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A Classical Instability of Reissner-Nordstr¨om Solutions

arXiv:hep-th/9111045v2 4 Dec 1991CU-TP-540FERMILAB-Pub-91/326-A&TA Classical Instability of Reissner-Nordstr¨om Solutionsand the Fate of Magnetically Charged Black HolesKimyeong Lee,a∗V.P.Naira, and Erick J. Weinberga,baPhysics Department, Columbia UniversityNew York, New York 10027bTheory Group and NASA/Fermilab Astrophysics CenterFermi National Accelerator LaboratoryP.O.Box 500, Batavia, Illinois 60510.AbstractWorking in the context of spontaneously broken gauge theories, we show that themagnetically charged Reissner-Nordstr¨om solution develops a classical instability if thehorizon is sufficiently small. This instability has significant implications for the evolutionof a magnetically charged black hole.

In particular, it leads to the possibility that such ahole could evaporate completely, leaving in its place a nonsingular magnetic monopole.This work was supported in part by the US Department of Energy (VPN and EJW),by NASA (EJW) under grant NAGW-2381 and by an NSF Presidential Young InvestigatorAward (KL). * Alfred P. Sloan Fellow.

The Reissner-Nordstr¨om solution to the coupled Einstein-Maxwell equations describesa spherically symmetric black hole endowed with electric or magnetic charge. Althoughthe solution makes mathematical sense in a theory involving only gravity and the electro-magnetic field, its physical motivation is somewhat tenuous unless the theory also containsparticles carrying such charges.

This remark has little consequence in the case of electriccharge, since one only need add a field whose elementary particles are electrically charged.The addition of magnetic charge to the theory is less trivial, but can be accomplished byenlarging the structure of the theory so that electromagnetism emerges as the unbrokensubgroup of a spontaneously broken gauge theory in which magnetic monopoles arise astopologically nontrivial classical solution [1]. The incorporation of these additional fieldsinto the Reissner-Nordstr¨om solution is rather straightforward [2], and changes neither themetric nor the magnetic field.

However, as we will show in this letter, the presence of thesefields can render this solution unstable. This instability arises at the level of the classicalfield equations and does not depend on any quantum mechanical process.

It has importantimplications for the ultimate fate of magnetically charged black holes.The magnetically charged Reissner-Nordstr¨om solution to the Maxwell-Einstein equa-tions has a radial magnetic field with magnitude QM/r2 and a metric which may be writtenasds2 = Bdt2 −Adr2 −r2dθ2 −r2 sin2θdφ2(1)whereB = A−1 = 1 −2MGr+ 4πGQ2Mr2≡BRN(2)There is a physical singularity at r = 0 which is hidden within a horizon atrH = MG +qM 2G2 −4πGQ2M(3)provided that the mass M is greater thanMcrit =√4π|QM|MP(4)where the Planck mass MP = G−1/2. If |QM| ≫1 (as will be the case for the weak gaugecoupling we will assume) the horizon of the critical Reissner-Nordstr¨om black hole is atrH ≫M −1P , and is thus in a region where quantum gravity effects can be neglected.

This solution is readily incorporated into a theory possessing classical magnetic mono-pole solutions. For definiteness we consider an SU(2) gauge theory which is spontaneouslybroken to U(1) by the vacuum expectation value of a triplet Higgs field Φ.

The action isS =Zd4x√−g−116πGR + LMatter(5)whereLMatter = −14Fµν · Fµν + 12DµΦ · DµΦ −V (|Φ|)(6)Fµν = ∂µAν −∂νAµ −eAµ × Aν(7)DµΦ = ∂µΦ −eAµ × Φ(8)and vector notation refers to the internal SU(2) indices. The potential V (|Φ|) is assumedto have a minimum at |Φ| = v; to avoid a cosmological constant, V must vanish at thisminimum.

This theory contains nonsingular monopoles with magnetic charge QM = 1/eand mass Mmon ∼4πv/e, provided that v <∼MP . (For v larger than this, the would-bemonopoles are so massive that they become black holes themselves [3,4].

)The metric for the Reissner-Nordstr¨om solutions of this theory is precisely the sameas that given above for the Maxwell theory. For vanishing electric charge and magneticcharge QM = n/e, the matter fields are, up to a possible gauge transformation,Φ = v ˆe(θ, φ)(9)Aµ = 1eˆe × ∂µˆe(10)where ˆe is a unit vector with winding number n; a convenient choice, which we adopthenceforth, is ˆe = (sin θ cos nφ, sin θ sin nφ, cos θ).

These imply thatFθφ = −Fφθ = ne sin θ ˆe(11)This lies entirely within the electromagnetic U(1) subgroup defined by the Higgs field andprecisely reproduces the radial magnetic field of the Maxwell-Einstein theory. All othercomponents of the field strength, as well as all of the covariant derivatives of Φ, vanish.

We now investigate the stability of these solutions, beginning with the case of per-turbations about the solution with unit magnetic charge. The problem can be simplifiedby considering only spherically symmetric configurations; this turns out to be sufficient todemonstrate instability.

For such configurations, the metric can be written in the form ofEq. (1), with B and A being functions only of r and t. By an appropriate gauge choice thematter fields can be brought to the formΦ = v ˆe(θ, φ) h(r, t)(12)Ai = 1e ˆe × ∂iˆe (1 −u(r, t))(13)where A0 = 0 because we are interested in electrically neutral solutions.

(With our choicefor ˆe, this reduces to the standard ansatz in flat space.) Substitution of this into the matterLagrangian givesLMatter = 1B ˙u2e2r2 + 12v2 ˙h2−1A"u′2e2r2 + 12v2h′2#−(u2 −1)22e2r4−u2h2v2r2−V (h) (14)with overdots and primes referring to derivatives with respect to t and r, respectively.

Thisleads to the equations1√AB∂∂t √AB ˙hB!−1r2√AB∂∂r r2√ABh′A!= −2hu2r2−1v2dVdh(15)and1√AB∂∂t √AB ˙uB!−1√AB∂∂r √ABu′A!= −u(u2 −1)r2−e2uh2v2(16)for the matter fields, as well as equations, whose explicit form we do not need, for the metriccoefficients A and B. To consider fluctuations about the Reissner-Nordstr¨om solution weonly need keep terms linear in u, h −1, δB ≡B −BRN, and δA ≡A −B−1RN.

Remarkably,the coupled equations separate. The equations for the metric components contain neither unor h−1, and thus cannot lead to unstable modes (otherwise there would be an instabilityin the pure Maxwell-Einstein case).

The perturbation of the scalar field enters only inthe linearization of Eq. (15), and can be shown not to lead to instability.

The remaining

fluctuation, u, is determined by the linearized version of Eq. (16).

If we define a variablex bydxdr =1BRN(r)(17)so that x ranges from −∞to ∞as one goes from the horizon to spatial infinity, then theequation for u may be written as0 = ¨u −d2udx2 + U(x)u(18)whereU(x) = BRN(r)(e2v2r2 −1)r2(19)and r is understood to be a function of x determined by Eq. (17).

Instability occurs ifthere are solutions of the form u(r, t) = f(r)eωt with real ω. Substitution of this givesa one-dimensional Schroedinger equation for a particle moving under the influence of thepotential U(x).

The unstable mode exists if this potential has a bound state. Since U(x)goes to the positive value e2v2 at x = ∞, although it goes to zero at x = −∞, it isnot entirely trivial to see for what range of parameters we have a bound state.

One canshow that a bound state exists if rH < c(ev)−1 where c is somewhat less than one, or,equivalently forM < cM 2P2ev + 2πvce(20)As M →Mcrit, c approaches unity. For M ≫Mcrit, we can bound c by a variationalcalculation.

Using the variational ansatz u = √r −rH exp(−λ(r−rH)/2), we find c > 0.32.The physical basis for this instability is easily understood. The classical monopolesolution has a core of radius ∼(ev)−1, inside of which the Higgs field deviates from itsvacuum value and the massive components of the gauge field are nonzero.

The effect ofthis core is to remove the singularity in the energy density which would arise from a pointmagnetic charge. Its radius is determined by the balancing of the energy needed to producethe nontrivial matter fields against the energy cost of extending the Coulomb magneticfield further inward.Similar considerations can be applied to solutions with horizons.

Here, however, weshould only consider the region outside the horizon since singularities are allowed, and

even expected, inside the horizon. Looking at the case of a Reissner-Nordstr¨om solutionwith rH <∼(ev)−1, we see that the Coulomb field has, in a sense, been extended inward toofar.

Energetically, it would be preferable to have a core region extending outward beyondthe horizon [5]. In fact solutions of this sort, which may be viewed as small black holeslying within larger magnetic monopoles, can be shown to exist if v is less than a criticalvalue vcr ∼MP and if the mass M is not too great [4].

When they exist, the horizonradius rH of these solutions is larger than that of the Reissner-Nordstr¨om solution withthe same value of M. Thus, these solutions appear to be the natural endpoints to whichthe instability of the Reissner-Nordstr¨om solution leads.We now turn to the case of multiple magnetic charge. The analysis is complicated bythe fact that in the SU(2) theory the only configurations with higher topological chargewhich are spherically symmetric (i.e., invariant up to a gauge transformation under spatialrotations) are the singular solutions given by Eqs.

(9) and (10) [6].There is thus nospherically symmetric case to which we can restrict our consideration; instead, we mustconsider the full perturbation problem.This can be done by expanding the action inpowers of the fluctuations about the Reissner-Nordstr¨om solution and examining the termsquadratic in these fluctuations. (The linear terms vanish because we are expanding abouta solution.) It is convenient to use the gauge freedom to require that the orientation ofthe scalar field remain the same as in the unperturbed solution, so that δΦ × ˆe = 0.

Itis also useful to decompose the fluctuation in the gauge field into parts orthogonal to andparallel to ˆe; thus, we write δAµ = aµ + cµˆe with ˆe · aµ = 0. The fact that Dµˆe = 0(here, and for the remainder of this discussion, Dµ is the covariant derivative defined bythe unperturbed vector potential) leads to the useful result that ˆe · Dµaν = 0.Several factors simplify the process of extracting the quadratic terms in the action.Because DµΦ, Frµ and Ftµ all vanish for the unperturbed solution, terms containingthe product of a metric perturbation and a matter perturbation can only arise from theFθφ · Fθφ term in LMatter; it is easy to see that the only matter field that can enter here iscµ.

Further, the cross terms between aµ and cν, between aµ and δh, and between cµ andδh all vanish. The result is that the quadratic part of the action may be decomposed asSquad(aµ, cν, δΦ, δgµν) = S1(cµ, δgµν) + S2(δΦ) + S3(aµ)(21)

Since cµ is the component of the fluctuation lying in the unbroken U(1) subgroup, S1describes an essentially Abelian problem; we therefore do not expect it to contain anyunstable modes. Similarly, S2 is simply the action for a neutral scalar field in a curvedReissner-Nordstr¨om background, and easily shown to give no instabilities.This leaves us withS3 =Zd4xr2 sin θ−14(Dµaν −Dνaµ) · (Dµaν −Dνaµ)+12e2v2 aµ · aµ + eFθφ · aθ × aφ(22)where indices are understood to be raised by unperturbed metric and Fθφ is the unper-turbed magnetic field.

Note first that stability would be manifest if it were not for thepresence of the last term in the integrand. Indeed, the instability of the n = 1 solutionsets in as soon as this driving term can be greater in magnitude than the mass terms foraθ and aφ just outside the horizon.

With the aid of the inequality|eFθφ · aθ × aφ| =nr4 sin θ|ˆe × aθ · aφ| ≤nr4 sin θ |aθ||aφ|(23)it is easily shown that the driving term cannot be dominant, and thus stability is assured, ifrH > √n/(ev). Conversely, exponentially growing solutions can be constructed wheneverrH < c√n/(ev).

An explicit example, which can be verified by substitution into the fieldequations derived from S3, is given by at = ar = 0 andaθ = un(r, t) sinn−1θ ∂θˆe × ˆeaφ = un(r, t) sinnθ ∂θˆe(24)where un(r, t) satisfies Eq. (18), but with e2v2 replaced by ne2v2 in the potential U(x).As expected, this solution is not spherically symmetric; under rotation, it transforms intoother linearly independent solutions.

Using Eq. (3), we can see that this instability ispresent whenever n <∼(MP /v)2 andM < Minst =√n cM 2P2ev+ 2πn3/2vce(25)Some physical understanding of the n-dependence of this result can be obtained byreturning to the flat space picture of a core region of radius R containing nontrivial Higgs

and charged boson fields, with only a Coulomb magnetic field extending beyond the core. Avariational argument shows that the value of R which minimizes the energy is proportionalto √n.This instability has significant implications for the evolution of a magnetically chargedblack hole.

A Reissner-Nordstr¨om black hole will lose mass through the emission of Hawk-ing radiation [7]. In the absence of the classical instability, this process would eventuallyturn offas M approached Mcrit, where the Hawking temperatureTH = M 2P2πpM 2 −M 2critM +pM 2 −M 2crit2(26)vanishes, unless it had lost its magnetic charge in the meantime.

Such a discharge could beaccomplished by the production of monopole-antimonopole pairs in the strong magneticfield outside the horizon, with one particle falling into the hole and the other moving outto spatial infinity [8]. Pair production of monopoles with magnetic charge 1/e becomessignificant only in magnetic fields of magnitude eM 2mon [9].

The field at the horizon ofa hole with charge n/e is this large only if M <∼Mpair ∼√nM 2P /v [10]. Since this is afactor of e smaller than Minst, pair production is significant only for black holes which arealready classically unstable [11].The classical instability changes this scenario.

Consider first the case of a hole withunit magnetic charge. Thus, suppose that a single magnetic monopole falls into a largeneutral black hole, which eventually settles down to a Reissner-Nordstr¨om solution.

Thehole begins to lose mass through the Hawking process. As the mass falls below Minstand the horizon contracts within the sphere r = c(ev)−1, the instability causes nontrivialmatter fields to begin to outside the horizon.

The black hole is now described by a solutionof the type found in Ref. 4.

Its horizon continues to contract, revealing more and more ofa monopole core. Its temperature, like that of a Schwarzschild black hole, increases mono-tonically.

While the question of its ultimate fate cannot be settled within the semiclassicalapproximation, the answer will be the same as for a Schwarzschild black hole. If the lattercan in fact evaporate completely, then so can our black hole.

When it does so, it leavesbehind a monopole identical to the one which had fallen in long before.

This picture is modified slightly if n > 1. Because the unstable modes are not spher-ically symmetric, the matter fields which emerge when M falls below Minst do not form auniform shell, but are instead localized about isolated points on the horizon.

A plausibleguess is that as the horizon contracts these grow into lumps which can eventually breakoffas unit monopoles, thus reducing the magnetic charge of the hole. Eventually only asingle charge is left, and the evolution proceeds as described above.Furthermore, if M > Mcrit and n > (MP/v)2, stability is assured since rH > √n/ev.Thus, a black hole could be stabilized by endowing it with a sufficiently large magneticcharge.

However, this stabilization is not absolute. Pair production, although stronglysuppressed, is not quite forbidden.

Eventually, enough of the magnetic charge will havebeen emitted for the monopole instability to emerge.While these results have been obtained in the context of an SU(2) gauge theory, theyclearly can be extended to other gauge theories containing magnetic monopoles. In sometheories with two stages of symmetry breaking it is possible to have more than one varietyof monopole; e.g.

a heavy singly- charged monopole and a somewhat lighter (and spatiallylarger) doubly-charged one [12].In such theories the Reissner-Nordstr¨om solutions ofhigher charge presumably become unstable when their horizon is comparable to the thesize of the lighter monopole, with the singly charged solution remaining stable until it hasshrunk to the size of the heavier one.One can also obtain magnetic monopole solutions in Kaluza-Klein models [13]. Thequestion of whether these lead to similar instabilities is an interesting one, but is beyondthe scope of this letter.Thus, the effect we have found leads to a remarkable new possibility for the evaporationof a black hole carrying a conserved magnetic charge.Previously, it had seemed thatif such a hole did not somehow lose its charge the Hawking process would terminatebefore complete evaporation was achieved.

We see now that charge conservation need notbe a barrier to complete evaporation, and that it is quite possible that a magneticallycharged black hole could evaporate completely, leaving in its place a nonsingular magneticmonopole.We thank Hai Ren for pointing out an error in a previous version of this paper.

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