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EUCLIDEAN BLACK HOLE VORTICES

arXiv:hep-th/9112065v1 20 Dec 1991FERMILAB-Pub-91/332-AEFI-91-70UMHEP-361December 1991EUCLIDEAN BLACK HOLE VORTICESFay DowkerNASA/ Fermilab Astrophysics Center, Fermi National Accelerator LaboratoryP.O.Box 500, Batavia, IL 60510, U.S.A.Ruth GregoryEnrico Fermi Institute, University of Chicago, 5640 S.Ellis Ave, Chicago, IL 60637, U.S.A.Jennie TraschenDepartment of Physics, University of Massachussetts, Amherst, MA 01002, U.S.A.abstractWe argue the existence of solutions of the Euclidean Einstein equationsthat correspond to a vortex sitting at the horizon of a black hole. We find theasymptotic behaviours, at the horizon and at infinity, of vortex solutions for thegauge and scalar fields in an abelian Higgs model on a Euclidean Schwarzschildbackground and interpolate between them by integrating the equations numer-ically.

Calculating the backreaction shows that the effect of the vortex is tocut a slice out of the Euclidean Schwarzschild geometry. Consequences of thesesolutions for black hole thermodynamics are discussed.

1. Introduction.The view that the quantum aspects of black hole physics will play an important rˆolein leading us towards a quantum theory of gravity has been strengthened recently, notonly by the discovery that some coset conformal field theories correspond to string the-ory in two-dimensional black hole geometries1, but also by the suggestion that the morefamiliar four-dimensional variety can carry “quantum hair”2,3.

This latter development isof particular interest to relativists, since the conventional wisdom is that powerful theo-rems imply that black holes are characterised only by their mass, angular momentum andelectric charge (and other charges that are associated with a Gauss’ law). Investigatingthese “no-hair” theorems, however, shows that whilst powerful, they are not omnipotent!In particular, the existing ‘no-hair theorem’ for the abelian Higgs model with the usualsymmetry breaking potential makes restrictive assumptions about the behaviour of thefields exterior to the horizon4,5, restrictions that are not obviously satisfied by all physi-cally interesting scenarios.

It has been shown that a black hole cannot be the source ofa non-zero, static, massive vector field6 but the jury is still out on the case where a U(1)gauge field acquires a mass through the Higgs mechanism. However since the expectationis that in this case too, black holes cannot support non-zero massive vector fields, apparentcontradictions are of great interest since they would limit the conditions of validity of arigorous no-hair theorem.It has been noted by Aryal et al.7 that black holes might have hair - quite literally- since they wrote down the metric for a black hole with a cosmic string passing throughit.

They used a distributional energy momentum source as the string, so one could notsay with confidence that this corresponds to a physical vortex spacetime since such alimit is not valid for line-like defects8. However, one might find this suggestive that ano-hair theorem would have to be limited to the case where no topological defects exist,thus reducing the physical relevance of such a theorem since defects will exist if they canexist.

It was also shown by Luckock and Moss9 that black holes could carry skyrmion hair,although they conjectured that such solutions were unstable.More recently, it was pointed out by Bowick et al.2 that there exists a family of2

Schwarzschild black hole solutions to the Einstein-axion equations labelled by a conservedtopological charge. Thus, in some sense, such black holes could be said to be carryingaxion hair.

It was then rapidly realised that the same fractional charge that could giverise to enhancement of proton decay catalysis by cosmic strings10 could potentially becarried by black holes3. The full ramifications of this type of quantum hair have beenmost eloquently argued by Coleman et al.11,12, who suggest that this charge might havedramatic implications for black hole thermodynamics.

Remarkably, their work implies thateven if a black hole does not carry discrete charge its temperature is still renormalised awayfrom the Hawking value. This means that if we are to believe in spontaneous symmetrybreaking and the existence of strings in nature, then we must take into account suchrenormalisation effects independently of whether discrete charge exists or not.All of these claims rest on the existence of a family of ‘vortex’ solutions which aresaddle points in some Euclidean path integral.These solutions are obviously outsidethe domain of standard no-hair arguments, being Euclidean, however they are static inthe sense that the metric is static and the energy-momentum tensor is time-independent(though not in the restricted sense of Gibbons5) and establishing existence would setbounds on the validity of future theorems.In this paper we will focus on the problem of existence of solutions of the abovesort.

The layout of the paper is as follows. We begin by setting up the general problem,discussing what is meant by a ‘vortex centered on a black hole’.

We then show that aperturbative analysis is justified for weakly gravitating vortices, after which we focus onthe specific example of a complex scalar (Higgs) field with a “Mexican hat” potential,coupled to a U(1) gauge field. We find numerically a vortex solution on a Schwarzschildbackground and describe its asymptotic behaviour.

We calculate the back-reaction on thegeometry to first order in GT, the energy per unit area of the vortex (in Planck units),and also calculate the Euclidean action of this geometry. We calculate the expectationvalue of the metric in a black hole state at a certain temperature and derive a relationbetween the mass and temperature without appealing directly to the partition function.We also calculate the expectation value of the area of the black hole.

We draw analogieswith cosmic string physics, and discuss problems with global charge.3

2. Einstein-matter equations: general formalism.We have said we are interested in finding vortex solutions to the abelian Higgs modelin a Euclidean black hole spacetime.

First we should discuss what we mean by a Euclideanblack hole spacetime.Recall that a Schwarzschild black hole metric has the formds2 = −1 −2GMrdt2 +1 −2GMr−1dr2 + r2(dθ2 + sin2 θdφ2). (2.1)We may formally Euclideanise this by setting t →iτ.

However, we now see that the formerLorentzian coordinate singularity at r = 2GM is in danger of becoming a real singularityin the Euclidean space, since the metric changes signature from four to zero for r < 2GM.This tells us that we must regard r > 2GM as the only region of relevance in our Euclideansection, and that therefore we must be able to include r = 2GM in a non-singular fashioninto our manifold. Changing variables to ρ2 = 16G2M 2r−1(r −2GM) we see thatds2 = ρ2d(τ4GM )2 + dρ2 + 4G2M 2dΩ2II(2.2)near r = 2GM, which shows that τ must be identified with period 8πGM, and that r and τare analogous to cylindrical polar coordinates on a plane.

Thus, we arrive at the conclusionthat Euclidean Schwarzschild has topology S2×IR2, with a periodic time coordinate, periodβ = 8πGM. The geometry of the t−r section of Euclidean Schwarzschild can be visualisedas the surface of a semi-infinite “cigar” with a smoothly capped end and tending to acylinder of radius 4GM as r →∞.In general there will be matter present as well as a black hole, therefore, assumingthat the matter is spherically symmetric and ‘static’ (i.e., cylindrically symmetric), wewill be looking for solutions to the Euclidean Einstein equations with topology S2×IR2,being spherically symmetric on the S2 sections, and cylindrically symmetric on the IR2sections.

(Note that we require only the energy momentum to have these symmetries. Itis quite possible that the constituent fields do not, for example, a Nielsen-Olesen vortex iscylindrically symmetric even though the Higgs field has a dependence on the azimuthalcoordinate.) The metric is then a function of just one variable, a radial coordinate in the4

IR2 plane. The presence of a black hole is indicated by the existence of a minimal value ofthe radial coordinate rs (= 2GM, say) at which the metric and curvature are nonethelessregular.

Following Garfinkle et al.13 we will write the metric in the formds2 = A2dτ 2 + A−2dr2 + C2(dθ2 + sin2 θdφ2)(2.3)where A(rs) = 0, τ is understood to be a periodic coordinate with period β, and C(rs)2 =A/4π is given in terms of the area of the event horizon. The regularity of the metric at rsimplies we can choose local cylindrical coordinates in which the metric is regularρ = BA(r)(2.4)where B = β/2π is used for convenience.

Regularity then implies (A2)′|rs = 2/B. Inprinciple we can leave the metric in terms of the period, β, and the area of the event horizon,A, however, for calculational simplicity we choose to use up the coordinate freedomr →ar + b,τ →a−1τ(2.5)to set B = 2rs and C(rs) = rs.

We may then re-interpret our coordinates if required. TheEinstein equations for this metric can then be written as:C′′ = 4πG CA2 (T 00 −T rr )(2.6a)((A2)′C2)′ = 8πGC2(2T θθ + T rr −T 00 )(2.6b)2AA′C′C−1C2 (1 −A2C′2) = 8πGT rr(2.6c)whereTab =2√g∂(L√g)∂gab(2.7)is the energy momentum tensor, which obeys the conservation lawT rr′ + A′A (T rr −T 00 ) + 2C′C (T rr −T θθ ) = 0(2.8)which is valid for a general spherical-cylindrical symmetric source.5

In order to complete our preliminaries on formulating the Einstein equations, we notethat since we expect the greatest variation of T ab to occur near the horizon, it may beexpedient to have a form of the Einstein equations in terms of the proper distance fromthe horizon. For convenience we also scale out the dimensional fall-offbehaviour of theenergy momentum tensor, rH say, to express quantities in terms of the dimensionlessparameterˆr = 1rHZ rrsdr′A .

(2.9)Setting ˆC(ˆr) = C/rs, and ǫ ˆT ab = 8πGT ab r2H, the boundary conditions at the horizon becomeˆC(0) = 1,ˆC′(0) = 0,ˆC′′(0) =12R2 + 12ǫ ˆT 00 |rs ,(2.10a)andA(0) = A′′(0) = 0 ,A′(0) = 12R ,(2.10b)where prime denotesddˆr and R = rs/rH is the ratio of the Schwarzschild radius to thevortex width. The Einstein equations are now(A′ ˆC2)′ = ǫ ˆC2A(2 ˆT θθ + ˆT rr −ˆT 00 )(2.11a) ˆC′A!′= 12ǫˆCA( ˆT 00 −ˆT rr )(2.11b)ˆC′ = −A′ ˆCA"1 −s1 +A2A′2 ˆC2 1R2 + ǫ ˆC2 ˆT rr#,(2.11c)where we have rearranged (2.6c) as a quadratic for ˆC′.

Regularity at the horizon fixes thesign of the root in (2.11c), which is then valid in some neighbourhood of the horizon.Having set up this formalism, we now turn to the problem of deciding under whatcircumstances we expect a vortex black hole to exist.3. Asymptotic solution of Einstein’s equations.We would like to show that solutions exist which correspond to a vortex at the horizonof the black hole.

However, rather than taking a specific field theory source for T ab , in this6

section we remain more general, investigating what minimal conditions T ab must satisfy inorder to have an asymptotically Schwarzschild metric. We naturally have in mind that T abhas some, as yet unspecified, field theory vortex solution as its source, therefore we expectT ab = E ˆT ab /r2H, where E is an energy per unit area characterising the source, ˆT ab is therescaled energy momentum referred to in (2.11) which is of order unity, and rH representsa cut-offscale of the vortex.Thus, for example, a Nielsen-Olesen vortex has E ∼η2and rH ∼1/√λη, where η is the symmetry breaking scale and λ the quartic self-couplingconstant.

Because we are in Euclidean space, we do not have a conventional set of energyconditions for T ab , but since we know that T ab is derived from a θ and φ independent fieldtheoretic lagrangian, we do have a modified dominant energy condition, namely thatL = −T θθ = −T φφ ≥|T 00 |, |T rr |. (3.1)Now, as we have already remarked, we are looking for a non-singular asymptoticallySchwarzschild metric.

This means that we do not expect C = 0, nor in fact do we expectA′ = 0 at any finite r. (We cannot make a similar statement concerning C′, since theeffect of the radial stresses can conspire to make C actually decrease near the horizon. )Inspection of (2.11a) shows that A′(ˆr) > 0 is guaranteed ifJ(ˆr) = ǫZ ˆr0ˆC2A(2 ˆT θθ + ˆT rr −ˆT 00 )dˆr′(3.2)converges, and its modulus is less than 1/2R.

What we will now prove is that if ǫ =8πGE ≪1 (the vortex is suitably weakly gravitating) and if the energy momentum satisfiescertain fall-offconditions then J is not only convergent, but is of order ǫ/R. By a fall-offcondition we mean that outside the core (ˆr ≥few) | ˆT ab | ≤K(ˆr−n) for some K of orderunity, n > 0.

Our aim is to find a value of n which will guarantee that we can integrateout the metric functions to large values of ˆr. This will then tell us what sort of energymomenta we expect well-behaved vortex solutions to have.

Since we are not, at this stage,trying to argue the existence of a full solution to the coupled Einstein-matter system, werestrict our attention to only two of the metric equations, (2.11a,c). The reason for thisis that the three Einstein equations implicitly contain the matter equations of motion,7

conservation of energy momentum being an integrability condition for (2.11a-c). Now letus turn to proving our claim - and finding the value of n.We start by assuming the contrary - that J is divergent.

Then there exists an ˆr0 atwhich J(ˆr0) = −1/4R, thus on [0, ˆr0] (2.11a) implies12R ≥A′ ˆC2 ≥14R. (3.3)Now, in order to use (2.11c) to bound ˆC, we must be sure that the sign of the root is fixed;this relies crucially onf(ˆr) = A′2 ˆC2A2+ 1R2 + ǫ ˆC2 ˆT rr(3.4)being positive.

Let ˆrf ≤ˆr0 be chosen so that f > 0 on [0, ˆrf]. Then, on this interval−√ǫ ˆC| ˆT rr |12 ≤ˆC′ ≤ 1R2 + ǫ ˆC2| ˆT rr | 12(3.5)using {1 −p|y| ≤√1 + x + y ≤1 +px + |y|} for x > 0, |y| < 1.Let us consider the implications of each bound in turn.

The lower bound on ˆC′ impliesˆC ≥exp{−√ǫZ| ˆT rr |12 } ≥e−α√ǫ(3.6)where α will be order unity if we use the fall-offassumption with n ≥4, (and so inparticular ˆC is always positive). HenceA′ ≤12Re2α√ǫ⇒A ≤ˆr2Re2α√ǫon [0, ˆrf].

(3.7)Using this bound and (3.3) we see thatA′2 ˆC4A2+ ǫ ˆC4 ˆT rr ≥e−2α√ǫ4ˆr2−ǫe−4α√ǫ| ˆT rr |(3.8)is strictly positive on [0, ˆrf] provided ǫ ≪1 and the previous fall-offassumption holds.Therefore ˆC2f > ˆC2/R2 on [0, ˆrf], and without loss of generality, we may choose ˆrf = ˆr0.8

Now we examine the upper bound on ˆC:ˆC′ ≤1R + √ǫ ˆC| ˆT rr |12 ≤e√ǫR| ˆT rr |12R+ √ǫ ˆC| ˆT rr |12 ,(3.9)which implies thatˆC ≤e√ǫR| ˆT rr |12 (1 + ˆrR). (3.10)BoundingR| ˆT rr |12 by α as before, we see that|J| ≤ǫZ ˆr0ˆr2Re4√ǫα(1 + ˆrR)2|2 ˆT θθ + ˆT rr −ˆT 00 |dˆr.

(3.11)This is readily seen to be convergent on [0, ˆr0] if n ≥5 in the fall-offassumption, and wemay write|J| ≤ǫγ2R(3.12)for some γ of order unity provided R ≥1. Therefore for R ≥1, J(ˆr0) cannot be equalto 1/4R thus contradicting the initial assumption about ˆr0.

Therefore we conclude thatno such ˆr0 exists, and provided that | ˆT ab | ≤Kˆr−5 we may (formally) integrate out themetric equations to infinity keeping A′, ˆC > 0. Note again that this argument only involves(2.11a) and (2.11c).We now use the following argument to conclude that if a solution does exist then it isasymptotically Schwarzschild.Note that the initial conditions imply thatR rs+δrs(r −rs)|T 00 −T rr |/A2dr is bounded.But then we use A > A(rs + δ) on (rs + δ, ∞) to conclude thatZ ∞rs+δ(r −rs)|T 00 −T rr |A2dr

(3.13)We may then use a theorem† from ordinary differential equations to conclude thatC ∼cr + das r →∞. (3.14)† The theorem states that ifR ∞0x|a(x)|dx is bounded, then the non-zero solutions ofthe 2nd order equation u′′ + a(x)u = 0 have the asymptotic form u ∼Ax + B where theconstants A and B cannot both be zero14.9

Examining (2.6b,c) as r →∞shows that c ̸= 0 and (2.6b) then implies (A2)′ →0 asr →∞, and a rearrangement of (2.6c) givesA2 ∼1c21 −(rs + I)ras r →∞,(3.15)whereI = 8πGZ rrsC2(2T θθ + T rr −T 00 )dr′ = rsRJ. (3.16)Thus we see that any solution must be asymptotically Schwarzschild.

We can also seethat the solution will be changed by O(ǫ) from exact Schwarzschild. Indeed,2AA′C2 = rs + I(= rs(1 + O(ǫ)))(3.17)impliesCA2 (T 00 −T rr ) = 2(C3T rr )′ −C2C′(T rr + 2T θθ )(rs + I)(3.18)using the equations of motion for T ab .

Then, using (2.6c) at the horizon to determineC′|rs = 1 + 8πGr2sT rr |rs, we may rewrite (2.6a) asC′(r) = 1 +ǫC3T rr(rs + I)E + ǫEZ C2(T rr + 2T θθ )(rs + I)ǫC3T rr(rs + I)E −C′dr −ǫ2E2ZC5T rr T 00(rs + I)2 dr→1 −ǫErsZ ∞rsC2(T rr + 2T θθ )dr + O(ǫ2)as r →∞(3.19)which gives the value of c to order ǫ.It is possible to write integral expressions for the changes in the Arnowitt-Deser-Misner (ADM) mass15 and period of the spacetime from their vacuum values. Recall from(3.14,15) that the asymptotic form of the metric isds2 = c−21 −rs + Irdτ 2 + c21 −rs + Ir−1dr2 + (cr + d)2dΩ2II.

(3.20)where c is given by (3.19). If c ̸= 1, then clearly the τ, r coordinates are not those of a‘Euclidean observer’ at infinity.

In order to identify the true period and ADM mass of thespace, we must rescale the r, τ coordinates so that A2 →1 at infinity. Thus we setτ ′ = τ/c;r′ = cr + d(3.21)10

to obtainds2 =1 −(rs + I)cr′ −ddτ ′2 +1 −(rs + I)cr′ −d−1dr′2 + r′2dΩ2II,(3.22)and henceβ′ = β/c = β1 +ǫErsZ ∞rsC2(T rr + 2T θθ )dr(3.23)M∞= c(rs + I(∞))/2G = rs2G1 −ǫrsEZ ∞rsC2T 00 dr.(3.24)are the period and ADM mass of the space to order ǫ.Thus to order ǫ, the period of the geometry decreases, whereas M∞may increase ordecrease according to the details of the specific vortex model chosen.The preceeding expressions give the modified period and ADM mass of the spacetime,if one knows what the solutions are. However, a perturbation expansion in ǫ for solutionsis justified if ǫ ≪1 and we will now give the solutions for the metric functions in theperturbative case.

One can solve for the sources T ab (r) as test fields on the Schwarzschildbackground. In the next section we will study the equations for the matter fields in theAbelian Higgs model, so for now let us assume that we have solved the equations and knowwhat the vortex sources are.

These solutions on the background are exact if ǫ = 0, i.e. thematter and gravity decouple.

The next step is to compute the corrections to the metriccoefficients when ǫ ̸= 0.One finds that to first orderC = C1 = r +Z rrsdr′I1(r′)(3.25)whereI1(r) =ǫErsr3T rr −Z rrsdr′r′2(T rr + 2T θθ ),(3.26)andA2 = A21 = 1 −rsr +Z rrsdr′ I(r′)r′2−2rsr′3Z r′rsdsI1(s)! (3.27)11

where I(r) is given by (3.16) with C replaced by r2.In equations (3.25) and (3.27)everything on the right hand side is known, in terms of the sources.For large r one can then extract the derivative of C and the ADM mass, to give themodifications to the period and mass which are just equations (3.23) and (3.24) with themetric functions in the integrals replaced by their Schwarzschild forms.4. An abelian Higgs vortex solution.We now examine the specific energy momentum source of an abelian Higgs vortexcentered on the horizon.

The lagrangian for the matter fields isL =14F 2µν + (Dµψ)∗Dµψ + λ4 (|ψ|2 −η2)2. (4.1)For a simple vortex solution we choose the variation of the phase of the ψ field to distributeitself uniformly over the periodic τ direction.

This is simply a gauge choice which allowsus to simplify the equations of motion by settingψ = ηX(r)eikτ/BAµ = 1Be(P(r) −k)∂µτ = 1Be(Pµ −k∂µτ) . (4.2)This implies that the lagrangian and equations of motion simplify toL =(P 2,r2e2B2 + η2X2,rA2 + η2 X2P 2A2B2 + λη44 (X2 −1)2)(4.3)1C2 (C2P,r),r = λη2νX2PA2(4.4a)1C2 (C2A2X,r),r = P 2XA2B2 + λη22 X(X2 −1) ,(4.4b)where ν = λ/2e2.It is straightforward to check that the asymptotic behaviour of the bounded solutionsto (4.4) isX ∝(r −rs)|k|/2P = k −α(r −rs)as r →rs(4.5a)12

where α = −Be/(4πr2s)RH A′τdS and1 −X ∝r−1e−√ληr/A∞P ∝r−1e−√ληr/√νA∞as r →∞(4.5b)where A∞is given by (3.15). The appearance of the square root in the dependency of Xon r near the horizon simply reflects the dependence on the local proper distance there.Note that at this level, there is no obvious obstruction to the fall-offcondition on ˆT ab beingsatisfied.If solutions to the coupled Einstein-Higgs equations exist, then we expect that thereis a perturbative limit as ǫ →0, as we have noted‡.

Indeed, many of the demonstrationsof the lack of ‘hair’ on Lorentzian black holes have shown that on a fixed Schwarzschildbackground the interaction between test fields and a source is extinguished as the sourceapproaches the horizon16.Therefore we first consider the question of the existence ofsolutions for the matter fields on a fixed Euclidean Schwarzschild background, settingC = r and A2 = 1 −rsr in (4.4). Rescaling the radial variable to ˜r = r−rsrHgives1(˜r + R)2(˜r + R)2P ′′ = X2Pν(˜r + R)˜r1(˜r + R)2 [˜r(˜r + R)X′]′ = P 2X4R2(˜r + R)˜r+ 12X(X2 −1)(4.6)The question is - is there a solution to (4.6) which connects the bounded behaviour atthe horizon (4.5a) to the bounded behaviour at infinity (4.5b with A∞= 1)?

Existence ofsuch solutions is similar to the difficult question of existence of abelian Higgs vortices in‡ This limit might seem problematic since it involves taking either G →0 or E = η2 →0.The former limit must be taken at finite rs in order to preserve the background geometry,this would mean that the Euclidean black hole would have a formally infinite “mass”.The latter limit is equivalent to sending the symmetry breaking scale to zero which wouldrequire sending the self coupling, λ and the charge, e, to infinity in order to keep rH and νfixed. Since, by rescaling the fields, one can express the equations in terms of ǫ, rH rs andν only, both limits are equivalent as far as the equations are concerned.

However, since Gis a measured physical constant, it may be easier to think of the limit as η →0.13

Minkowski spacetime, first investigated by Nielsen and Olesen17. To see this, consider flatspace and make the ansatz (4.2) with ρ and θ replacing r and τ respectively, where ρ and θare cylindrical polar coordinates in the plane perpendicular to the infinitely long straightstatic Nielsen-Olesen vortex.

Setting XNO and PNO as the Nielsen-Olesen solutions, theequations of motion that these satisfy can be readily seen to be(ρX′NO)′ = XNOP 2NOρ+ 12ρXNO(X2NO −1)P ′NOρ′= X2NOPNOνρ(4.7)Existence of solutions to these equations was shown numerically by Nielsen and Olesen,and their stability properties discovered by Bogomoln’yi18.Much is known about thebehaviour of Nielsen-Olesen vortices, or cosmic strings. In particular, Bogomoln’yi showedthat for a special value of ν, ν = 1, the second order equations in (4.7) reduce to two firstorder equations:ρX′NO = XNOPNO;P ′NO/ρ = 12XNO(X2NO −1).This is often referred to as the supersymmetric limit, since the model is supersymmetrisablefor this value of ν.

The above relations also have the direct consequence that the radialand azimuthal stresses, T ρρ , T θθ , vanish identically. For ν ̸= 1, these stresses become non-zero changing sign according to the value of ν.

This idea will be important in our laterdiscussions of the mass and entropy. However, for the moment, let us just note that forν ≤1 vortex solutions are stable for all values of the winding number k, whereas for ν > 1,solutions with k ≥2 are unstable.In order to see the similarities (and differences) between our problem and the Nielsen-Olesen case we have just discussed, let z = ρ2/4R, then (4.7) becomes1R[zX,z ],z = XP 24Rz + 12X(X2 −1)P,zz = RzX2Pν(4.8)14

The two sets of equations (4.6) and (4.8) become identical as z, ˜r ≪R. However, far fromthe horizon, z, ˜r ≫R, the equations are very different, and we cannot simply infer theexistence of well-behaved solutions to (4.6) from the Nielsen-Olesen case.We do not currently have an analytic proof of the existence of regular solutions to(4.6), however, we have integrated the equations numerically using a relaxation technique,and these results show that the bounded eigenfunctions at the horizon do indeed integrateout to the exponentially decaying eigenfunctions at infinity.

Figure 1 shows a plot of Xand P with k = 1, ν = 1 and R = 2, compared with the Nielsen-Olesen solutions. Theradial coordinate is ˜r for the Schwarzschild case and ρ for the Nielsen-Olesen case.

Thedifference in behaviours at the origin reflects the fact that for the Schwarzschild case r isnot the coordinate in which the metric near the horizon looks flat. At r = 0, X′SCHW = ∞,P ′SCHW = −1.92, X′NO = 1.37 and P ′NO = 0.Having justified the existence of a background solution, let us remark on the behaviourof a fully coupled system.

Settingˆρ = ρ/rH = 2RA(r),a local cylindrical coordinate, we find1ˆρr2s(rs + I)2C4(ˆρX′)′ = XP 2ˆρ2+ 12X(X2 −1) −ǫˆρX′(2 ˆT θθ + ˆT rr −ˆT 00 )(4.9a)1ˆρr2s(rs + I)2C4(P ′/ˆρ)′ = X2Pνˆρ2 −ǫP ′ˆρ (2 ˆT θθ + ˆT rr −ˆT 00 ),(4.9b)or, alternativelyˆρ( ˆT rr )′ + ( ˆT rr −ˆT 00 ) + [O(ǫ) + O(ˆρ2R−2)]( ˆT rr −ˆT θθ ) = 0,(4.10)where I = O(rsǫ) is given by (3.16).Now, noting that C = rs(1 + O(ǫ) + O(R−2)) for ˆρ ≪R, from (3.6,10), we readily seethe similarity of (4.9) with (4.7). We also see that the matter equations can be writtenas some background piece plus an order ǫ piece coming from the interaction of the vortexwith the geometry.

This then justifies the iterative procedure for the matter part of thefully coupled system.15

To zeroth order, the space is Euclidean Schwarzschild,C = r, A2 =1 −rsr,ˆρ = 2Rr1 −rsr . (4.11)In order to calculate the back reaction we will focus on thin vortices, since these are morephysically relevant.

This limit corresponds to R ≫1, and we therefore expect our solutionsto be well approximated by the Nielsen-Olesen solution for ˆρ ≪R, of the exponential form(4.5b) for ˆρ > R, and having some transitionary nature from ˆρ-exponential decay to r-exponential decay for intermediate radii. We will in fact assume R−2 ≪ǫ to facilitatethe following analysis, keeping in mind that for a typical GUT vortex ǫ ∼10−6 wouldonly require rs ≫103rH ∼10−26cm!Since R is so very large, the energy momentaare negligibly small for ˆρ ≥R, so as far as the Einstein equations are concerned we canessentially ignore corrections from the Nielsen-Olesen form for ˆρ ≥R as well, and we willsimply setX0 = XNO(ˆρ);P0 = PNO(ˆρ)(4.12)where it is understood that X0 and P0 have O(R−2) corrections which do not contributeto the order in perturbation theory (O(ǫ)) to which we will be working.The results of section 3 allow us to now calculate the back-reaction on the metric quitestraightforwardly.

In what follows we will suppress the suffix 0 on the energy momentumtensor for clarity. Settingˆµ = −Zˆρ ˆT θθ dˆρ(4.13)the normalised energy per unit area of the vortex, and ˆp = −Rˆρ ˆT rr dˆρ, an averaged scaledpressure, we see that (3.18) implies that, to first order in ǫ,C′(∞) = 1 + ǫ(ˆµ + 12 ˆp).

(4.14)Then, noting from (4.10) thatZˆρ( ˆT 00 + ˆT rr )dˆρ = O(ǫ),(4.15)16

the ADM mass parameter from (3.23) isM∞= rs2G(1 −12ǫˆp)(4.16)to first order in ǫ. Thus, making the coordinate transformation defined in (3.21):r′ = (1 −ǫ(ˆµ + 12 ˆp))r;τ ′ = (1 −ǫ(ˆµ + 12 ˆp))τ(4.17)the asymptotic metric takes the formds2 =1 −2GM∞r′dτ ′2 +1 −2GM∞r′−1dr′2 + r′2dΩ2II .

(4.18)Therefore our asymptotic solution takes the form of Schwarzschild, with an adjusted periodβ′ = β(1 −ǫ(ˆµ + 12 ˆp))= 8πGM∞(1 −ǫˆµ)(4.19)and mass parameter M∞, adjusted that is, relative to the ‘expected’ mass-period relation-ship derived at the horizon. Note also that the area of the black hole is now related to theADM mass viaA = 4πr2s = 16πG2M 2∞(1 + ǫˆp).

(4.20)Note some similarities with a self-gravitating cosmic string. There the IR2 sectionsperpendicular to the string acquire an asymptotic ‘deficit angle’18 δθ = −(2π).4Gµ, whereµ = 2πη2Zˆρ ˆT ρρ dˆρ = 2πη2ˆµ(4.21)is the energy per unit length of the cosmic string in its rest frame.

Here we see that our‘deficit angle’ is δτ = −8πGM∞ǫˆµ = −(8πGM∞).4Gµ. Since we expect the period of τto be 8πGM∞(as we expect the 2π period in θ), we see that the form of the correctionin both cases is the same.

Thus, the gravitational effect of the vortex is to ‘cut’ a wedgeor slice out of the Euclidean black hole cigar outside the vortex. In figure 2 we show aschematic representation of the black hole vortex geometry.17

As we remarked at the end of the previous section, the vortex always decreases the pe-riod compared to its Schwarzschild value for a black hole of a given horizon area. The ADMmass, on the other hand, can be larger than, smaller than or equal to its Schwarzschildvalue for fixed horizon area, depending on ˆp.

Existing results for a self gravitating cosmicstring19 indicate that for ν > (<) 1, ˆp > (<) 0. These results were numerically obtainedand so may only be true to a certain order, however they indicate that there is some criticalvalue of ν, close to 1, for which the average pressure, ˆp, changes sign.

Now, in our case, thebackground is flat space only to zeroth order in R−2 so we expect that the critical valueof ν, νC, differs from the flat space value by O(R−2) and thus is still close to 1.5. Actions, temperature and entropy.Having calculated the gravitational effect of the vortex, it is instructive to calculatethe Euclidean action:IE =Z LM −R16πG √gd4x −18πGZΣ(K −K0)√hd3x(5.1)where K is the trace of the extrinsic curvature of Σ - a boundary “at infinity”, calculatedin the true geometry and K0 the extrinsic curvature trace calculated for Σ isometricallyembedded in flat space.

For our asymptotically flat geometry, C ∼r′, A2 = 1 −2GM∞r′+O(r′−2), this boundary term has the valueIΣ = 12β′M∞. (5.2)For the pure vortex source, we may use the Einstein equations to deduce that the Ricciscalar R = 16πGLM −8πG(T rr + T 00 ).

However, from (4.15) we see thatZC2(T rr + T 00 )dr = 1GO(ǫ2). (5.3)ThusZ LM −R16πG √gd4x = 1GO(ǫ2).

(5.4)18

Therefore, we come to the conclusion that, to first order in ǫ, the Euclidean action is, aswith Schwarzschild, equal to its boundary term, 12β′M∞. However, reading offthe relationbetween β′ and M∞from (4.19), we see thatIE =β′216πG(1 + ǫˆµ) + 1GO(ǫ2)(5.5)in terms of the period.

However, note thatβ′2ǫˆµ16πG = −β′2rsZ ∞rsC2T θθ dr =β′4πrsZLM√gd4x =ZLM√gd4x + O(ǫ2) . (5.6)HenceIE(β′) =β′216πG +ZLM√gd4x = I0(β′) + IM(β′),(5.7)to first order in ǫ, where I0(β′) is the action of Schwarzschild with period β′ and IM(β′)is the action of the X0, P0 solution in the background of Schwarzschild with period β′.Therefore, taking into account the back-reaction of the vortex on the geometry, we confirmthe value of the Euclidean action used by Coleman et al.11The interest of computing the Euclidean vortex solutions is that their actions con-tribute to the gravitational path integral.

In the path integral one must decide whichfields to include in the sum. One prescription is to include all metrics and matter fieldswith a particular fixed period, β, and this describes ”a system at temperature 1/β”.

Herewe compute what follows from such a prescription. Other boundary conditions are possible,which will be explored in further work.Having calculated the vortex geometry we are in a position to directly calculate theexpectation value of the mass of a black hole of temperature 1/β using< gab >= (1 +X±C±e−I±)−1[g0ab +X±C±e−I±g±ab] + O(e−2I±)(5.8)where g0ab is the Schwarzschild metric with period β, g+ab = g−ab are the k = ±1 vortexgeometries with period β and I+ = I−are the matter parts of their actions.

C+ = C−arethe determinants of quadratic fluctuations about the vortices.19

This formula is derived from a Euclidean path integral and must be used with cautionsince the metric is not a gauge invariant quantity. One must add the metrics at the samepoint of the space-time manifold, which concept has no diffeomorphism invariant meaning.However, in this case, since the metrics are all asymptotically flat, we can fix coordinatesin the asymptotic region and only use the formula (5.8) there.

In each case we choosecoordinates such that g00 →1, and the area of the two-spheres is 4πr2 as a function of rat infinity.Since the geometries for k = ±1 are identical, setting C = C+ + C−yields< g00 > ∼(1 + Ce−IM )−11 + Ce−IM −2Gr (M + Ce−IM M∞)< grr > ∼(1 + Ce−IM )−11 + Ce−IM + 2Gr (M + Ce−IM M∞)< gθθ > = < gφφ >sin2 θ∼r2(5.9)as r →∞, where IM = I± andM =β8πG,M∞=β8πG(1 + ǫˆµ).Substituting in for the masses we obtain< g00 >∼1 −β4πr(1 + ǫˆµCe−IM )< grr >∼1 +β4πr(1 + ǫˆµCe−IM ). (5.10)Thus we have< M(β) >=β8πG[1 + Ce−IM ǫˆµ](5.11)as the predicted value of the mass of a black hole with temperature β−1.

Noting that, fork = ±1, ǫˆµ = 4Tstring in the notation of Coleman et al., this is readily seen to agree withtheir expression for the modified Hawking temperature of the black hole11.20

The horizon is another place where we can make sense of (5.8). It is a two-sphere andfor each metric in (5.8) we know its area, A, in terms of the period, giving< A >= β24π1 + Ce−IM (2ǫˆµ + ǫˆp)(5.12)for the expectation value of the area of the black hole.

We compare this with the entropy,S(β), calculated from the partition function, Z(β), viaS = β2 ∂∂β−β−1lnZ. (5.13)Approximating the Euclidean path integral for Z(β) semiclassically yieldsZ(β) = e−β216πG 1 + Ce−IM (5.14)and thus4GS(β) =˜β24π1 + 2ǫˆµCe−IM −Ce−IM .

(5.15)We find that the central formula S =14GA in black hole thermodynamics has now appar-ently been violated, and depending on the specifics of the vortex (i.e. the size and sign ofˆp) S can either be greater than or less than14G < A >.

Note that the result (5.12) couldnot be obtained from the partition function since it contains an ǫˆp term.6. Conclusions.To summarise: we have argued the existence of solutions of the coupled Einstein-vortexequations by showing that under suitable fall-offconditions of the energy-momentum ofa weakly gravitating vortex a perturbative analysis is justified.

We have demonstrateda suitable vortex for beginning an iterative procedure by numerically obtaining a vortexsolution of the abelian Higgs model in a Schwarzschild background. We calculated themass-period-area relations for the corrected geometry to first order in ǫ, the gravitationalstrength of the vortex and used these results to derive the renormalised mass of a blackhole of a certain temperature.

We also found that the expected value of the horizon areais not related to the entropy of the black hole in the usual way.21

Our work also provides a potential ‘no-go’ argument for global vortices. In the cosmicstring scenario, local strings have asymptotically conical spacetimes whereas static globalstring spacetimes are singular21, the energy momentum tensor having only a 1/r2 fall-offin flat space.

In our Euclidean case, the energy of a global vortex in the Schwarzschildbackground would have no fall-offdue to the fixed circumference (β) of r, θ, φ =constcircles. Therefore, drawing an analogy between these two situations, if static global cosmicstrings are singular we do not expect global black hole vortices to be otherwise.Nothaving asymptotically flat geometries, they would therefore not contribute to the partitionfunction.We mentioned the effect of varying the parameter ν on the results obtained.Forthe flat space Nielsen-Olesen vortex, the critical value of ν is exactly 1.In that case,ν > 1 means that a string with winding number k ≥2 is unstable17, alternatively, thatthe vortices repel one another, whereas ν < 1 implies that they attract.

Since we haveargued that just such a critical value of ν, νC close to 1, exists for the black hole vortices,it is interesting to speculate that, for ν > νC, the k ≥2 solutions are unstable, i.e. arenot minima of the Euclidean action.

In that case the k ≥2 solutions that we have foundwould not contribute to a Euclidean path integral. It seems plausible to suppose thatstable solutions of the matter equations on a Schwarzschild background do exist, whichwould consist of two separate string world sheets sitting opposite each other (τ2 −τ1 = 12β)at finite distance from the horizon, where any further loss of energy due to moving furtheraway would be balanced by an increase in energy due to increase in the area of the worldsheets.

Such a solution would not be cylindrically symmetric and its action would differfrom the form calculated in (5.6), although presumably the difference would be small.However, it would be interesting to investigate such types of solutions.Our derivation of the geometry not only enabled us to confirm the results of Colemanet al., but we were also able to calculate the expected area of the black hole. We obtainedwhat looks to be a discrepancy in the usual area-entropy relationship, though, in this case,virtual string world sheets “dress” the black hole around the horizon and perhaps oneshould not expect the area-entropy relation to survive.

However, it is the pressure, ratherthan some combination of energy and pressure, that is contributing to the discrepancy and22

this result certainly merits further thought.Acknowledgements.We would like to thank Peter Bowcock, Gary Gibbons, Stephen Hawking and RobertWald for useful discussions. We are especially grateful to Jerome Gauntlett and DavidKastor for assistance during the early stage of this work and to JeffHarvey for usefulsuggestions.

F.D. was supported in part by the DoE and NASA grant NAGW-2381 atFermilab, R.G.

by the NSF grant PHY 89-18388 and the McCormick Fellowship fund atthe Enrico Fermi Institute, University of Chicago and J.T. by the NSF.References.

[1]E.Witten, Phys. Rev.

D44 314 (1991). [2]M.Bowick, S.Giddings, J.Harvey, G.Horowitz and A.Strominger, Phys.

Rev. Lett.

612823 (1988). [3]L.Krauss and F.Wilczek, Phys.

Rev. Lett.

62 1221 (1989). [4]S.Adler and R.Pearson, Phys.

Rev. D18 2798 (1978).

[5]G.Gibbons, Self gravitating magnetic monopoles, global monopoles and black holes (R90/31). [6]J.Bekenstein, Phys.

Rev. D5 1239 (1972).

[7]M.Aryal, L.Ford and A.Vilenkin, Phys. Rev.

D34 2263 (1986). [8]R.Geroch and J.Traschen Phys.

Rev. D36 1017 (1987).

[9]H.Luckock and I.Moss, Phys. Lett.

176B 341 (1986). [10]Ph.de Sousa Gerbert, Phys.

Rev. D40 1346 (1989).M.Alford, J.March-Russell and F.Wilczek, Nucl.

Phys. B328 140 (1989).

[11]S.Coleman, J.Preskill and F.Wilczek, Mod. Phys.

Lett. A6 1631 (1991).

[12]S.Coleman, J.Preskill and F.Wilczek, Phys. Rev.

Lett. 67 1975 (1991 1991).

[13]D.Garfinkle,G.Horowitz and A.Strominger, Phys. Rev.

D43 3140 (1991). [14]R.Bellman, Stability Theory of Differential Equations(McGraw-Hill, New York,1953).

[15]R.Arnowitt, S.Deser and C.Misner in Gravitation: an Introduction to Current Re-23

search (ed. L.Witten, Wiley, New York, 1962).

[16]C.Teitelboim, Phys. Rev.

D5 2941 (1972).J.B.Hartle, Phys. Rev.

D1 394 (1970). [17]H.B.Nielsen and P.Olesen, Nucl.

Phys. B61 45 (1973).[18]E.B.

Bogomoln’yi, Sov. J. Nucl.

Phys. 24 449 (1976).

[19]D.Garfinkle, Phys. Rev.

D32 1323 (1985).R.Gregory, Phys. Rev.

Lett. 59 740 (1987).

[20]P.Laguna-Castillo and R.Matzner, Phys. Rev.

D36 3663 (1987). [21]R.Gregory, Phys.

Lett. 215B 663 (1988).G.Gibbons, M.Ortiz and F.Ruiz-Ruiz, Phys.

Rev. D39 1546 (1989).24

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