Geometry and Thermodynamics of Black Holes in Magnetic Fields ...

Geometry and Thermodynamics of Black Holesin Magnetic Fields7th Crete Regional Meeting in String Theory20th June 2013Chris PopeWe investigate the global structure of rotating black holes immersedin an external magnetic field; i.e. a “Magnetised Kerr-Newman Spacetime.” In general, there is an ergoregion extendingto infinity, but this is avoided in a special case. We discuss thethermodynamics, and also extensions to black holes in supergravity.Based on work with Gary Gibbons, Abid Mujtaba, and more recentwork with Yi Pang, Mirjam Cvetič and Zain Saleem.

Astrophysical Black Hole in an External Magnetic Field• Black holes are known to occur at the centre of most galaxies.These will in general have large angular momentum, owingto the accretion processes involving infalling matter.• Blandford and Znajek proposed a mechanism in which themagnetic field generated by the circulation of charged particlesin an equatorial accretion disc leads to electrostatic fieldsthat accelerate charged particles near the horizon, generatingradiation that itself gives rise to electron-positron pair production.This in turn can lead to a net extraction of energyfrom the black hole via a Penrose-type process, thus possiblyproviding a powerful energy source at the galactic nucleus.• Wald considered a neutral rotating black hole (Kerr solution)in the background of an external magnetic field. As withBlandford and Znajek, the magnetic field was treated as a“test field”; i.e. the back-reaction of the magnetic field onthe geometry was neglected.• Wald argued that if the black hole has mass m and angularmomentum j = am, then in the presence of an external magneticfield B it becomes energetically favourable for the blackhole to acquire an electric charge Q, given byQ = 2jB .This could occur via a vacuum breakdown through pair production.

Exact Solution with Back Reaction?• The notion of a black hole immersed in an “originally uniformmagnetic field” is a little problematic in general relativity,since the total electromagnetic energy will be infinite, andspacetime will necessarily become highly non-Minkowskian atinfinity.• An exact solution that is perhaps the closest to this is provided,in the absence of rotation, by the Schwarzschild-Melvinmetric, withds 2 = H [−fdt 2 + f −1 dr 2 + r 2 dθ 2 ] + r2 sin 2 θHf = 1 − 2m r , H = (1 + 1 4 B2 r 2 sin 2 θ) 2 ,A = Br2 sin 2 θ√Hdφ .dφ 2 ,• This is asymptotic to the Melvin universe at large r. (i.e.when r >> m). It approaches the Minkowski metric nearthe axis (θ = 0 or θ = π), but is highly non-Minkowskianaway from the axis. This is most clearly seen in cylindricalcoordinatesρ = r sin θ , z = r cos θ .

The Melvin Universe• In cylindrical coordinates, Melvin solution (m = 0) isds 2 = H (−dt 2 + dz 2 + dρ 2 ) + H −1 ρ 2 dφ 2 ,F = dA = B H −1 ρdρ ∧ dφ , H = (1 + 1 4 B2 ρ 2 ) 2 .• If B 2 ρ 2

Rotating Generalisations?• Rotating generalisations have been obtained before. The solutionsare too complicated to work with reliably by hand,so it is useful first to reconstruct from scratch, using algebraiccomputing. We used a solution-generating techniqueto “magnetise” Kerr-Newman, via a Kaluza-Klein reductionto three dimensions. The starting point is Einstein-Maxwelltheory in four dimensions:L 4 = R − F 2 .We then assume a metric with a spacelike Killing vector ∂/∂φand reduce to three dimensions:ds 2 4 = e2ϕ d¯s 2 3 + e−2ϕ (dφ + 2Ā) 2 ,A = Ā + χ (dφ + 2Ā) .This gives the three-dimensional Lagrangian¯L 3 = ¯R − 2(∂ϕ) 2 − 2e ϕ (∂χ) 2 − e −4ϕ ¯F 2 − e −2ϕ ¯F 2 ,where ¯F = dĀ and ¯F = dĀ + 2χ dĀ.• Dualising Ā and Ā; e −2ϕ ¯∗ ¯F = dψ and e −4ϕ ¯∗ ¯F = dσ − 2χ dψgives 3D gravity coupled to a nonlinear sigma model:L 3 = ¯R − 2(∂ϕ) 2 − 2e ϕ [(∂χ) 2 + (∂ψ) 2 ] − 2e 2ϕ (∂σ − 2χ∂ψ) 2 .

SU(2, 1) Transformations, and Magnetisation• The scalar sigma model metricdΣ 2 = dϕ 2 + e 2ϕ [dχ 2 + dψ 2 ] + e 4ϕ (dσ − 2χdψ) 2˜is CP 2 = SU(2, 1)/U(2), the non-compact (negative curvature)version of CP 2 . Defining matrices E b a with just a 1 atrow a, column b, and H = E 0 0 − E 2 2 , we can parameterise acoset representative, and SU(2, 1) element, asV = e ϕH e −2i σE 0 2 e√2χ(E0 1 +E 2) 1 e −i √ 2ψ(E 1 0 −E 2) 1 .We haveV † ηV = η , η =⎛⎜⎝0 0 −10 1 0−1 0 0⎞⎟⎠ ,with η being the invariant metric of SU(2, 1). The Lagrangiancan be written as L 3 = R − tr(M −1 ∂M) 2 where M = V † V.L 3 is manifestly invariant under M −→ M ′ = U † MU where Uis any constant SU(2, 1) matrix, obeying U † ηU = η.

Magnetisation of Kerr-Newman Black Hole• We start from the rotating, charged Kerr-Newman black hole:ds 2 4 = ( −fdt2 + R 2 dr2 )∆ + dθ2A = Φ 0 dt + Φ 3 (dφ − ωdt) ,where+ Σ sin2 θR 2 (dφ − ωdt) 2 ,R 2 = r 2 + a 2 cos 2 θ , ∆ = (r 2 + a 2 ) − 2mr + q 2 ,ω = a(2mr − q2 ), Σ = (r 2 + a 2 ) 2 − a 2 ∆ sin 2 θ ,Σf = R2 ∆Σ , Φ 0 = qr(r2 + a 2 ), Φ 3 = − aqr sin2 θΣR 2 .• We now reduce to three dimensions on ∂/∂φ, apply the magnetisingtransformation, which turns out to beU =⎛⎜⎝1 0 0B√21 0B 24⎞⎟⎠ ,B√ 1 2and then retrace the steps back to four dimensions.

Further SU(2, 1) Transformations• Amongst the other possible transformations within the SU(2, 1)symmetry group are electrifications (adding an external electricfield):U =⎛(B+i E)⎜⎝(B 2 +E 2 )41 0 0⎟⎠ .(B−i√E)1 2√21 0• Electric/magnetic duality transformations, which in four dimensionstake the formF −→ F ′ = F cos α + ∗F sin α ,This is implemented in 3 dimensions by the SU(2, 1) matrix⎛e − i ⎞3 α 0 0U =⎜ 0 e 2i3 α 0.⎝0 0 e − 3 i ⎟α ⎠⎞

Magnetisation of Kerr-Newman Black Hole• Applied to Schwarzschild, it indeed gives Schwarzschild-Melvin.• Applied to the Kerr-Newman metric (with electric and magneticcharges q and p, it gives a rather indigestible result:wherewithds 2 4 =[ ( drH − fdt 2 + R 2 2 )]∆ + dθ2A = Φ 0 dt + Φ 3 (dφ − ωdt) ,H = 1 + H (1)B + H (2) B 2 + H (3) B 3 + H (4) B 4R 2H (1) = 2aqr sin 2 θ − 2p(r 2 + a 2 ) cos θ ,H (2) = 1 2 [(r2 + a 2 ) 2 − a 2 ∆ sin 2 θ] sin 2 θ + 2˜q2 3 (a 2 + r 2 cos 2 θ) ,+ Σ sin2 θH R 2 (dφ − ωdt) 2 ,H (3) = −pa 2 ∆ sin 2 θ cos θ − qa∆r [r2 (3 − cos 2 θ) cos 2 θ + a 2 (1 + cos 2 θ)] + aq(r2 + a 2 ) 2 (1 + cos 2 θ)2r− 1 2 p(r4 − a 4 ) sin 2 θ cos θ + q˜q2 a[(2r 2 + a 2 ) cos 2 θ + a 2 ]− 12r2 p˜q2 (r 2 + a 2 ) cos 3 θ ,H (4) = 1 16 (r2 + a 2 ) 2 R 2 sin 4 θ + 1 4 ma2 r(r 2 + a 2 ) sin 6 θ + 1 4 ma2˜q 2 r(cos 2 θ − 5) sin 2 θ cos 2 θ+ 1 4 m2 a 2 [r 2 (cos 2 θ − 3) 2 cos 2 θ + a 2 (1 + cos 2 θ) 2 ]+ 1 8˜q2 (r 2 + a 2 )(r 2 + a 2 + a 2 cos 2 θ) sin 2 θ cos 2 θ + 116˜q4 [r 2 cos 2 θ + a 2 (1 + sin 2 θ) 2 ] cos 2 θand we have defined ˜q 2 ≡ q 2 + p 2 .The function ω is given by

whereω = (2mr − ˜q2 )a + ω (1) B + ω (2) B 2 + ω (3) B 3 + ω (4) B 4,Σω (1) = −2qr(r 2 + a 2 ) + 2ap∆ cos θ ,ω (2) = − 3 2 a˜q2 (r 2 + a 2 + ∆ cos 2 θ) ,ω (3) = 4qm 2 a 2 r + 1 2 ap˜q4 cos 3 θ − 1 2 qr(r2 + a 2 )[r 2 − a 2 + (r 2 + 3a 2 ) cos 2 θ]+ 1 2 ap(r2 + a 2 )[3r 2 + a 2 − (r 2 − a 2 ) cos 2 θ] cos θ + 1 2 q˜q2 r[(r 2 + 3a 2 ) cos 2 θ − 2a 2 ]+ 1 2 ap˜q2 [3r 2 + a 2 + 2a 2 cos 2 θ] cos θ − am˜q 2 (2aq + pr cos 3 θ)+qm[r 4 − a 4 + r 2 (r 2 + 3a 2 ) sin 2 θ] − apmr[2R 2 + (r 2 + a 2 ) sin 2 θ] ,ω (4) = 1 2 a3 m 3 r(3 + cos 4 θ) − 1 16 a˜q6 cos 4 θ − 1 8 a˜q4 [r 2 (2 + sin 2 θ) cos 2 θ + a 2 (1 + cos 2 θ)]+ 116 a˜q2 (r 2 + a 2 )[r 2 (1 − 6 cos 2 θ + 3 cos 4 θ) − a 2 (1 + cos 4 θ)] − 1 4 a3 m 2˜q 2 (3 + cos 4 θ)+ 1 4 am2 [r 4 (3 − 6 cos 2 θ + cos 4 θ) + 2a 2 r 2 (3 sin 2 θ − 2 cos 4 θ) − a 4 (1 + cos 4 θ)]+ 1 8 am˜q4 r cos 4 θ + 1 4 am˜q2 r[2r 2 (3 − cos 2 θ) cos 2 θ − a 2 (1 − 3 cos 2 θ − 2 cos 4 θ)]+ 1 8 amr(r2 + a 2 )[r 2 (3 + 6 cos 2 θ − cos 4 θ) − a 2 (1 − 6 cos 2 θ − 3 cos 4 θ)] .• This solution is often assumed to be asymptotic to the Melvinuniverse. In general, it isn’t. Look at g tt :• At large r, for generic θ, the metric component g tt does behavejust like in Melvin, withg tt −→ − 116 B4 r 4 sin 4 θ .But, near to the z axis at large distance, g tt goes (arbitrarily)positive, signaling an ergoregion extending out to infinity.

• This is evident in the cylindrical coordinates ρ = r sin θ andz = r cos θ where, for fixed ρ, a large z expansion givesg tt −→ 16B6 (q + amB) 2 ρ 2W (ρ) 2 z 2− 4B6 (q + amB)[8qm + aB(q 2 + 4m 2 )]ρ 2W (ρ) 2 z + O(z 0 ) ,So not asymptotically Melvin near the axis, and an infiniteergoregion.• The energy of a particle with 4-momentum p µ , measured withrespect to a future-directed timelike Killing vector K, isE = −K µ p µHere, we are thinking of K = ∂/∂t. If K becomes spacelike,i.e. if there is an ergoregion, then the energy can becomenegative. Can then extract energy by the Penrose Process,in which a particle decays into two, one with negative energythat falls into the black hole, and the other, with more energythan the original particle, that escapes.• In the original Kerr-Newman black hole there is a uniquechoice of asymptotically-timelike Killing vector (i.e. K =∂/∂t). In the magnetised solution we find that providedwe take q = −amB, there is a family of possible choicesK = ∂/∂t + Ω ∂/∂φ, if the angular velocity Ω lies in an appropriaterange.

q = −amB Magnetised Kerr-Newman• With q = −amB, the problem of an ergoregion at infinity canbe avoided. Using the coordinate ˜φ = φ − Ω t, first look atthe off-diagonal metric component g t˜φat large z:g t˜φ = 2(8Ω + 12am2 B 4 + a 3 m 2 B 6 )ρ 2(4 + a 2 m 2 B 4 + B 2 ρ 2 ) 2 + O( 1 z ) .Choosing Ω = Ω s , wherewe find at large z thatΩ s = − 1 8 am2 B 4 (12 + a 2 B 2 ) ,g t˜φ = −8amB2 (4 + a 2 m 2 B 4 )ρ 2(4 + a 2 m 2 B 4 + B 2 ρ 2 ) 2 1z + O( 1 z 2) ,g tt = − 116 (4 + a2 m 2 B 4 + B 2 ρ 2 ) 2 + O( 1 z ) ,and so the metric near the axis is then genuinely asymptoticto the static Melvin universe.

Asymptotically-Melvin Magnetised Kerr-Newman• In the asymptotically static Melvin frame with timelike Killingvector K = ∂/∂t + Ω s ∂/∂φ, the q = −amB magnetised Kerr-Newman-Melvin solution will still have an ergoregion in theform of an oblate spheroid outside the outer horizon (verylike in the Kerr solution):• In the Kerr solution, ∂/∂t is the unique Killing vector thatis timelike at infinity, and hence it is the unique choice asgenerator of time translations:• In the magnetised Kerr-Newman solution there is in fact arange of angular velocities Ω around Ω = Ω s for which theergoregion is of only finite extent, and confined to the vicinityof the black hole.

Conserved Charge and Angular Momentum• These can be calculated most simply in the KK reduced 3Dlanguage. The physical charge is given by the Gaussian integral∫∫Q = 1∆φ∗F =4π S2 4π S 2 e−4ϕ ¯∗ ¯F ,= ∆φ ∫dψ = ∆φ [ ] θ=πψ = q(1 − 14π 4π4 q2 B 2 ) + 2amB .θ=0∆φ is the period of the azimuthal coordinate φ, determinedby requiring no conical singularities at N and S poles of S 2 :]∆φ = 2π[1 + 3 2 q2 B 2 + 2aqmB 3 + (a 2 m 2 +16 1 q4 )B 4 .• For angular momentum, following Wald, for every Killing vectorξ the Noether method gives a conserved chargeQ[ξ] = 116π∫S 2 ∗P , P = dξ + 4(ξµ A µ ) F .Taking the azimuthal Killing vector with ξ = ∂/∂ ˜φ, where˜φ = (2π/∆φ) φ has canonical 2π period, then as a 1-form,and in the 3D language, we have

ξ = ∆φ2π e−2ϕ (dφ + 2Ā) and henceP = ∆φ [e −2ϕ ¯F + 2χ ¯F − (e −2ϕ dϕ − 2χdχ) ∧ (dφ + 2Ā)πThis gives∗P = ∆φ []dσ ∧ (dφ + 2Ā) − ¯∗(dϕ − 2e 2ϕ χdχ) ,πand so the angular momentum isJ = 116π∫S 2 ∗P = ( ∆φ4π) 2 ∫dσ =( ∆φ4π) 2 [ ] θ=πσθ=0= am + 1 2 q3 B + 9 2 amq2 B 2 + 1 2 qB3 (12a 2 m 2 − q 4 )+ 1 16 amB4 (48a 2 m 2 − 11q 4 ) .(Note: This includes the electromagnetic field contributionas well as the purely gravitational “Komar” contribution from∗dξ. Each by itself is singular in the Melvin background, butthe sum is non-singular.)].

Thermodynamics• We can expect that the first law should take the formdE = T dS + ΩdJ + ΦdQ − µdB ,We have calculated the conserved charges Q and J, and thecalculation of the entropy S and temperature T is straightforward.Unusual asymptotics of the Melvin background meanthat calculating the mass E, the angular velocity Ω the electrostaticpotential Φ, and the magnetic moment µ is trickier.• The mass is given by (??) E = (∆φ/2π) m. We can seeksolution of the first law for Ω, Φ and µ. This works, and isnon-trivial since we have 4 equations for 3 unknowns.• Solution is complicated. At leading order, we haveµ = aq(1 + a 2 m 2 B 4 ) + O(q 2 ) ,Ω =ar+ 2 + − 2qBr +a2 r+ 2 + + a2 O(B2 ) ,Φ = qr +r 2 + + a2 −3aq2 B2(r 2 + + a2 ) + O(B2 ) .Magnetic moment µ ∼ aq ∼ JQ/M, so at leading order werecover the well-known black hole gyromagnetic ratio g ∼ 2.• Satisfies Smarr-type relation E = 2T S + 2ΩJ + ΦQ + µB.

Magnetised Black Holes in STU Supergravity• We can magnetise black holes in supergravities too. Currentlywe (Cvetič, Gibbons, Saleem, CNP) are looking in the fourdimensionalSTU model (N = 2 supergravity coupled to 3vector multiplets). Now have four independent U(1) gaugefields, so four charges and four magnetic fields. We no use theO(4, 4) symmetry of the associated KK reduced 3-dimensionalsigma model to generate the magnetised solutions.• As well as magnetising electric black holes, it is also nowof interest to magnetise magnetically-charged black holes.Consider static black holes for simplicity. The metric isds 2 = H ds 2 3 + H−1 sin 2 θ (dφ − ωdt) 2 ,(ds 2 3 = −r(2 − 2m)dt2 + R 1 r 2 r 3 r 4 dθ 2 + R )1r 2 r 3 r 4r(r − 2m) dr2ω =4∑i=1[− p iB i+ p iB 1 B 2 B 3 B 4 [r i + (r − 2m) cos 2 ]θ]r,2r i 8B i r iH 2 1 ∏ 4 (=(1 + 1r 1 r 2 r 3 r2 B ip i cos θ) 2 + B2 i r 1r 2 r 3 r 44 i=1ri2r i = r + 2m sinh 2 δ i , p i = 2m sinh δ i cosh δ i .,]sin 2 θ ,

Conical Singularities and Force Balancing• To avoid conical singularity at the North (South) pole, φshould be identified with different periods at each:N pole :S pole :∆φ = 2π ∏ i∆φ = 2π ∏ i(1 + 1 2 B ip i ) ,(1 − 1 2 B ip i ) .In general, there is no single choice of period ∆φ that eliminatesboth conical singularities. This reflects the fact thatthere is a net force on the black hole, and it must thereforebe supported by a “strut,” which is described by an energymomentumtensor with a delta-function singularity on theaxis. This happens, for example, for a magnetic Reissner-Nordström black hole in an external magnetic field. (Or theS-dual, an electric black hole in an external electric field.)• With four separate gauge fields in the STU model we cantune the fields and the charges to achieve a force balance:∏(1 + 1 2 B ip i ) = ∏ (1 − 1 2 B ip i ) .iiThis provides families of non-singular magnetic black holesin external magnetic fields. These provide interesting modelsfor investigating thermodynamics of systems in externalbackgrounds.

Conclusions and Open Problems• A rotating generalisation of the asymptotically-Melvin magnetisedSchwarzschild solution does exist, but it must carry avery specific electric charge, related to its mass and angularmomentum and the external B field. Otherwise, it has anergoregion extending to infinity close to the axis.• The thermodynamics is quite subtle, owing to the unusualasymptotic behaviour of the Melvin type background. Chargeand angular momentum can be calculated from conservedcurrents, but ab initio computation of angular velocity, electricpotential and magnetic moment is more difficult. Framedraggingof the magnetic field is rather complex.• More extensive analysis of the global structure is needed.Also, a detailed investigation of “asymptotically Melvin” boundaryconditions for fields.• Melvin/CFT type dualities?• Magnetisation of black holes in supergravities opens up newpossibilities; non-singular magnetised magnetic black holes,etc.• Recent work (Cvetič, Guica, Saleem) has shown how relatedsolution-generating transformations can be used to interpolatebetween black hole backgrounds and “subtracted geometries”that can be used for studying the microscopic entropy.We are investigating these further.