The Meissner Effect for weakly isolated horizons

Norman Gurlebeck∗

ZARM, University of Bremen,Am Fallturm, 28359 Bremen, Germany

DLR Institute for Space SystemsLinzer Str. 1, 28359 Bremen, Germany

Martin Scholtz†

Institute of Theoretical Physics, Charles University,V Holesovickach 2, 162 00 Prague, Czech Republic

(Dated: November 6, 2018)

Black holes are important astrophysical objects describing an end state of stellar evolution, whichare observed frequently. There are theoretical predictions that Kerr black holes with high spinsexpel magnetic fields. However, Kerr black holes are pure vacuum solutions, which do not includeaccretion disks, and additionally previous investigations are mainly limited to weak magnetic fields.We prove for the first time in full general relativity that generic rapidly spinning black holes includingthose deformed by accretion disks still expel even strong magnetic fields. Analogously to a similarproperty of superconductors, this is called Meissner effect.

PACS numbers: 04.70.Bw; 04.20.Cv; 98.62.Nx; 95.30.Sf;

I. INTRODUCTION

Black holes, as a final state of stellar evolution,are nowadays considered standard astronomical objects.Their existence is predicted by general relativity, sup-ported both by strong theoretical arguments [1] and ob-servational evidence, most directly in the recent detectionof gravitational waves [2]. If they are not surrounded bymatter, they are treated as Kerr–Newmann black holescharacterized by their mass, spin and charge alone. Thisfundamental prediction of general relativity is known asthe no-hair theorem for black holes [3], although three-hair theorem would be a better name. Additionally, thecharge is usually neglected in astrophysical environments.

The black hole’s spin is successfully measured using thecontinuum fitting and the iron line method [4–6] possi-bly augmented by gravitational lensing. These methodsrequire the presence of an accretion disk, which does notcomply with the aforementioned assumptions of the no-hair theorem. The masses of the accretion disk, which aresmall compared to the black hole’s mass, are typically as-sumed to yield negligible perturbations. Yet, if tests ofthe no-hair theorem are carried out, as suggested for fu-ture observatories like the Event Horizon Telescope [7, 8],the admittedly small effects by the disk may become non-negligible. To estimate such effects, it is prudent to treatblack holes in a more general setting allowing for devia-tions from the Kerr geometry caused by additional mat-ter in general relativity as it was recently started for theno-hair theorem in [9]. Naturally, this raises the questionwhich other properties of Kerr black holes are universally

∗ norman.guerlebeck@zarm.uni-bremen.de† scholtz@utf.mff.cuni.cz

holding for any black hole and which are sensitive to apossible accretion disk.

We will show for one important property – the so-calledMeissner effect – that it is universal. The Meissner effectdescribes the property of black holes to expel any mag-netic field if they become extremal, i.e., if they have amaximal spin. This is especially interesting, since obser-vations suggest that many supermassive black holes arealmost extremal [6, 10]. If the spin would exceed thisthreshold, the singularity inside the black hole would be-come naked and visible to distant observers, which isbelieved to be unphysical and, thus, prohibited as sum-marized in the cosmic censorship conjecture [11].

On the theoretical side, an analogy between black holesand thermodynamics emerged quite early in works ofBekenstein and Hawking [12, 13]. In particular, theyfound that the surface gravity κ of a black hole plays therole of its temperature T via T = κ/2π, where we choosegeometrical units in which G = ~ = c = 1. The spin aand the mass M of a Kerr black hole in turn determine itssurface gravity κ =

√M2 − a2/(2M(M +

√M2 − a2)).

For extremal Kerr black holes, where a = M , the surfacegravity and, hence, the temperature vanish.

The analogy with thermodynamics can be carried fur-ther. In particular, extremal black holes, for which thetemperature vanishes, expel external magnetic test fieldsmuch like superconductors [14–17]. In light of these sim-ilarities, the effect was dubbed Meissner effect. It hasbeen investigated for electromagnetic fields coupled tothe gravitational field around Kerr–Newmann black holes[18], for special exact models containing magnetic fields[19–23], and in string and Kaluza-Klein theory [24]. Arelation between the Meissner effect and entanglementwas also discussed [25].

From this theoretical treatment, one might be led tobelieve that the Meissner effect has consequences on the

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production efficiency η of jets via the Blandford–Znajekprocess [26]: η ∝ a2 Φ2

BH, where a is the spin of theblack hole and ΦBH is the time-averaged magnetic flux[27–29]. Faster rotating black holes are expected to pro-duce jets more efficiently. However, a simultaneous de-crease of the magnetic flux, as predicted by the Meissnereffect, might counter-balance this behavior. This con-clusion rests, however, on the assumption that there isno matter in the vicinity of the horizon. In contrast theauthors in [30] did not assume vacuum electrodynamicsbut rather force-free electrodynamics as it is suitable foraccreting black holes. In that case, the Meissner effectwas found to have no effect on the jet creation, for othermodels allowing matter crossing the horizon, see [31, 32].Based on the aforementioned results it is generally be-lieved that the presence of matter suppresses the Meiss-ner effect. Indeed, this conclusion is corroborated by theobservation of the black hole GRS 1915+105, which hasa spin of a = 0.98 ± 0.01 and still creates jets efficiently[33, 34].

In contrast to these approaches, we assume vacuumin an arbitrarily small vicinity of the horizon to showanalytically that the Meissner effect is a general propertyof isolated black holes in general relativity.

II. THE MEISSNER EFFECT FORASTROPHYSICAL BLACK HOLES

Since we wish to discuss the Meissner effect for moregeneral black holes than those described by the Kerr met-ric, we use the quasi-local definition of a weakly isolatedhorizon (WIH). It describes horizons in equilibrium, i.e.,currently no matter or radiation falls in [35]. However,the WIH can be penetrated by electric and magneticfields. The WIHs play also a role in loop quantum grav-ity [36]. Note also that the thermodynamics of WIHs wasdeveloped [37]. In what follows, we assume a station-ary and axially symmetric spacetime in a neighborhoodof an uncharged WIH, which corresponds to a genericblack hole in equilibrium. Moreover, we assume that suf-ficiently close to the WIH we have no matter, i.e., wehave electrovacuum. Let us stress the facts that we nei-ther assume the symmetries globally nor do we make anyassumption whatsoever about the matter further out, inparticular, about a possible accretion disk.

We assume that the space-time contains a WIH [37],i.e. a non-expanding null hypersurface H on which theEinstein-Maxwell equations are satisfied, equipped withthe normal `a; by definition, there is no flux of matteror radiation through the horizon. For the description ofthe space-time, we employ the Newman–Penrose (NP)formalism [38] in which the main geometrical quantitiesare the spin coefficients, the Weyl scalars and the mat-ter is described by the scalar projections of the energy-momentum tensor of an electromagnetic field. The nullnormal `a is necessarily tangent to the geodesics gen-erating the horizon and satisfies D`b = κ(`) `b, where

D = `a∇a and the constant κ(`) is the surface gravity ofthe WIH. κ(`) vanishes for extremal horizons.

We take the notation, the coordinate system and themetric of such an arbitrary black hole as in [39]. Addi-tionally, we use standard spherical coordinates θ and ϕ onthe topological 2-spheres foliating the horizon. In thesecoordinates, the intrinsic geometry of the 2-spheres isgiven by the metric conformal to a unit Euclidean spherewith the conformal factor R = R(θ, ϕ),

ds2 = R2(dθ2 + sin2 θ dϕ2

). (1)

At any point of such a 2-sphere there are exactly two nullfuture-pointing directions: `a is tangent to the horizonand we denote the other one by na and fix its scaling by`an

a = 1. We complete these vectors to a full NP nulltetrad by introducing two complex null vectors ma andma satisfying mam

a = −1 which span the tangent spaceof the sphere. The intrinsic connection compatible withthe metric (1) is encoded in the complex spin coefficient

a(0) = α(0) − β(0) = maδma, (2)

where δ = ma∇a on H is given by

δ =1√2R

(∂θ +

i

sin θ∂ϕ

), (3)

The transformation

ma 7→ eiχma (4)

is called spin, where χ is an arbitrary real parameter. Itcorresponds to a rotation in the tangent space of a 2-sphere. A quantity η is said to have the spin weight s ifit transforms as

η 7→ ei s χη (5)

under the spin (4). For a spin s quantity η one definesthe spin raising and lowering operators ð and ð by

ðη = δη + s a(0) η, ðη = δη − s a(0) η. (6)

Following [39], we extend vectors comprising the nulltetrad off the horizon by conditions

∆na = ∆`a = ∆ma = 0, (7)

where ∆ = na∇a. In terms of the spin coefficients, con-ditions (7) imply

γ = ν = τ = 0 (8)

everywhere in the neighborhood of the horizon.The full space-time geometry on the horizon and in its

neighborhood is a solution of a characteristic initial valueproblem with the initial data given on two intersectingnull hypersurfaces. The first one is the horizon H, theother one is an arbitrarily chosen null hypersurface Ntransversal toH and intersectingH in a 2-sphere S0. The

3

free data on the sphere S0 consists of the aforementionedfunction R, the values of the spin coefficients π(0), a(0) =

α(0)− β(0), λ(0) and µ(0), the Weyl scalars Ψ(0)2 and Ψ

(0)3

and the electromagnetic scalar φ(0)1 , where we use the

notation of [38, 39]. The real and imaginary part of φ(0)1

are the flux densities of the electric and magnetic fieldthrough the sphere S0, respectively. The Weyl scalar

Ψ(0)2 determines the multipole moments of the horizon

[40], namely, its real part is related to the mass and itsimaginary part is related to the angular momentum of the

horizon. The functions Ψ(0)2 , a(0), π(0) and φ

(0)1 are not

independent but they are constrained by the equations

<Ψ(0)2 = |a(0)|2 − 1

2(δa(0) + δa(0)) + |φ(0)1 |2, (9)

=Ψ(0)2 = −=ðπ(0); (10)

the spin weight of π(0) is −1. Finally, the spin coeffi-cients λ(0) and µ(0) describe the extrinsic curvature ofthe horizon. In order to have a fully determined initialvalue problem, the Weyl scalar Ψ4 and the electromag-netic scalar φ2 must be specified on the null hypersurfaceN .

Next, imposing the aforementioned symmetries, we re-quire that, in the neighborhood of H, there exists a time-like Killing vector Ka which equals `a on the horizon. Anecessary condition for the existence of such a Killingvector field is given by [41]

∇c∇aKb = RabcdKd, (11)

In addition, we introduce the axial Killing vector ηa

which acquires the form ηa = (∂ϕ)a [42]. The Killingequation (A3f) (with K replaced with η) then implies

a(0) = − 1√2R2

(R′ +R cot θ) . (12)

The electromagnetic field Fab is assumed to possess thesame symmetries, i.e. we impose

£ηFab = 0, £KFab = 0, (13)

so that the electromagnetic NP quantities do not dependon the coordinates v and ϕ. In the electrovacuum case,the anti-self dual part of Fab

Fab =1

2(Fab + i ?Fab) (14)

is a closed form which, together with (13) implies thatthe 1-form

Ja = Fabηb (15)

is also closed

∇[aJb] = 0. (16)

Writing the conditions (11) and (16) in the NP formal-ism and restricting them to the horizon we arrive at theconstraints

κ(`) λ(0) = ðπ(0) +

(π(0)

)2, (17a)

κ(`) φ(0)2 = ðφ(0)1 + 2π(0) φ

(0)1 , (17b)

where π(0) and φ(0)1 have spin weights −1 and 0, respec-

tively. Additionally, Eq. (17a) implies that the spin co-efficient λ(0) is time-independent, as can also be inferredfrom substituting (17) into the expression for λ in [39].The instructive but tedious calculations showing the va-lidity of Eqs. (17) for Kerr black holes will be presentedelsewhere.

Subsequently, we prove the Meissner effect for un-charged black holes, i.e., we show that the magnetic fluxacross extremal, axially symmetric and stationary hori-zons vanishes. This is done by determining φ1 explicitly.For extremal horizons [43], where we have κ(`) = 0, Eq.(17a) can be solved in terms of the free function R:

π(0)(θ) =R(θ) sin θ

cπ +√

2θ∫0

R2(θ) sin θ dθ

, (18)

where cπ is a complex integration constant. Now, thesolution of Eq. (17b) reads

φ(0)1 =

cφ(cπ +

√2θ∫0

R2(θ) sin θ dθ

)2 , (19)

where cφ is another complex integration constant. Thetotal electric charge Q and magnetic charge Q? of theblack hole, which are not restricted at this stage, arethen given by

Q+ iQ? =2√

2π cφ

cπ +√

2π∫0

R2(θ) sin θ dθ

. (20)

Requiring that both charges vanish we get cφ = 0. Thisin turn means that the magnetic and electric flux den-

sity encoded in φ(0)1 vanish everywhere at the horizon,

thereby, proving the Meissner effect.It is worth mentioning that the symmetries were essen-

tial for our derivation. For a general WIH, φ(0)1 is part of

the free, unconstrained data, showing that for the Meiss-ner effect some symmetry is necessary. Indeed, it wasshown that specific non-axially symmetric magnetic testfields penetrate the horizon of an extremal Kerr blackhole, see [16]. Thus, the Meissner effect does not hold inthis case. On the other hand, as those authors point out,the test field they consider is, in fact, not the limit of astationary electromagnetic field [44]. Hence, the station-arity might be the crucial of the two symmetries.

We also emphasize that the existence of the Killing vec-tors was assumed only in the neighborhood of the hori-zon, not in the entire space-time.

4

III. THE EXPULSION OF THE MAGNETICFIELD FROM THE HORIZON

The proof given above shows that the magnetic fluxacross any part of the horizon vanishes for strictly ex-tremal black holes. For the understanding of the Meiss-ner effect, the transition from the non-extremal case tothe extremal one is important. We depict it in Fig. 1 fora specific deformation of the Kerr black hole. We fix the

deviation by choosing φ(0)2 as the spin-weighted spherical

harmonic −1Y2,0 [45],

φ(0)2 = C −1Y2,0, (21)

with an arbitrary non-vanishing constant C, leaving theother quantities, including the mass, unchanged. Foreach value of κ(`), we solve Eq. (17b) numerically andcalculate the electromagnetic field in the neighborhood ofthe black hole using the NP field equations [38]. Finally,we plot the level sets of the magnetic and electric flux

density, i.e., the imaginary and real part of φ(0)1 rescaled

by C−1, respectively. We choose the contours of the con-stant rescaled dimensionless magnetic flux density to beequidistant, with the difference between two neighboringcontours being 10−2.

Fig. 1 and 2 clearly show that the lines of constantnon-vanishing flux density are penetrating the horizon fora/M < 1 (under-extremal case) and are expelled in thetransition a/M → 1 (extremal case). In Fig. 3 we plotthe magnetic field lines for different spins of the blackhole.

For the visualization, we transform the spherical coor-dinates r, θ to the Cartesian ones by the usual relationsx = r sin θ, y = r cos θ in all figures.

IV. JET CREATION EFFICIENCY

Although our analysis and Figs. 1–3 show that theMeissner effect holds for generic black holes in equilib-rium in general relativity, the impact on the jet creationefficiency has still to be assessed. As we explained inthe introduction, the Meissner effect does not operate inthe presence of matter and the Blandford–Znajek pro-cess requires an influx of accreting matter through theblack hole horizon, while we assumed the black hole to beisolated in our approach. Nevertheless, since the Meiss-ner effect plays a role only in the limit of maximal spin,it will be interesting to see how strongly it could affectthe jet creation efficiency. In order to do so, we assumehere that the accretion influx, while powering the jet viathe Blandford-Znajek process, is negligible for solving thefield equations. The physically more viable setting, force-free electrodynamics rather than electro-vacuum, wouldindeed probably increase the jet creation efficiency, see[30]. Hence, the idealized situation treated here, yields alower bound.

The efficiency of the Blandford–Znajek process is givenin geometrical units by [28, 29]

η =κ4πx2⟨

Φ2BH (M M2)−1/2

⟩(1 + 1.38x2 − 9.2x4),

(22)

where κ is a constant depending on the geometry of themagnetic field, x is a variable given in terms of the di-mensionless spin parameter a/M :

x =a/M

2(1 +√

1− (a/M)2). (23)

ΦBH is the flux of the magnetic field through a hemi-sphere of the horizon, M is the accretion rate, and 〈. . . 〉is a time average.

In order to investigate the behavior of the jet produc-tion efficiency independently of a particular model of ac-cretion, we vary the spin a keeping all other parametersfixed. The result is depicted in Fig. 4, where we chose thesame deformation as for Fig. 1. Other deformations givequalitatively the same result. From Fig. 4 any particularmodel can be recovered by a simple rescaling.

As Fig. 4 shows, the efficiency is increasing up toa/M ≈ 0.89. For higher spins, the efficiency drops. How-ever, it deviates from the maximum value only by about17% for a/M = 0.95 and by 50% for a/M = 0.98. Foreven higher spins, it decays rapidly to zero. Estimatesof the maximal expected spin of a black hole with an ac-cretion disk depend on the particular model chosen andranges from a/M ≈ 0.9 to a/M ≈ 0.95 for magneto-hydrodynamic simulations of thick disks [46, 47], whichwould still admit a high efficiency for the jet creation.For thin disks with low viscosity [48] and for models tak-ing only radiation into account [49] the limit can be ashigh as a/M ≈ 0.9994 and a/M ≈ 0.998, respectively.

Our result suggests that even if the Meissner effectwould not be suppressed by the presence of matter cross-ing the horizon, it quenches the jet creation significantlyonly for black holes with spins higher than a/M ≈ 0.98.

ACKNOWLEDGMENTS

N.G. and M.S. thank the Friends of ZARM for theirfinancial support and hospitality. M.S. was financiallysupported by the Albert Einstein Center, grant no. 14-37086G by GACR. N.G. gratefully acknowledges sup-port from the DFG within the Research Training Group1620 “Models of Gravity”. Partial support comes alsofrom NewCompStar, COST Action MP1304. The au-thors thank D. Giulini for helpful discussions. We arealso grateful to the referees for valuable suggestions im-proving the paper.

We dedicate this work to Jirı Bicak on the occasionof his 75th birthday. He pioneered the investigation ofthe Meissner effect during the last decades and we aregrateful to him for very enlightening discussions.

5

FIG. 1: Lines of equal magnetic flux density for a given value of a/M . Dashed lines represent vanishing fluxdensity.

a/M = 0.5 a/M = 0.9 a/M = 1.0

FIG. 2: Electric flux density. The lines of equal electric flux density around the black hole for a given value of thespin parameter a/M . Dashed thick lines represent the lines of zero flux density.

Appendix A: Killing equations

Let Ka be a Killing vector of a spacetime. We expandit as

Ka = K0 na +K1 `a −K2 ma − K2ma, (A1)

so that the spin weights of K0,K1,K2 and K2 are 0, 0, 1and −1, respectively. Then, the projections of the Killingequations

∇aKb +∇bKa = 0 (A2)

6

a/M = 0.8 a/M = 0.9 a/M = 1

FIG. 3: Field lines of the magnetic field Ba measured by an observer with the four-velocity ua = (`a + na)/√

2,i.e., Ba = ?Fabu

b, where ?Fab is the dual of the electromagnetic field tensor defined by φ0, φ1 and φ2 [50].

0.0 0.2 0.4 0.6 0.8 1.0

a/M

0.0

0.2

0.4

0.6

0.8

1.0

η/η

max

η = ηmaxa/M = 0.89

a/M = 0.98η = 0.5 ηmax

a/M = 0.95η = 0.83 ηmax

a/M = 0.998η = 0.07 ηmax

Jet creation efficiency

FIG. 4: Efficiency of the jet production η/ηmax depending on the spin a, which varies from 0 to a = M . Themaximal value ηmax is acquired for a/M ≈ 0.89.

onto the null tetrad read

DK0 = (ε+ ε)K0 − κK2 − κK2, (A3a)

DK1 + ∆K0 = (γ + γ)K0 − (ε+ ε)K1+

+ (π − τ)K2 + (π − τ)K2, (A3b)

DK2 + δK0 = π + α+ β)K0 − κK1+

+ (ε− ε− ρ)K2 − σ K2, (A3c)

∆K1 = −(γ + γ)K1 + ν K2 + ν K2, (A3d)

∆K2 + δK1 = ν K0 − (β + τ + α)K1+

+ (γ − γ + µ)K2 + λK2, (A3e)

ðK2 = λK0 − σK1, (A3f)

ðK2 + ðK2 = (µ+ µ)K0 − (ρ+ ρ)K1. (A3g)

In the paper we employ two Killing vectors. The sta-tionary Killing vector Ka reduces to `a on the horizon,i.e.

K0 = 0, K1 = 1, K2 = 0, on H, (A4)

and the Killing equation (A3d) together with Eq. (8)implies K1 = 1 everywhere in the neighborhood of thehorizon.

The axial Killing vector ηa satisfies the Killing equa-tions (A3) in whichK has to be replaced by η everywhere.On the horizon we have ηa = (∂ϕ)a and hence

η0 = η1 = 0 on H. (A5)

For the choice (3) we have

η2 = − iR sin θ√2

on H. (A6)

7

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