We consider the nonlinear stability of the Kaluza-Klein monopole viewed as
the static solution of the five dimensional vacuum Einstein equations. Using
both numerical and analytical methods we give evidence that the Kaluza-Klein
monopole is asymptotically stable within the cohomogeneity-two biaxial Bianchi
IX ansatz recently introduced in \cite{bcs}. We also show that for sufficiently
large perturbations the Kaluza-Klein monopole loses stability and collapses to
a Kaluza-Klein black hole. The relevance of our results for the stability of
BPS states in M/String theory is briefly discussed.Comment: 4 pages, 4 figure

Bizoń, Piotr

Chmaj, Tadeusz

Gibbons, Gary

English

arXiv

We consider the nonlinear stability of the Kaluza-Klein monopole viewed as the static solution of the five dimensional vacuum Einstein equations. Using both numerical and analytical methods we give evidence that the Kaluza-Klein monopole is asymptotically stable within the cohomogeneity-two biaxial Bianchi IX ansatz recently introduced in \cite{bcs}. We also show that for sufficiently large perturbations the Kaluza-Klein monopole loses stability and collapses to a Kaluza-Klein black hole. The relevance of our results for the stability of BPS states in M/String theory is briefly discussed

Bizon, Piotr

Gibbons, Gary W.

Max Planck Society eDoc Server

Nonlinear perturbations of the Kaluza-Klein monopole

Bizon, P.

Chmaj, T.

Gibbons, G.

MPG.PuRe

PRL 96, 231103 (2006) P H Y S I C A L R E V I E W L E T T E R Sweek ending16 JUNE 2006Nonlinear Perturbations of the Kaluza-Klein MonopolePiotr Bizoń,1,2 Tadeusz Chmaj,3 and Gary Gibbons41M. Smoluchowski Institute of Physics, Jagiellonian University, Kraków, Poland2Max-Planck-Institut für Gravitationsphysik, Albert-Einstein-Institut, Golm, Germany3H. Niewodniczanski Institute of Nuclear Physics, Polish Academy of Sciences, Kraków, Poland4Department of Applied Mathematics and Theoretical Physics, Cambridge University, Cambridge, United Kingdom(Received 10 April 2006; published 16 June 2006)0031-9007=We consider the nonlinear stability of the Kaluza-Klein monopole viewed as the static solution of thefive-dimensional vacuum Einstein equations. Using both numerical and analytical methods, we giveevidence that the Kaluza-Klein monopole is asymptotically stable within the cohomogeneity-two biaxialBianchi type-IX ansatz recently introduced by Bizoń, Chmaj, and Schmidt [Phys. Rev. Lett. 95, 071102(2005)]. We also show that for sufficiently large perturbations the Kaluza-Klein monopole loses stabilityand collapses to a Kaluza-Klein black hole. The relevance of our results for the stability of Bogomol’nyi-Prasad-Sommerfield states in M or string theory is briefly discussed.DOI: 10.1103/PhysRevLett.96.231103 PACS numbers: 04.50.+hIntroduction.—The Taub-NUT instanton is the completeRiemannian Ricci-flat 4-manifold with the metric [1]ds2 1md2 1m2d2  sin2d2m21m1d  cosd2; (1)where   0, 0    , 0    2, 0    4, andm> 0 is a free parameter which sets the scale. The Taub-NUT instanton is topologically R4 with the surfaces ofconstant  being the squashed three-spheres. Althoughcomplete, the metric (1) is not asymptotically Euclidean,because the S1 fibers approach a constant circumference atinfinity (rather than growing like ). This kind of asymp-totic behavior is called asymptotically locally flat.By adding the trivial termdt2, one obtains from (1) theregular static solution of the five-dimensional vacuumEinstein equationsds2dt21md21m2d2sin2d2m21m1d cosd2: (2)This metric has been used in the past to obtain the four-dimensional magnetic monopole via a Kaluza-Klein reduc-tion along the coordinate  [2], and since then it is usuallycalled the Kaluza-Klein (KK) monopole. To the best of ourknowledge, the role of the KK monopole in the dynamicsof nonasymptotically flat initial data, and, in particular, itsnonlinear stability, has not been studied, probably becausethis problem appeared to require an analysis of the fullfive-dimensional Einstein equations which is beyond cur-rently available numerical tools, let alone the analytic ones.This situation has changed recently with the introductionof a new cohomogeneity-two symmetry reduction of thefive-dimensional vacuum Einstein equations (referred to06=96(23)=231103(4) 23110below as the BCS ansatz [3]) which provided a frameworkfor investigating the dynamics in a simple 11-dimensional setting. Since the KK monopole falls withinthe BCS ansatz, the question of its nonlinear stability nowbecomes tractable and is the subject of this Letter.The KK monopole plays an important role in M or stringtheory. For example, its metric product with six flatEuclidean dimensions gives a Ricci-flat 11-dimensionalLorentzian metric, which when reduced to ten spacetimedimensions may be interpreted as a D6-brane solution oftype IIA string theory [4]. As such, it is an example of whatis called a supersymmetric or Bogomol’nyi-Prasad-Sommerfield (BPS) solution. BPS solutions of supergrav-ity theories are believed by many string theorists to beabsolutely stable, both classically and quantum mechani-cally. However, until now, this belief has never been testedin a fully nonlinear setting. Our results show that theclassical stability is not absolute since for sufficiently largeperturbations a gravitational collapse to a black hole ispossible. The implications of our results for BPS stateswill be discussed briefly in the conclusions.Preliminaries.—The BCS ansatz has the form [3]ds2  Ae2dt2  A1dr2  14r2e2Bd2  sin2d2 14r2e4Bd  cosd2; (3)where A, , and B are functions of t and r. In contrast tospherical symmetry, this ansatz possesses a dynamicaldegree of freedom, the field B, which measures the defor-mation of sphericity of the angular part of the metric.Substituting the metric (3) into the vacuum Einstein equa-tions, one gets the following system of equations:A0  2Ar23r4e2B  e8B  2re2A1 _B2  AB02;(4)3-1 © 2006 The American Physical SocietyPRL 96, 231103 (2006) P H Y S I C A L R E V I E W L E T T E R Sweek ending16 JUNE 20060  2re2A2 _B2  B02; (5)eA1r3 _B  eAr3B00  43ere2B  e8B  0;(6)where primes and dots denote derivatives with respect to rand t, respectively. The initial value problem for thissystem was investigated in Ref. [3] in the case of asymp-totically flat initial data.The KK monopole (2) expressed by the ansatz (3) takesthe formB0 13ln1m; A0 143m21m8=3; e20  A0;(7)wherer  2m1=21=21m1=6: (8)Note that scaling invariance excludes the existence of non-trivial solutions which are everywhere regular and asymp-totically flat. The KK monopole is the unique nontrivialregular static solution of the system (4)–(6). To see this,note that static solutions which are analytic at the originform a one-parameter family with the following asymp-totic behavior for r! 0:Br 	 br2; Ar 	 1 4b2r4; 	2b2r4;(9)where b is the shooting parameter. Since the system (4)–(6) is scale invariant, it follows from the uniqueness of so-lutions of ordinary differential equations that, up to scaling,there is only one static solution which is analytic at theorigin. It is easy to verify that the solution (7) satisfies theregularity conditions (9) with b  1=12m2. For conve-nience, hereafter we choose units in which m  1.Linear stability.—Before discussing numerical results,we first demonstrate that the KK monopole is linearlystable within our ansatz. To this end, we seek solutionsof the system (4)–(6) in the formBt; r  B0  B1t; ;At; r  A0  A1t; ;t; r  0  1t; ;(10)where  is a function of r determined implicitly by (8).Substituting (10) into (4)–(6), linearizing and separatingthe time dependence B1t;   expitv, we getthe eigenvalue equation for the spectrum of small pertur-bations around the KK monopole23110Av  2v;A  11 dd2dd181 3 42:(11)Although this equation cannot be solved analytically ingeneral, there is an explicit solution for   0v0 3 4: (12)This is the zero mode corresponding to the scaling free-dom—it can be obtained by the action of the scalinggenerator r ddr on B0r. The zero mode (12) has nozeros, which implies by the standard Sturm-Liouville the-ory that the operator A has no negative eigenvalues. Thus,the KK monopole is linearly stable within our ansatz. Notethat the zero mode is not a genuine eigenfunction becauseit is not square integrable. More precisely, on the Hilbertspace L20;1; 1 d, the self-adjoint operator Ahas the purely continuous spectrum 2 2 0;1.Nota bene, the Taub-NUT instanton (1) is known to belinearly stable against transverse traceless perturbations.This follows from the work of Hawking and Pope [5], whoshowed, using two linearly independent covariantly con-stant spinors which the Taub-NUT metric admits, that thespectrum of the Lichnerowicz operator, which governs thelinearized transverse traceless perturbations, is the same(apart from zero modes) as for the scalar Laplacian andtherefore non-negative.Numerical results.—Using a fourth-order accurate finitedifference code, we have solved the system (4)–(6) nu-merically for several families of regular initial data whichrepresent various perturbations of the KK monopole. Theoverall picture does not depend on the specific choice of afamily. The results shown below were produced for initialdata of the form (using P  eA1 _B)B0; r  B0r; P0; r  prR4erR4=s4 ; (13)where the amplitude p was varied and the parameters Rand s were kept fixed. Note that, although the perturbation(13) is exponentially localized, the induced perturbation ofthe function A has a 1=r2 tail as follows from theHamiltonian constraint (4).Small perturbations.—We have found that for smallamplitudes p the perturbation is scattered off to infinityand the solution returns to equilibrium; i.e., it settles downto the KK monopole with the same parameter m. This isshown in Fig. 1. One might wonder why the parameter mdoes not change under perturbation. The reason is simple:The perturbations considered by us have finite energy,while a change of m would require infinite energy. Hereby energy we mean the energy measured with respect to agiven KK reference background as described by Deser andSoldate [6]. This energy is determined by the next to3-2-0.2 0 0.2 0.4 0.6 0.8 1 0  2  4  6  8  1011.0r15.0 19.08.06.0B(t,r)5.03.0 4.0 4.5t=0.0 1.0 2.0FIG. 1. Asymptotic stability of the Kaluza-Klein monopole.For initial data (13) with a small amplitude (p  0:1, R  3, s 1), we plot a series of snapshots of the function Bt; r (where t iscentral proper time). The dashed line shows the unperturbed KKmonopole. During the evolution, the excess energy of the per-turbation is clearly seen to be radiated away to infinity and thesolution returns to equilibrium.PRL 96, 231103 (2006) P H Y S I C A L R E V I E W L E T T E R Sweek ending16 JUNE 2006leading order term in the asymptotic expansion at spatialinfinity.The qualitative picture of pointwise convergence to theKK monopole is shown in Fig. 2. The details of thisprocess, in particular, the role of quasinormal modes andthe rate of decay of a tail, will be pursued elsewhere.We have found that a sufficiently strong kick may de-stabilize the KK monopole. In order to understand what isthe fate of a strongly perturbed KK monopole, we need tomake a digression.Kaluza-Klein black holes.—The KK monopole (2) is aspecial case of the two-parameter family of solutions of thevacuum Einstein equations in five dimensions [7]-12-10-8-6-4-2 0 0  5  10  15  20  25  30ln|P(t,r0)|tFIG. 2. The convergence to the Kaluza-Klein monopole. Forthe same initial data as in Fig. 1, we plot the time serieslnjPt; r0j at r0  2. The asymptotic power-law tail is seen atlate times.23110ds2  dt2 d2  d2  sin2d2 4P2d  cosd2; (14)where   3MM2  2P2p;  MM2  2P2p:Here   0, the parameterM is positive, and the parameterP (usually called the magnetic charge) has the range 0<P  2M. For P< 2M, the metric (14) has a regular hori-zon at   0, so after Ref. [7] we shall refer to this solutionas the Kaluza-Klein black hole. For P  2M, the horizondegenerates to a point and one gets the KK monopole (2)with m  4M.Translating (14) into the ansatz (3), we obtainB 13ln2P; A r2323623; e2 A;(15)where r  42=3P1=31=61=2. Note that the leading orderasymptotic behavior for large r does not depend on M andis determined only by PA	43P9r; e2B 	r4P: (16)Although the KK black holes are known in closed form, forour purposes it is helpful to look at them from the view-point of the shooting procedure. Assuming that there is anondegenerate horizon at r  rH, we obtain from (4)–(6)the following behavior:Br 	 2e2  e84e2  e8rrH 1;Ar 	234e2  e8rrH 1;(17)hence, for each rH > 0 there is a one-parameter family oflocal solutions with a regular horizon parametrized by BrH> 0. Shooting these local solutions towards infinity,we get the KK black holes (15) where 13lnMM2  2P2p2P;r6H  44P2MM2  2P2p3MM2  2P2p3:(18)Large perturbations.—Now we return to the discussionof numerical results. We have found that for sufficientlylarge perturbations the inner part of the KK monopolecollapses and a horizon forms at some r  rH > 0.Outside the horizon, the solution settles down to the KKblack hole (see Fig. 3). The perturbations considered by ushave finite energy, and, therefore, they do not change theleading order asymptotic behavior (16). In this sense, the3-3 0 0.2 0.4 0.6 0.8 1 0  2  4  6  8  103.1991r3.19948 3.1995043.191873.052A(t,r) 2.66t=0.0 1.0 2.0FIG. 3. Instability of the Kaluza-Klein monopole for largeperturbations. For initial data (13) with a large amplitude (p 0:3, R  3, s  1), we plot a series of snapshots of the functionAt; r (where t is central proper time). During the evolution,At; r drops to zero at r  rH  1:68, which signals the for-mation of an apparent horizon there. Outside the horizon, thesolution relaxes to the Kaluza-Klein black hole (15) (shown bythe dashed line) with M  0:502.-4.5-4-3.5-3-2.5-2-1.5-27 -24 -21 -18 -15ln(rH)ln(p-p*)γ = 0.332numerical datalinear fitFIG. 4. Critical behavior. For supercritical solutions, we plotthe horizon radius vs the distance to the threshold on the log-logscale. The fit of the power law lnrH  =2 lnp p const yields   0:33, which agrees (up to numerical errors)with the critical exponent obtained in Ref. [3]. The echoingperiod   0:47 (read off from the period of wiggles around thelinear fit above and computed independently from the spatialshift between the echoes) is also the same.PRL 96, 231103 (2006) P H Y S I C A L R E V I E W L E T T E R Sweek ending16 JUNE 2006magnetic charge P can be viewed as the constant of mo-tion. For the KK monopole, P  2M  m=2, so all ourconfigurations have the same magnetic charge P  1=2(recall that we use units in whichm  1) and the end statesof evolution differ only by M.As we approach the threshold of collapse from above,the parameter M decreases towards the limiting valueM 1=4 corresponding to the KK monopole. At the same time,as follows from (18), both  and rH tend to zero. In thislimit, close to the horizon the KK black hole is very wellapproximated by the Schwarzschild black hole. Near thethreshold, we observe the type II discretely self-similarcritical behavior governed by the same critical solution asin the asymptotically flat situation [3] (see Fig. 4). This isnot surprising in view of the fact that critical collapse is alocal phenomenon and, as such, does not depend on the farfield behavior.Conclusions.—We have shown that, although it is clas-sically stable against small perturbations, the KK mono-pole is classically unstable against perturbations which arelarge enough to allow gravitational collapse to form a blackhole. Classically, the black hole is probably stable.Quantum mechanically, one expects it to emit Hawkingradiation, which can carry off energy but not magneticcharge. In the long run, therefore, one might expect thesystem to settle down to the original KK monopole. If thisis so, then the common intuition about BPS states would becorrect, provided one bears in mind that the stability canhold only by virtue of quantum mechanical effects.Finally, we remark that, besides the KK monopole, theBCS ansatz incorporates other nonasymptotically flat so-23110lutions. In particular, an exact regular time dependentsolution is known [8]ds2  dt2 tmmd2 tmm2d2 sin2d2 m2tmm1d  cosd2:(19)This solution can be viewed as the time evolving KKmonopole whose S1 fibers pinch off to a point for t! 1.It would be interesting to determine the role of this solutionin dynamics.P. B. acknowledges the hospitality of the Isaac NewtonInstitute in Cambridge during work on this Letter. Theresearch of P. B. and T. C. was supported in part by thePolish Ministry of Science Grant No. 1 PO3B 012 29.3-4[1] S. W. Hawking, Phys. Lett. 60A, 81 (1977).[2] R. D. Sorkin, Phys. Rev. Lett. 51, 87 (1983); D. J. Grossand M. J. Perry, Nucl. Phys. B226, 29 (1983).[3] P. Bizoń, T. Chmaj, and B. G. Schmidt, Phys. Rev. Lett.95, 071102 (2005).[4] P. K. Townsend, Phys. Lett. B 350, 184 (1995).[5] S. W. Hawking and C. N. Pope, Nucl. Phys. B146, 381(1978).[6] S. Deser and M. Soldate, Nucl. Phys. B311, 739 (1989).[7] G. W. Gibbons and D. L. Wiltshire, Ann. Phys. (N.Y.) 167,201 (1985); 176, 393(E) (1987).[8] G. W. Gibbons, H. Lü, and C. N. Pope, Phys. Rev. Lett. 94,131602 (2005).

Nonlinear perturbations of the Kaluza-Klein monopole

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