moss [ 13 ] showed that the Schwarzschild black hole can support chiral hair and argued that such configurations might be stable. Glendenning et al. [14] studied gravitating skyrmions of large winding number and found that they cannot be models ofbaryonic stars because of their instability. More systematic investigation of the ES model was undertaken very recently by Droz, Heusler and Straumann [ 15 ], who solved numerically the static spherically symmetric ES equations and found globally regular solutions (with the winding number one ) and black holes with chiral hair. They also discussed the stability of these solutions [ 16 ]. We show that the whole spectrum of static spherically symmetric solutions in the ES model is surprisingly rich. It consists of two fundamental branches (one of which was found in ref. [ 15 ] and a family of excitations (whose number depends on a value of the coupling constant ot defined below). We give a plausibility argument which accounts for such structure of the spectrum. Finally the question of stability is addressed. We demonstrate that the bifurcation point at which the two fundamental branches of solutions coalesce is the transition point between stable and

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unstable configurations (for regular solutions with winding number one and black holes with baryon number less than one). To our knowledge, the existence of stable black holes with chiral hair provides the first known counter example to the weak no-hair conjecture (i.e. uniqueness, within any theory, of stable black holes for given global charges at spatial infinity) since at spatial infinity such black holes are are indistinguishable from the Schwarzschild solution. An extended version of this letter containing a qualitative analysis and comparison of gravitating monopoles and skyrmions will be published elsewhere.

The matter part of the Einstein-Skyrme system consists of a chiral field U(x) which is an SU(2)valued scalar function on spacetime (M, gu~). The dynamics is given by the action

f

M

d"x

(') -

R

,

get the following reduced hamiltonian ~2 (, = d / d r ) :

H[A, m, F] =fA

-lm'+g

dr+~m(oo),

(5)

with g = 4zr(f 2 ( ½r2NF t2..~ sin 2F) 1 sin2F

)

+ e-i r---f - (½sin2F+r2NF '2) ,

(6)

where N = 1 - 2m/r. Note that Hhas a standard form of a pure constraint volume term plus a boundary term at infinity. It is convenient to introduce dimensionless variables

2. Field equations and boundary conditions

s=

24 December 1992

( 1)

where

x=efr,

12(x)=efm(r).

(7)

Then the field equations derived from (5) are ~3 are (' = d / d x )

F g,= CtL ½x2NF t2.~_sin2F + sin 2F (ME'2 + sin 2F~]

'

(8)

LM = -~f2 Tr (V/, UVuU -1 ) [ (x2+2

+ ~

Tr[ (VuU)U-', ( V , U ) U - ' ] 2 .

(2)

Here Vu is the covariant derivative associated with the metric gu~. The two coupling constants f2 and e 2 have dimensions: if2 ] = g c m - 1 and [e 2] = g- 1cm- 1 (we use units where c= 1 ). We concentrate on static spherically symmetric solutions and therefore we assume the metric to be

ds2=--A2(r)(1

2m~r))d/2+ (1

+rX(dO2+sin2~ d~ 2) .

2m~r).)- ~dr2 (3)

For the SU (2) chiral field we make the hedgehog ansatz

U=exp[ tr.~F( r) ] ,

(4)

where e are the Pauli matrices and ~ is a unit radial vector. Inserting (3) and (4) into (1) and (2) we 56

sinEF)ANF']

'

sin2F~ =A sin 2F 1 + N F ' 2 + - - - ~ ] ,

A'=a(x+ 2sin2F~F a .

(9) (10)

Here a = 47~Gf 2 is a dimensionless effective coupling constant. When a = 0 (decoupling) eqs. ( 8 ) - ( 1 0 ) reduce to the Skyrme model in flat space (if#(x) =0) or in Schwarzschild background (if #(x)=const. > 0). Note that using (10), A can be eliminated from (9). Asymptotically fiat solutions of eqs. (8)- (10), i.e. #2 Actually, to obtain (5) one has to add a suitable boundary term to (I). #3 The fact that the critical points of the reduced hamiRonian

( 5) are the critical points of the action ( I ) can be checked directly, but it also followsfrom general theorems (cf. refs. [17,18]).

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solutions with A ( o o ) = c o n s t . and r n ( ~ ) = c o n s t . have to satisfy F(oo) = 0 ( m o d zt) ,

( 11 )

and split into two classes of non-singular solutions which differ by the interior boundary conditions: globally regular solutions (we shall refer to them as to solitons) and black holes. For solitons, to ensure regularity at the origin, we require that as x--.0 /.t(x)=O(x3),

F(x)=kTt+O(x2),

Black holes are defined for X>>-XHwhere xH, the radius of the (outer) horizon, is the largest zero of N(x). We assume that F a n d F ' are regular at x=xM. The fact that F(xH) need not be a multiple ofrt has important topological implications. To see this, let us recall the topological classification of flat space skyrmions. The condition ( 11 ) effectively compactifies ~3 to S 3, so, for fixed time, the space of chiral fields U(x) splits into homotopy classes of ~z3( SU ( 2 ) ) = Z, labeled by the integer winding number B. This winding number (usually interpreted as a baryon number) can be expressed as

(13)

where B u is the topological current B~,= ~u~p T r [ ( U - 1 V ~ U ) ( U - 1 V'*U) 24r¢ 2 X ( U -~ VPU)] .

(14)

For the hedgehog ansatz (4) this gives x/~Bo

= - 2-~2r2 s i n 2 F F ' ,

ing the condition U ( h o r i z o n ) = l , i.e. F ( x n ) = 0 ( m o d rt). This however implies, via eq. (9), that F ( x ) - O. Therefore, all black holes in the ES model are topologically trivial.

3.Numerical results

3. I. Solitons

k~7_. (12)

B= f d 3 x x / ~ B o ,

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(15)

so for the boundary conditions ( 1 1 ) and (12) we have B = ( 1/n) I F ( 0 ) - F ( o o ) ]. This topological classification obviously carries over to gravitating solitons, however for black holes the situation is different. The point is that black holes are defined in the domain Z which (under condition ( 11 ) ) is topologically S 3 minus a ball, so the space of maps U(x): Z--,SU(2) is topologically trivial (because Z is contractible). To recover a nontrivial topology one could try to glue the hole in S 3 by impos-

The formal power-series expansion about x = 0 is

F(x) =k~z-bx+O(x 3) ,

(16)

2/1(x) = c~b2( 1 + b2)x3 +O(x 5) ,

(17)

where all higher terms are determined by b. Thus local solutions of eqs. ( 8 ) - ( 1 0 ) (whose existence we assume) form a one-parameter family labeled by b=-F'(0). We solve eqs. ( 8 ) - ( 1 0 ) using the shooting method, i.e. we try to choose such b that a local solution extends to a global solution with F(oo) = 0. It turns out that for every winding number B= k there is a critical value otkax beyond which there is no such global solution (for B = 1 this was first observed in ref. [ 15 ] ). As k grows, otka~ scales approximately as am~x/k ~ 2, where alex---_0.040378. The occurrence of am,~ is by no means surprising and has a simple heuristic explanation. Namely, since the typical mass is M~k2f/e, for large ot (i.e. oL~ 1/8nk 2) the Schwarzschild radius Rschw= 2GMis of the order of the actual radius R ~ 1 / e f and a configuration must collapse. Our main new result is that for a < Otm~xwe find two branches of fundamental solutions. They are shown in fig. 1 for a few values of the winding number B. We shall refer to these two branches as to the lower and upper one, accordingly to their position on fig. 1 (as we already mentioned the lower branch solutions for B = 1 were found previously by Droz et al. [ 15 ] and our numbers confirm their results). On both branches the shape function F(x) decreases monotonically while the metric function N has exactly one minimum. Table 1 shows how masses and Nmin depend on a. The point o~= Otm~ is a bifurcation point at which the two branches coalesce. Note that the critical solution corresponding to Ot=Ctm~x is perfectly regular, in particular Nmln(am~) is quite large 57

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for or--,0 (see fig. 1 ) is due to the fact that the units chosen in (7) are not suitable to take the limit f--, 0. To examine this limit we define new dimensionless variables

25

20

er ,n

x

15

era(r)

10

5

I~(x)

In these variables the eqs. ( 8 )- ( 10 ) become 0

.01

.02

.03

]z'= a ( ½ycENF'2+ sin 2F)

.04

+sinaF(NF,2+ Fig. 1. Two fundamental branches of soliton solutions. The shooting parameter b= - F ' ( 0 ) as a function of a for B= 1 (solid line), B=2 (dashed line), and B--3 (dotted line).

[ (a)~:+ = A sin

Table 1 Masses (in units f~ e) and Nminversus a of the two fundamental branches of soliton solutions with B=I. Note that mADM=(4n.a)#(~)fie. In units defined by ( 18 ), the total mass mADMof the upper branch solutions tends for a--,0 to ~(oo) = 0.8286 (the mass of the n= 1 BM solution). a

0 0.0001 0.01 0.02 0.04 0.04030 0.040375 0.040378

lower branch

upper branch

mADM

Nmia

mADM

groin

72.92 72.90 70.53 68.05 62.41 62.301 62.2726 62.2715

1 0.999 0.933 0.859 0.616 0.597 0.5857 0.5838

oo 1042 109.4 80.7 62.43 62.302 62.2726 62.2715

0.242 0.244 0.305 0.359 0.551 0.568 0.5795 0.5814

which means that staticity is lost well before an apparent horizon forms. To understand the occurrence of two branches of solutions it is useful to consider the limit a--,0. Recall that a = 4nGf 2, so ot = 0 corresponds either to G = 0 (with f fixed) or to f = 0 (with G fixed). The case G = 0 corresponds to decoupling of gravity and in fact as a--,0 the solutions on the lower branch go continuously to the flat space skyrmion with the same winding number. The case f = 0 has no clear physical interpretation but mathematically is well defined. The apparent blowing-up behaviour on the upper branch 58

(18)

-sin - ~ 2F'~ -],

(19)

sin:F)ANF']' 2F(a + NF': + -sinEF~ --~j,

A'= ot.~+ ~2 s i n : F )~A- ,r2

.

(20)

(21)

The key observation is that for or=0 eqs. ( 1 9 ) - ( 2 1 ) are identical to the static spherically symmetric magnetic EYM equations, if we substitute w = cos F (eqs. ( 3 ) - ( 5 ) in ref. [ 1 ] ). Thus when a = 0, eqs. ( 1 9 ) (21 ) have a countable family of Bartnik-Mckinnon regular solutions which interpolate between w (0) = 1 and w(oo) --- ( - 1 )', where n is the number of zeros of w. In the present context the solutions with n odd have B = 1, while solutions with n even are topologically trivial ( B = 0 ) . Therefore, in the B = 1 sector, the n = 1 BM solution is the limiting solution o f eqs. ( 1 9 ) - ( 2 1 ) for a--,0. For B > 1 the situation is more complicated since the function F, for B = k > 1, crosses a multiple of zc at ( k - 1 ) points and for a--,0 these points become singular points of eq. (20). Thus we have a kind of singular perturbation problem. It turns out that the first point Xo at which F()~o) crosses a multiple of ~, goes to infinity when a - , 0 , so that on any finite interval (0, Xo) all B > 1 solutions on the upper branch also tend to the n-- 1 BM solution. To recapitulate, although the ES equations depend on the single parameter a, the limit a--,0 can be taken in two ways giving two different limiting theories with distinct solutions (but satisfying the same boundary conditions). The perturbations (in a ) of these two

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limiting solutions (the flat space skyrmion or the n = 1 BM solution) account for the existence o f two branches of solutions (the lower a n d u p p e r one, respectively), at least for sufficiently small a . It is clear from the above discussion that for sufficiently small a there should also exist solutions ofeqs. ( 1 9 ) - ( 2 1 ) which are related to n > 1 BM solutions. This is indeed the case: we have f o u n d numerically such excited solutions. In the two lowest topological sectors the structure o f excitations is as follows. Let us set F ( 0 ) = z ~ so that F ( ~ ) = z t for B = 0 a n d F ( ~ ) = 0 for B = 1. Then there exists a decreasing sequence o f bifurcation points {ao, a l , a2, ...} such that for a < a i there appears a pair o f new solutions with i + 1 crossings o f zc/2. At a 0 = a m ~ x ~ 0 . 0 4 0 3 7 8 the f u n d a m e n t a l solutions appear, at a 1 ~ 0 . 0 0 1 4 7 the topologically trivial excitations appear, etc. Finally let us note that for the a b o v e classical solutions the neglect of q u a n t u m effects is justified if the "fine structure" constant he 2 is small since then the C o m p t o n radius ~ h e / f is much less than the effective radius ~ 1 ~el

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Table 2 Maximal value ofxa versus a (B< 1). Quadratic fit for the last three values gives x ~ ' = 0 for a=0.040378 (=am,., for B=I solitons).

xff'~

0,0005 0.010 0.030 0.035 0.037 0.039

0.1436 0.1262 0.0560 0.0306 0.0196 0.00815

3

2.8

~,~ 2.6 2.4 2.2

3.2. Black holes In this case, it is easy to show that the formal power series about x = x n is uniquely d e t e r m i n e d by FH=F(xH) so we again have one free p a r a m e t e r to shoot global solutions satisfying F ( ~ ) = 0 . It turns out that such solutions exist only if ( k - ½)zc < F n < kzt and ot ~ 0 ). The masses of black holes decrease with x8 and converge to the masses o f solitons for xH--,0 (see table 3 a n d c o m p a r e with table 1 for a = 0 . 0 1 ). As or--,0 the solutions on the lower branch (larger values o f FH) go continuously to the Schwarzschild solution with chiral hair, while the solutions on the u p p e r branch go continuously to the n = 1 colored

a

2

I .15

.1

.05 Xh

Fig. 2. Two fundamental branches of black hole solutions, The shootingparameterFa=F(xH) as a function ofxH for a--0.0005 (solid line), a=0.01 (dashed line), and or=0.03 (dotted line). Table 3

Masses (in unitsf/e) of black holes versus XHfor c~= 0.01 (B< 1) XH

lower mAD M

m ~

0.001 0.01 0.05 0.10 0.120 0.125 0,126 0,1262

71.09 76.18 98.63 126,14 136,77 139,34 139.845 139.9445

109,53 110.53 116.33 129.21 137.10 139.37 139.848 139.9450

black hole solution described in ref. [2 ]. There is an infinite family o f excitations related to n > 1 colored black holes. The p a t t e r n o f bifurcations is analogous to the regular case. 59

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As we already mentioned, black hole configurations are topologically trivial. However, we can still compute the baryon number given by (13) to obtain a non-integer value B = ( 1/rt) [ F ( X H ) -- ½sin (2FH) ]. One may interpret this as having part of the baryon (s) hidden inside the horizon. Let us also note that for black holes with XH<< 1 which may be viewed as small black holes inside skyrmions, the classical description breaks down since the radius of the horizon is of the order of the Compton radius.

4. Stability analysis Now, let us address the issue of stability of the above static solutions. Here by stability we mean dynamical stability in the Lyapunov sense. An equilibrium configuration is said to be stable if all linear perturbations about it remain bounded (in some suitable norm) with time. Below we restrict our attention to the fundamental branches (all excited solutions are obviously unstable). 4.1. Solitons

First, let us note that the solutions with B> 1 are unstable against single-particle ( B = I ) decay because mADM(B= k) > km~a~M(B= 1 ). This reflects the fact (well known for flat space skyrmions) that for B > 1, the total mass is minimized by nonspherical configurations. Nora bene this property disqualifies (on the classical level) gravitating skyrmions as models of baryonic stars (cf. ref. [ 14 ] ). Now let us consider the stability of B = 1 solutions. In this case the nonspherical instabilities are not expected, so the stability properties can be determined by studying radial perturbations about an equilibrium configuration. At the bifurcation point the equation for radial perturbations would have a zero-eigenvalue mode which is a hint for the stability-instability transition. From comparison of total masses (see table 1 ) it is expected that the lower branch solutions are stable and the upper branch solutions are unstable. Now we prove that this is indeed the case. As was shown in ref. [ 16 ] the evolution of an initially small radial perturbation of the ES equilibrium 60

24 December 1992

configuration is governed by a single equation for the chiral field perturbation 8F, - (AN~')'+ANU~=tr2(AN)-t~,

(22)

where I

8F(x, t)= ~

sin2F ~(x)e . . . . i~

and Uis a complicated function of N, A, Fand x (eq. (22) in ref. [16]). Introducing a new radial coordinate p defined by d p / d x = ( A N ) - l we obtain from (22) a radial SchriSdinger equation _ d2 ~-~ + Verr(p))~= a2~,

(23)

with the effective potential Vo~r=A2N2U. As imaginary a correspond to exponentially growing modes, the instability manifests itself in the presence of a negative eigenvalue for cr2 (bound state). It turns out that V=- V~ff- 2 / p 2 is a smooth bounded function, so eq. (23) is effectively the p-wave radial SchrSdinger equation with the bounded potential V which vanishes at infinity. In this case a simple way to determine the number of bound states is to count the nodes of the zero-energy ( a 2 = 0 ) solution of eq. (23) (which vanishes at p = 0 ) [19]. We have found that for all values of a < c~m~,, on the lower branch the zero-energy solution has no nodes, while on the upper branch there is exactly one node. At the bifurcation point this node goes to infinity signalling the existence of the normalized static perturbation. This proves that, as expected, there is a change of stability at the bifurcation point; the lower branch solitons are stable and the upper branch solitons are unstable #4 4.2. Black holes

In this case the stability analysis proceeds along similar lines as above. The only difference is that now the coordinate p defined by d p / d x = ( A N ) - t range f r o m p = - o o (horizon) t o p = +oo (infinity), hence #4 It was mistakenly asserted in ref. [ 16 ] that stability is lost below the bifurcation point at ~_~0.038. This error was probably due to the not sufficient numerical accuracy of the Levinson criterion which was used in ref. [ 16l to determine the existence of bound states.

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eq. (23) is a one-dimensional Schr6dinger equation. The potential Veer(P) is the smooth bounded function which vanishes at p--, + oo. We have found that for all a
24 December 1992

be inferred by using the Morse-theory argument (first applied in field theory by Taubes and Manton [ 20 ] ). Namely, consider the set of paths in ~ ' connecting the global and the local minimum. On each path there is a point of maximal mass and by minimizing over these masses one gets the saddle point solution. Admittedly, due to the non-compactness and infinite dimensionality of ~¢¢' this mountain pass argument is far from being rigorous. Nevertheless, in view of our numerical results, we believe that this reasoning is basically correct. In particular, by the mini-max construction the saddle point solution should have exactly one unstable mode which agrees with our stability analysis. Thus we claim that the upper branch black hole is a sphaleron solution which lies on the top of the energy barrier separating two minima. It is plausible that the existence of higher mass saddle points corresponding to excited solutions (both regular and black holes) could also be explained by a suitable generalisation of the mountain pass argument (cf. ref. [ 18] ). Finally, notice that since the lower branch black hole is a local minimum of mass it would decay to the Schwarzschild black hole by quantum tunneling through the barrier and, at high temperature, by overbarrier transitions. This baryon-number violating process may have important cosmological implications.

Acknowledgement P.B. thanks Peter Aichelburg for many stimulating discussions. T.C. thanks Nordita, where part of the computations was performed, for hospitality. The work of P.B. was supported in part by the Fundaci6n Federico. The work of T.C. was supported in part by the KBN Grant No 2-0054-91-01.

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[ 5 ] B.A. Campbell, N. Kaloper and K.A. Olive, Phys. Lett. B 263 (1991) 364. [ 6] P. Breitenlohner, P. Forg~lcs and D. Maison, Max-PlanckInstitut preprint MPI-Ph/91-91 ( 1991 ). [7] M. Ortiz, Phys. Rev. D 45 (1992) R2586. [ 8 ] K. Lee, V.P. Nair and E.J. Weinberg, Phys. Rev. D 45 (1992) 2571. [ 9 ] P. van Nieuwenhuizen, D. Wilkinson and M.J. Perry, Phys. Rev. D 13 (1976) 778. [10] P. Hajicek, Proc. R. Soc. A 386 (1983) 223; J. Phys. A 16 (1983) 1191. [ 11 ] R. Friedberg, T.D. Lee and Y. Pang, Phys. Rev. D 35 (1987) 3640, 3658. [ 12 ] S.B. Selipsky, D.C. Kennedy and B.W. Lynn, Nucl. Phys. B 321 (1989) 430.

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[ 13 ] H. Luckok and I. Moss, Phys. Lett. B 176 (1986) 341. [ 14] N.IC Glendenning, T. Kodama and F.R. Klinkhamer, Phys. Rcv. D 38 (1988) 3226. [ 15 ] S. Droz, M. Heuslcr and N. Straumann, Phys. Lett. B 268 (1991) 371. [ 16] M. Hcusler, S. Droz and N. Straumann, Phys. Lett. B 271 (1991) 61. [17] R.S. Palais, Commun. Math. Phys. 69 (1979) 19. [ 18 ] D. Sudarsky and R.M. Wald, University of Chicago prcprint

(1991). [ 19 ] M. Reed and B. Simon, Methods of modern mathematical physics, Vol. IV (Academic Press, New York, 1978). [20] N.S. Manton, Phys. Rev. D 28 (1983) 2019; C.H. Taubes, Commun. Math. Phys. 86 (1982) 267,299.