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Horizons Inside Classical Lumps

arXiv:hep-th/9207070v2 23 Jul 1992UMHEP-374July, 1992Horizons Inside Classical Lumps⋆David Kastor and Jennie TraschenDepartment of Physics and AstronomyUniversity of MassachusettsAmherst, Massachusetts 01003ABSTRACTWe investigate the possibility of having horizons inside various classical fieldconfigurations. Using the implicit function theorem, we show that models satisfyinga certain set of criteria allow for (at least) small horizons within extended matterfields.

Gauge and global monopoles and Skyrmions satisfy these criteria. Q-ballsand Boson stars are examples which do not and can be shown not to allow forhorizons.

In examples that do allow for horizons, we show how standard ‘no hair’arguments are avoided.⋆This work was supported in part by NSF grant NSF-THY-8714-684-A01

1. IntroductionBlack holes are intriguing objects and worth studying in all their possiblevarieties.

In this paper we will study the possibility of having black holes insidevarious classical field configurations.Examples we consider include gauge andglobal monopoles, Skyrmions, Q-balls [1], and Boson stars [2].Besides the basic search for black hole solutions, there are a number of physi-cally motivated questions one can ask in this context. For instance, what happenswhen you drop such an object into a Schwarschild black hole?

For a gauge or globalmonopole the result should be a black hole with the appropriate kind of hair, sincethese both involve non-trivial behavior of the fields at infinity. But in the case of aSkyrmion, one might think that the only possibility would be its vanishing withouta trace.

Our results show that there is another possibility, at least for horizonsvery small compared to the Skyrmion radius†. In the case of gauge monopoles,Lee et.al.

[7] have argued that, besides the Reissner-Nordstrom type solutions [8],there also exist, for sufficiently small horizon radius, solutions in which the Higgsfield and gauge field behave more like an extended monopole outside the horizon.Our results confirm their arguments, and show that global monopoles can also havehorizons inside them.Horizons inside extended field configurations may also be relevant in the latestages of black hole evaporation by Hawking radiation. Lee et.al.

[9] have shownthat extreme, magnetic Reissner-Nordstrom type black holes are unstable in a the-ory with extended monopole solutions. They conjecture that the extended solutiondiscussed above is stable and that evaporation of the black hole proceeds throughthis configuration, leaving a non-singular magnetic monopole as the end state.Perhaps a Schwarschild black hole in a Skyrmion theory, for example, similarlybecomes unstable (or metastable) when its radius is less than the characteristic† numerical results on extended Skyrmion fields around a black hole are given in references[3,4,5,6].2

Skyrmion radius.The evaporation process may then leave behind other stableremnants.Finally, in the literature Q-stars (large Q-balls) [10,11] and Boson stars (see[12] and references therein), as well as strange matter [13] and other types of non-topological solitons, are discussed as candidates for compact astrophysical objects.We can ask what the possible final collapsed states of such matter are.2. Existence of Solutions with HorizonsWe will be looking for static, spherically symmetric solutions to Einstein’sequation, which have nonsingular, nontrivial matter fields outside a horizon.

Theform of the metric will be taken to beds2 = −B(r)dt2 + A(r)dr2 + r2dΩ2(2.1)It is often convenient to define the function m(r) by1A(r) = 1 −2Gm(r)r.(2.2)A horizon occurs at coordinate rH if2Gm(rH) = rH. (2.3)When a horizon is present, one also expects that mo ≡m(0) ̸= 0, so that themetric is not well behaved at the origin.

This is like having a seed mass at theorigin.Let us agree to call a star a configuration of matter fields φ (not necessarilyscalar) such that the stress-energy is static, spherically symmetric, and localized.Suppose a particlar matter field theory has star type solutions, without gravity.There is some force balance, without gravity, which keeps the field configuration3

from either collapsing to a point or expanding to infinity. One might expect thatweakly gravitating solutions would then exist, and that even placing a small seedmass inside the star, wouldn’t disturb the balance too much.

This can be mademore precise by considering the Oppenheimer-Volkoff(OV) equation of hydrostaticequilibrium, which states (in the case when the three principal pressures are notnecessarily the same)dpˆrdr = −G(m(r) + 4πr3pˆr)r(r −2Gm)(ρ + pˆr) + 2r(pˆφ −pˆr)(2.4)In the absence of gravity, only the second term on the right hand side is presentand for weak gravity, this term may still dominate. However, from the first termin (2.4), we see that at a horizon the sum of the radial pressure and the energydensity must vanish.

In a normal, burning star, both these quantities are positiveand a horizon is not possible. On the other hand for many field theories, it happensquite naturally that (ρ + pˆr)|rH = 0.Our main result will be to show that given a matter theory which (1) has startype solutions without gravity and (2) satisfies (ρ + pˆr)|rH = 0 “automatically”(in a sense defined below), then there exist star solutions when the matter theoryis coupled weakly to gravity, and there also exist solutions with horizons inside.More precisely, the non-gravitating matter theory is described by a LagrangianLm.A star solution is found by evaluating the action on field configurationsconsistent with a particular static, spherically symmetric ansatz.

The Lagrangianrestricted to this class of fields will be written Lm(φ). We will assume that −Lm(φ)is positive definite.

When the matter theory is coupled to gravity, we will assumethat the sum of the energy density and radial pressure is given by(pˆr + ρ) = 1AK(φ),(2.5)Where K is a functional of the matter fields only. Then there exist regular startype solutions to the Einstein equation, and there also exist star type solutions4

horizons, which have nontrivial, nonsingular matter fields outside the horizon, forG and rH sufficiently small. The argument, as follows, is an application of theimplicit function theorem.First define new gravitational variables,ex =rBAandey =√AB.

(2.6)The action for fields outside a horizon is then taken to be S = ˜SE + Sm, with˜SE(x, y) = −18πG∞ZrHdry′((r −rH)ey −rex)Sm(φn, x, y) =∞ZrHdrr2eyLm(2.7)˜SE differs from the usual Einstein action by a boundary term, which has beenchosen so that varying ˜SE imposes the correct boundary condition at the horizon(see reference [14]). Varying the action with respect to x and y gives the equationsof motiony′ = −8πGrey−xδLmδx(2.8)ddr(r(ey −ex)) = −r2ey(ey−xδLmδx + Lm + δLmδy )(2.9)and the boundary conditionrex|rH = rrBA |rH = 0.

(2.10)Note that from the definition of the stress tensor −2δLmδx = pˆr + ρ. Equations (2.8)and (2.9) can be used to solve for the gravitational fields x and y in terms of thematter fields alone if and only if pˆr + ρ = 1AK(φ), where K is a function of the5

matter fields alone. This was one of our assumptions.

This is equivalent to thematter lagrangian having the formLm(φ) = −1AK(φ) −U(φ, AB)(2.11)We can then define a positive definite functional of the fields, E(φ, G, rH), byE(φ; G, rH) = −S =∞ZrHdrey(r(r −rH)K + r2U)(2.12)In (2.12), y(r) is given in terms of the matter fields byy(r) = −8πG∞Zrdr′r′K(φ)(2.13)Note that for a given configuration of the fields φ, E(φ, G, rH) is a continuous,differentiable function of G and rH.We assume that for G = rH = 0, the functional E(φ, 0, 0) has a minimum ¯φo.This is our non-gravitating star. For G and rH nonzero, we seek solutions ¯φ toF((¯φ; G, rH)) ≡δEδφ = 0,(2.14),which by construction will satisfy the equation of motion with the correct boundaryconditions.By assumption F(¯φo; 0, 0) = 0.The implicit function theorem forBanach spaces⋆[15] can then be used to show that for G and rH sufficiently closeto zero, there exist functions ¯φ(G, rh) satisfying (2.14), such that ¯φ(0, 0) = ¯φo.This can be seen by expanding (2.14),0 = δFδφ · (¯φ −¯φ0) + ∂F∂G · G + ∂F∂rH· rh + .

. .

,(2.15)with all the derivatives evaluated at φ = ¯φ0, G = 0, and rH = 0. There will bea solution for ¯φ as long as the operator δFδφ in (2.15) is an isomorphism between⋆In the appendix we sketch a finite dimensional version of the theorem, which illustrates therelevant points.6

two Banach spaces H1 and H2, and the two functions ∂F∂G and ∂F∂rh belong to thespace H2. The choice of particular function spaces depends on the system underconsideration.

However, roughly speaking, we can see that this will be true ingeneral given that the flat space solution ¯φ0 is a minimum of the energy functional(2.12) , which is equivalent toδFδφ(¯φ0,0,0)· δφ > 0. (2.16)Hence δFδφ has no zero modes and is invertable.

In the next section we indicate howto choose appropriate function spaces for global monopoles.The OV equation implied that (pˆr + ρ) ∝1A at a horizon. Above, we foundthat this same condition was needed to integrate out the metric coefficients Aand B from the action.

This allowed us to use the existence of non-gravitatingsolutions to imply via the implicit function theorem the existence of gravitatingsolutions and solutions with horizons.If we take a theory, such as Q-balls, inwhich, as we will see below, A and B cannot be eliminated from the action, thento use the implicit function theorem, one would have to compute the variationincluding all the dependent functions, φ, A and B. But knowledge of the flat spacesolutions gives us no information analogous to (2.16) about variations in the A orB directions, so the argument can’t proceed.3.

Global MonopolesIn this section we demonstrate the use of the implicit function theorem andselection of appropriate function spaces for global monopoles. The matter fieldtheory for the basic global monopole is given by an SO(3) invariant Lagrangianfor a triplet of scalar fields φa,L = 12∇µφa∇µφa −12λφaφa −v22 ,(3.1)where ∇µ is the covariant derivative operator.

The scalar field configuration for7

the monopole has the spherically symmetric formφa = vφ(r)ˆra. (3.2)For solutions without horizons φ(r) interpolates between 0 at the origin and 1at infinity.

Evaluated on such field configurations (with the covariant derivativeoperator appropriate for the spherically symmetric metric (2.1)) the lagrangian hasthe form Lm = 1AK + U, where the kinetic and potential terms are given byK = 12v2φ′2,U = v2φ2r2+ 12λv4(φ2 −1)2,(3.3)Here φ′ = dφ/dr. The equations of motion for the metric coefficients arem′(r) = 4πr2( 1AK + U),(AB)′(AB) = 16πGrK.

(3.4)The flat space global monopole solution has the following asymptotic behavior¯φ0(r) ∼ ar,r →0;1 −12λv2r2,r →∞,(3.5)where a and b are constants (the slope a at the origin must be determined numer-ically). From (3.5) and (3.3), one can see that the energy density for the globalmonopole falls offonly as 1/r2, so that the total energy of a global monopolediverges,limr→∞m(r) = 4πv2r.

(3.6)Hence the spacetime of a global monopole is not asymptotic to flat spacetime, butrather to flat spacetime minus a missing solid angle [16],limr→∞1A = 1 −8πGv2. (3.7)In order to avoid a horizon at large radius (which is not of the sort we are interestedin), we will keep 8πGv2 < 1.8

The quantity δFδφ in (2.15) for the global monopole is given byδFδφ δφ = −ddrr2 ddrδφ+2 + r2 6¯φ2 −2δφ(3.8)Here we have rescale lengths by a factor√λv2. The variations ∂F∂G and ∂F∂rh evaluatedon the background solution can be seen to have the forms∂F∂G ∼ r,r →0;1r2,r →∞,∂F∂rh∼ const,r →0;1r3,r →∞.

(3.9)If we take the variation δφ to have the assymptotic behaviorδφ ∼ const,r →0;1r4,r →∞,(3.10)(with the standard L2 norm in three dimensions), then we can accomodate thevariations induced by (3.9). This can be seen by examining the asymptotic behaviorof δFδφ in (3.8).

We then have to show that the operator L = δFδφ is an isomorphismbetween these spaces. Since the operator is elliptic, this will be the case if neitherit nor its adjoint have zero modes.

Suppose that L has a zero mode, then we canwrite0 =∞Z0dr−f ddrr2 ddrf+ r2δ2Uδφ2 f2. (3.11)Integration by parts yields0 = −r2 f ddrf∞0+∞Z0drr2( ddrf)2 + δ2Uδφ2 f2.

(3.12)The boundary term vanishes for functions f having the behavior (3.10). Equation(3.12) then leads to a contradiction ifδ2Uδφ2(¯φ0,0,0)≥0(3.13)holds everywhere.

We have checked numerically that (3.13) is satisfied for the flatspace monopole. Therefore the operator L, which is self-adjoint has no zero-modes.9

4. ExamplesThree examples of field configurations which allow horizons inside are Skyrmions,gauge monopoles, and global monopoles.

These three examples span a range oftypes: gauge monopoles have both a long range magnetic field and topologicalwinding, global monopoles have only the topological constraint, and the Skyrmionfield winds but is not topological. These all have Lm(φ) of the form (2.11), andso satisfy the condition (ρ + pˆr)|rH = 0 at a horizon.

The implicit function the-orem argument shows that solutions with hair exist for G and rH in some rangeabout zero, but gives no information about how large this range is. One can de-duce more information about the range from arguments based on the traditionalpositive ‘no-hair’ integrals, which we do below in Section 5.Field configurations which cannot support horizons include Q-Balls [1] andboson stars [2].

Q-Balls are star type configurations that exist without gravity [1],but, as we will see, fail to satisy the condition (ρ + pˆr)|rH = 0 at a horizon. Thesimplest Q-balls occur in the theory of a single complex scalar field [1].

The Q-ballfield has the form φ = f(r)e−iωt where f(r) vanishes at infinity. The lagrangianevaluated on such configurations isLQ = 12A(f′)2 + 12(m2 −ω2B f2) + U(f2),(4.1)where the mass-term in the potential has been separated out.

The frequency ωmust satisfy ω2 > m2 for stability. From the definition of the stress tensor we thenhavepˆr + ρ = −2AδLmδ1/A + 2BδLmδ1/B = −1A(f′)2 −1B ω2f2.

(4.2)We see that to satisfy (ρ + pˆr)|rH = 0, f must vanish at a horizon⋆. But thismeans that the field is in its vacuum both at the horizon and at infinity, which isnot a Q-Ball type solution.⋆We assume that the volume element√AB is well behaved at a horizon, which implies thatB ∼r −rH near the horizon.10

Boson stars (see [12] for a review) are localised scalar field configurations whichexist only with gravity. The matter lagrangian again has the form (4.2) (with differ-ent potential terms and with ω2 > m2).

Hence Boson stars satisfy (ρ + pˆr)|rH = 0only for f(rH) = 0, implying again that the field be in its vacuum at the horizon,as well as at infinity.A third example which probably does not allow hair is the Abelian-Higgs model[17]. If the scalar field has the form f(r) and the gauge field is given by At(r),then the matter lagrangian isLAH = 12A(f′)2 −1AB (A′t)2 −12B e2(At)2f2 + λ2(f2 −v2)2(4.3)This again is not of the form (2.11), and satisfying (ρ + pˆr)|rH = 0 requires thatA2t f2 = 0 at r = rH.

While this in itself is not enough to rule out solutions, itclearly makes it “harder” to satisfy the equations given this additional conditionon the fields. Indeed, the ‘no-hair’ integrals discussed in section 5 further implythat if At(rH) = 0, then the fields are in their vacuum states everywhere outsidethe horizon.

Adler and Pearson [17] explicitly analyzed the Einstein equation forthis system further, and have shown that this is indeed the case.Finally, it is interesting to think about the case of a Coulombic electric fielddue to a point charge. This is outside the framework of the present discussion,because the non-gravitating configurations are singular, At = q/r.

However, theReissner-Nordstrom charged black holes are solutions with nonzero, nonvacuum,regular matter fields outside the horizon†. In this case, it is easy to check that theE&M Lagrangian reduces toLEM =1AB (A′t)2(4.4)which has the form (2.11) and that, in fact, the combination pˆr + ρ vanisheseverywhere.† Visser [18] has independently studied the condition (ρ + pˆr)|rH = 0 in the context of variousrecent black hole solutions in field theories, such as dilatons and axions, coupled to gravity.He has also looked at the thermodynamics of such solutions.11

In looking at these various examples, one notices that different kinds of massterms play quite different roles. A “true” mass, or any potential U which is inde-pendent of the metric, makes no contribution to the sum pˆr + ρ, as in Inflation.

Adynamical mass which comes from the coupling to the time component of a gaugepotential, contributes a term to pˆr + ρ ∝1Bf2A2t, which tends to rule out hair. Adynamical mass which comes from coupling to the spatial components of a gaugefield contributes zero, and contributes a winding term ∝1r2 to pˆφ −pˆr, which isimportant in the OV equation (2.4).5.

‘No-Hair’ IntegralsIt is interesting to see how the black hole solutions discussed above avoid beingruled out by standard ‘no-hair’ arguments. In the case of extended gauge monopolesolutions, this was discussed in ref.

[7]. We will see that Skyrmions and globalmonopoles escape in basically the same way.

Necessary conditions for the existenceof black hole solutions in a given field theory can be derived by constructing energyintegrals from the equations of motion (see e.g. [17,19]).

If the action in the regionoutside the horizon is given byS = −∞ZrHdrJ(r),(5.1)an extremum occurs whenddrδJδφ′ = δJδφ,(5.2)with the boundary conditions δJ/δφ′ = 0 at r = rH and the fields going to theirvacuum values at infinity. Therefore∞ZrHdrφ′ δJδφ′ + (φ −φ∞)δJδφ= (φ −φ∞) δJδφ′rH= 0(5.3)Consider the case at hand (2.12), where S = −E and J is the positive definiteintegrand.

Since we are assuming that regular solutions exist when G = rH = 0,12

the above is true with rh = 0 and ey ≡1 in J. Since typically the gradient termin the integrand is of the form C2(φ)(φ′)2, this requires that as r ranges from zeroto ∞, there are positive and negative contributions to the potential (the second)term in the integrand.

Now, if the lower limit is taken to be rH, there is still apossibility for positive and negative contributions to sum to zero above, if rH issmall enough. This point was discussed in [7] in reference to gauge monopoles,noting that the fields had to be Reissner-Nordstrom outside the horizon if rH weresufficiently large.

For Skyrmions, the structure of the no-hair integrals dependson what the response is of the Skyrmion field to gravity. But assuming that theeffect of gravity is to further concentrate the energy density, again there will bea critical value of rH, such that if the horizon is larger, the field must be in itsvacuum outside the horizon.

On the other hand, global monopoles have no suchrestriction on the value of rH.Acknowledgements: We would like to thank Karen Uhlenbeck for helpful and in-formative discussions and the Aspen Center for Physics for its hospitality duringpart of this work.APPENDIX AHere we recall the arguement for the implicit function theorem for a system ofN equations in N unknowns, and the limit as N becomes a continuous variable.Let g be the independent variable, and π , i = 1, ..., N be N dependent variables. (These are numbers, not functions.) We seek solutions π = ¯φi(g) to the systemFj(π, g) = 0 , j = 1, ..., N ,(A.1)given that ¯φio is a solution when g = 0, F(¯φio, 0) = 0.

Let π −¯φio = δπ anddenote the matrix of first derivatives with respect to the independent variables byOji = −∂Fj∂π , evaluated at ¯φio, g = 0. Then Taylor expanding the equation F = 0,13

to linear order one needs to solveOjiδπ = −∂Fj∂g · g(A.2)There is a solution δπ for any “source” on the right hand side of (A.2) if the matrixOij has no zero eigenvectors, i.e.,Oijvivj ̸= 0 , for all vi(A.3)For an implicit functional theorem, we would like the limit where the discreteindex i becomes a continuous variable x, with Fj →F(x), π →φ(x). Let {Pi(x)}be a set of basis functions, and let φ(x) = ΣiAiPi(x) and δφ(x) = ΣiδAiPi(x).Then in this limit,Σi∂Fj∂π δπ →ZdyδF(x)δφ(y) δφ(y) = ΣiδFδAiδAi.

(A.4)Hence for a solution one needs that this last quantity, evaluated at the knownsolution, has no zero modes. In the main part of the paper, this condition wasmet since the second variation of the energy functional was nonzero, at the non-gravitating solutions.14

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