Test of the weak cosmic censorship conjecture with a charged scalar field and dyonic Kerr-Newman black holes
aa r X i v : . [ g r- q c ] J u l Test of the weak cosmic censorship conjecture witha charged scalar field and dyonic Kerr-Newmanblack holes
G´abor Zsolt T´oth ∗ Institute for Particle and Nuclear PhysicsWigner Research Centre for Physics, Hungarian Academy of SciencesKonkoly Thege Mikl´os ´ut 29-33H-1121 Budapest, Hungary
Abstract
A thought experiment considered recently in the literature, in which it isinvestigated whether a dyonic Kerr-Newman black hole can be destroyed byovercharging or overspinning it past extremality by a massive complex scalartest field, is revisited. Another derivation of the result that this is not possible,i.e. the weak cosmic censorship is not violated in this thought experiment, isgiven. The derivation is based on conservation laws, on a null energy condition,and on specific properties of the metric and the electromagnetic field of dyonicKerr-Newman black holes. The metric is kept fixed, whereas the dynamicsof the electromagnetic field is taken into account. A detailed knowledge of thesolutions of the equations of motion is not needed. The approximation in whichthe electromagnetic field is fixed is also considered, and a derivation for thiscase is also given. In addition, an older version of the thought experiment, inwhich a pointlike test particle is used, is revisited. The same result, namely thenon-violation of the cosmic censorship, is rederived in a way which is simplerthan in earlier works. ∗ email: [email protected] Introduction
If a spacetime contains a singularity not hidden behind an event horizon (a “naked”singularity), then far away observers can receive signals coming from this singularity.However, initial conditions cannot be specified at a singularity, therefore a singularitythat is not behind an event horizon prevents predictability in the spacetime thatcontains it. For this reason, it is conjectured that naked singularities cannot beproduced in a physical process from regular initial conditions, if the matter involvedin the process has reasonable properties. This conjecture, first stated by Penrose [1], isknown as the weak cosmic censorship conjecture (WCCC) (for a textbook exposition,see e.g. section 12.1 of [2]), and it is one of the major unsolved problems of classicalgeneral relativity to decide whether it is correct.In the absence of a general proof, the validity of the WCCC has been checked inseveral special cases by studying the evolution of initially regular physical systems. Apossible test is to throw a small particle at a Kerr-Newman black hole and to see if anoverextremal Kerr-Newman spacetime, which contains a naked singularity, can ariseafter the particle has been absorbed by the black hole. Wald [3] was the first whoconsidered this thought experiment, and he showed that an extremal Kerr-Newmanblack hole cannot be overcharged or overspin by throwing a pointlike test particlewith electric charge into it. A simpler derivation of this result was given by Needham[4]. Hiscock [5] and Semiz [6] extended Wald’s result to the case of dyonic Kerr-Newman black holes, which are rotating black holes with both electric and magneticcharge. The derivations presented in [5, 6] are generalizations of the derivation in[3]. Recently Semiz [7] also studied the case when a complex scalar test field is usedinstead of a test particle, and found that the WCCC is not violated. Other resultssupporting the WCCC were obtained by several authors, and there are also claimsthat the WCCC can be violated, for example by starting from a slightly sub-extremalblack hole and “jumping over” the extremal case [17]-[37]. The signatures of nakedsingularities for the observational verification of their existence was also investigated,e.g. in [38, 39]. Reviews on the status of the cosmic censorship conjecture can befound in [40]-[44].A dyonic Kerr-Newman black hole can be characterized by four parameters, whichare the mass M , the angular momentum per unit mass a , the electric charge Q e andthe magnetic charge Q m . The angular momentum of the black hole is J = aM , and Q m = 0 corresponds to a usual Kerr-Newman black hole. The metric of the dyonicKerr-Newman black hole spacetime with parameters ( M, a, Q e , Q m ) is the same asthe Kerr-Newman metric with parameters ( M, a, e ), e = Q e + Q m , where e denotesthe electric charge parameter of the Kerr-Newman metric. The parameters have tosatisfy the inequality η = M − Q e − Q m − a ≥ , (1)2therwise the spacetime contains a naked singularity. The black hole is called extremalif η = 0. Under certain conditions, the dyonic Kerr-Newman black holes are the onlystatic and asymptotically flat black hole solutions of the Einstein-Maxwell equations[9, 10].Under a small change ( dM, dJ, dQ e , dQ m ) of the parameters of the black hole, thechange of η is dη = 2 M + a M (cid:18) dM − aM + a dJ − Q e MM + a dQ e − Q m MM + a dQ m (cid:19) . (2)In the thought experiments discussed in [3, 4, 5, 6, 7] it is assumed that initiallyone has an extremal dyonic Kerr-Newman black hole, which then absorbs a smallamount of matter, and finally settles down in another dyonic Kerr-Newman statewith slightly different parameters. If one calculates the change ( dM, dJ, dQ e , dQ m ) ofthe parameters in this process, one should find dη ≥
0; a result dη < dM, dJ, dQ e , dQ m ) ofthe parameters were calculated in the approximation that the metric is fixed duringthe process and, as mentioned above, the result dη ≥ dM, dJ, dQ e , dQ m ). In [7], however, where the case of the testfield is considered, the electromagnetic field is also taken to be dynamical, althoughwith the restriction that free electromagnetic radiation that is not tied to the electriccurrent is not present.In this paper we revisit the thought experiment studied by Semiz [7], in whichit is investigated whether a dyonic Kerr-Newman black hole can be destroyed byovercharging or overspinning it past extremality by a massive complex scalar test field.We give a different and simpler derivation of the result that dη ≥ The metric of the dyonic Kerr-Newman black hole spacetime with parameters(
M, a, Q e , Q m ) can be given in a standard form as ds = g µν dx µ dx ν = − (cid:18) ∆ − a sin θ Σ (cid:19) dt − a sin θ ( r + a − ∆)Σ dtdφ + Σ∆ dr + Σ dθ + (cid:18) ( r + a ) − ∆ a sin θ Σ (cid:19) sin θ dφ , (3)where Σ = r + a cos θ (4)∆ = r + a + Q e + Q m − M r. (5)The signature of this metric is ( − + ++).The electromagnetic field of a dyonic Kerr-Newman black hole has the vectorpotential A = Q e A e + Q m A m , (6)4here A e = − r Σ dt + ar sin θ Σ dφ (7)and A m = a cos θ Σ dt + (cid:20) ˜ C − cos θ r + a Σ (cid:21) dφ. (8)The electromagnetic field derived from A m is dual to the electromagnetic field derivedfrom A e . The electromagnetic field does not depend on the constant ˜ C , which canbe used, by setting ˜ C = 1 or ˜ C = −
1, to eliminate the Dirac string singularity of A m along the positive or negative z axis ( θ = 0 and θ = π ), respectively. We set ˜ C tozero for a reason that is explained below.In the following sections various quantities will be considered at the future eventhorizon. Since the Boyer-Lindquist coordinates ( t, r, θ, φ ) do not cover the futureevent horizon, Eddington–Finkelstein-type ingoing horizon-penetrating coordinates,denoted by ( τ, r, θ, ϕ ), will be used. These coordinates can be introduced by thetransformation τ = t − r + Z dr r + a ∆ (9) ϕ = φ + Z dr a ∆ . (10)The future event horizon is located in these coordinates at the constant value r + = M + p M − ( a + Q e + Q m ) (11)of r , and the metric is non-singular in these points. In the extremal case r + = M. (12)The ( τ + r, θ, ϕ ) = constant lines are ingoing null geodesics, and there exists an r < r + such that the τ = constant hypersurfaces are spacelike in the domain r < r .The r component ( A e ) r of A e with respect to the coordinates ( τ, r, θ, ϕ ) is singularat the event horizon, but this singularity can be eliminated by the gauge transforma-tion A e → A e − r ∆ dr . After this gauge transformation A e = − r Σ dτ + ar sin θ Σ dϕ − r Σ dr. (13)The r component of A m with respect to the coordinates ( τ, r, θ, ϕ ) is also singularif ˜ C = 0, therefore we set ˜ C = 0. Nevertheless, in order to treat the Dirac string5ingularity of A m , we introduce an explicit gauge parameter into it by adding Cdϕ ,where C is a real constant. Thus A m = a cos θ Σ dτ + (cid:20) C − cos θ r + a Σ (cid:21) dϕ + a cos θ Σ dr. (14) A m has a string singularity along the z axis (which corresponds to θ = 0 and θ = π )because dϕ is singular here, and its coefficient ( A m ) ϕ does not cancel this singularity.However, in the special cases C = 1 and C = − z axis ( θ = 0) or along the negative z axis ( θ = π ), respectively. Thestring singularity can therefore be avoided by using two domains that cover the wholespacetime region of interest in such a way that one of the domains contains the entirepositive z axis but is well separated from the negative z axis and the other onecontains the entire negative z axis but is separated from the positive z axis. In thefirst domain the C = 1 gauge is used then, and in the second domain the C = − r < r , 0 ≤ θ ≤ π/ r < r , π/ < θ ≤ π ,for example. These domains will be denoted by D + and D − . The transition betweenthe two domains involves a gauge transformation, which has to be kept in mind inparticular calculations. This approach to treating the string singularity of A m wasproposed in [8] and was also taken in [6, 7].In the rest of the paper we use only the coordinates ( τ, r, θ, ϕ ), and we also usethe notation ω for the one-form dr (the exterior derivative of the coordinate function r ). A e , A m and A will denote (13), (14) and A = Q e A e + Q m A m , respectively. ∂/∂τ and ∂/∂ϕ are Killing fields; ∂/∂τ is the generator of time translations and ∂/∂ϕ is the generator of rotations around the axis of the black hole. A e and A m arealso invariant under these symmetries. The Killing field χ = ∂∂τ + Ω H ∂∂ϕ (15)is null at the event horizon, with Ω H = ar + a , (16)which is called the angular velocity of the event horizon. At the event horizon wealso have ( A e ) µ χ µ = − r + r + a , ( A m ) µ χ µ = C Ω H , (17)and ω µ is parallel to χ µ . The relation between ω µ and χ µ at the event horizon is ω µ = r + a r + a cos θ χ µ , (18)6hus ω µ is also future directed. We introduce the quantity Φ H asΦ H = r + Q e r + a . (19)In the case of Kerr-Newman black holes, Φ H is known as the electrostatic potentialof the horizon.Both D + and D − are contractible domains, therefore any gauge transformationtakes the form A → A + d Φ on D + or on D − , where Φ denotes a real valued function. If d Φ is invariant under the time translation and rotation symmetries of the spacetime,then Φ takes the form Φ ( r, θ ) + c τ + c ϕ , where c and c are constants, thusthe corresponding gauge transformation changes the τ and ϕ components of A onlyby adding the constants c and c . This shows that if A (or A e , A m ), understoodhere to be fixed up to gauge transformations, is required to be invariant under timetranslation and rotation, then the τ and ϕ components of A are determined uniquelyup to additive constants. These constants can also be fixed by requiring that A τ should tend to 0 as r → ∞ and A should not have a Dirac string singularity. If the τ and ϕ components of A are fixed, then the quantity A µ χ µ is also fixed, because itdepends only on these components of A . In this section we consider the thought experiment in which a pointlike test particleis thrown at a dyonic Kerr-Newman black hole. The extremality of the black holeis not assumed; we derive an inequality that holds for any values of the black holeparameters, and that becomes the desired inequality dη > m , electric charge q and zero mag-netic charge is L = 12 mg µν dx µ ds dx ν ds + qA µ dx µ ds , (20)and its conserved energy and angular momentum are E = − mg τµ dx µ ds − qA τ (21) L = + mg ϕµ dx µ ds + q ( A ϕ − Q m C ) . (22)7he − qQ m C term on the right hand side is added to cancel the dependence of A ϕ onthe gauge parameter C . This is important because C has different values in the twodomains D + and D − .By multiplying (22) by Ω H and then subtracting it from (21), and taking intoaccount (15), (16), (17) and (19), it follows immediately that if the particle doescross the event horizon, then − m dx µ ds χ µ = E − Ω H L − Φ H q (23)holds at the point where the crossing takes place. At this point dx µ ds χ µ < dx µ ds is a timelike future directed vector in the case of a massiveparticle, and χ µ is a future directed null vector at the event horizon. Thus, E − Ω H L − Φ H q > . (24)We note that dx µ ds χ µ is just the r component of dx µ ds multiplied by a positive number,as can be seen from (18), and the r component of dx µ ds is clearly negative or zero fora particle moving inward into the black hole.The change of the black hole parameters dM , dJ and dQ e in (2) can be identifiedwith E , L and q , respectively, and dQ m = 0. The inequality (24) together with therelations dM = E , dJ = L , dQ e = q and dQ m = 0 imply dM − Ω H dJ − Φ H dQ e > . (25)In the extremal case Ω H = aM + a and Φ H = MQ e M + a , thus in this case (2) can bewritten as dM − Ω H dJ − Φ H dQ e = M M + a ) dη , therefore in the extremal case (25)gives dη >
0, which indicates that the black hole is not destroyed by the absorptionof the test particle, i.e. no violation of the WCCC occurs.
In this section we consider a similar thought experiment as in section 3, but with testfields instead of a pointlike test particle. We discuss two different settings, in sections4.1 and 4.2, respectively.In section 4.1 we consider the process in which a small amount of electricallycharged matter, described by a complex scalar field ψ , falls into a dyonic Kerr-Newman black hole. As in section 3, we make the approximation in the calculationof dM , dJ and dQ e that the metric and the electromagnetic field do not change, i.e.we take the scalar test field to evolve in the fixed gravitational and electromagneticbackground fields of the black hole. 8n section 4.2 a similar process is considered, with the difference that only thegravitational field is kept fixed, which means that the test matter also has an elec-tromagnetic field component and the effect of the scalar field on the electromagneticfield is taken into account. This is the setting that was considered in [7], althoughin [7] the difference of the total electromagnetic field and the electromagnetic field ofthe black hole is tied to the electric current (see [7] for details). We do not imposesuch a restriction on the electromagnetic field.Our reason for considering also the first case, in which the electromagnetic fieldis fixed, is that this is the one that is obtained from the thought experiment de-scribed in section 3 if the pointlike test particle is replaced by a scalar test field in astraightforward manner, and it also has technical differences from the second case.One of the technical differences between the two settings is that the Einstein-Hilbert energy-momentum tensor is conserved and can be used to obtain the conservedenergy and angular momentum currents only in the second setting. For this reason inthe first setting we use Noether’s theorem to find the conserved currents. Noether’stheorem can be used in the second setting as well; the currents obtained in this waydiffer only in divergence terms from the currents obtained from the Einstein-Hilbertenergy-momentum tensor, and these terms do not give any contribution to dM and dJ .In the same way as in section 3, the extremality of the black hole is generally notassumed in sections 4.1 and 4.2. The action of the scalar field in fixed dyonic Kerr-Newman gravitational and electro-magnetic fields is S = R √− g L dτ drdθdϕ , with the Lagrangian density L = − g µν ( ∂ µ − ieA µ ) ψ ∗ ( ∂ ν + ieA ν ) ψ − m ψ ∗ ψ, (26)where A µ is the vector potential of the electromagnetic field of the black hole as givenin section 2.The energy and angular momentum Noether currents corresponding to the sym-metries generated by ∂/∂τ and ∂/∂ϕ are √− g E µ = √− g T µν ( ∂/∂τ ) ν = √− g T µτ (27)and √− g J µ = √− g T µν ( ∂/∂ϕ ) ν = √− g T µϕ , (28)where T µν is T µν = − ∂ L ∂ µ ψ ∂ ν ψ − ∂ L ∂ µ ψ ∗ ∂ ν ψ ∗ + δ µν L = ( ∂ µ − ieA µ ) ψ ∗ ∂ ν ψ + ( ∂ µ + ieA µ ) ψ∂ ν ψ ∗ + δ µν L (29)9see Appendix A for more details on Noether’s theorem). The conservation laws forthese currents are ∂ µ ( √− g E µ ) = 0 and ∂ µ ( √− g J µ ) = 0. T µν is introduced only fornotational convenience.The Noether current corresponding to the ψ → e iα ψ , α ∈ R global U (1) symmetryis √− gj µ , where j µ = ie [ ψ ∗ ( ∂ µ + ieA µ ) ψ − ψ ( ∂ µ − ieA µ ) ψ ∗ ] (30)is the electric current. j µ also satisfies the equality j µ = ∂ L ∂A µ . T µν , E µ , J µ and j µ are quantities that transform as proper tensor and vectorfields, respectively, under coordinate transformations. The conservation laws can bewritten, of course, as ∇ µ E µ = 0, ∇ µ J µ = 0 and ∇ µ j µ = 0.Defining ˆ T µν asˆ T µν = ( ∂ µ − ieA µ ) ψ ∗ ( ∂ ν + ieA ν ) ψ + ( ∂ µ + ieA µ ) ψ ( ∂ ν − ieA ν ) ψ ∗ + g µν L , (31)we have T µν = ˆ T µν + A ν j µ , (32)and E µ = ˆ T µ τ + A τ j µ , J µ = ˆ T µ ϕ + A ϕ j µ . ˆ T µν and j µ are gauge invariant and A τ does not depend on the gauge parameter C , therefore E µ is also independent of C . A ϕ does depend on C , however, thus J µ also depends on it. For this reason we takethe modified definition J µ = ˆ T µ ϕ + ( A ϕ − Q m C ) j µ (33)for J µ , which eliminates its dependence on C . The conservation of J µ is not affectedby this modification, because j µ is conserved. The independence of E µ and J µ of C is important because the value of C is different in the domains D + and D − .The charge flux through the event horizon into the black hole is dQdτ = − Z H √− g j r dθdϕ , (34)where H denotes the two-dimensional surface of the black hole (which is the relevanttime slice of the event horizon), and the energy and angular momentum fluxes are dEdτ = Z H √− g h ˆ T rτ + A τ j r i dθdϕ (35) dLdτ = − Z H √− g h ˆ T rϕ + ( A ϕ − Q m C ) j r i dθdϕ , (36)where the quantities in the brackets are E r and J r , respectively. The total energy,angular momentum and charge that falls through the event horizon is R ∞−∞ dEdτ dτ , R ∞−∞ dLdτ dτ and R ∞−∞ dQdτ dτ , respectively. 10ne of the main assumptions of the thought experiment is that the final stateof the physical system is again a dyonic Kerr-Newman state, which means that allmatter that does not fall through the event horizon is assumed to escape eventu-ally to infinity. In particular, it is assumed that the energy, angular momentumand charge of the matter contained in the domain r + ≤ r ≤ r m , given by the inte-grals − R r m r + dr R √− g E τ dθdϕ , R r m r + dr R √− g J τ dθdϕ , R r m r + dr R √− g j τ dθdϕ , go to 0as τ → ∞ for any fixed value of r m . Under this assumption dM , dJ and dQ e canbe identified with R ∞−∞ dEdτ dτ , R ∞−∞ dLdτ dτ and R ∞−∞ dQdτ dτ , i.e. the change of the mass,angular momentum and electric charge of the black hole equals to the total energy,angular momentum and electric charge that falls through the event horizon.From the equations (34), (35), (36) above and from (15), (16), (17) and (19) itfollows immediately that Z H √− g ˆ T µν ω µ χ ν dθdϕ = dEdτ − Ω H dLdτ − Φ H dQdτ . (37)Taking into account the relations dM = R ∞−∞ dEdτ dτ , dJ = R ∞−∞ dLdτ dτ and dQ e = R ∞−∞ dQdτ dτ , Z ∞−∞ dτ Z H √− g ˆ T µν ω µ χ ν dθdϕ = dM − Ω H dJ − Φ H dQ e (38)is obtained from (37). The right hand side in (38) is the same as the left hand side in(25), and in the extremal case it is M M + a ) dη . Thus the sign of dη depends, in theextremal case, on the sign of R ∞−∞ dτ R H √− g ˆ T µν ω µ χ ν dθdϕ .From the fact that χ µ is a null vector at the event horizon and from the form (31)of ˆ T µν it is obvious that at the event horizon ˆ T µν satisfies the inequality ˆ T µν χ µ χ ν ≥ A µ and ψ functions. Taking into consideration (18), this also means thatˆ T µν ω µ χ ν ≥ T µν ω µ χ ν on the left hand side of (38), hence dM − Ω H dJ − Φ H dQ e ≥ . (39)In particular, in the extremal case dη ≥
0, indicating that the WCCC is not violated.The inequality (39) is almost identical to (25), the only minor difference is that in(39) equality is allowed.Regarding the condition of strict equality in (39), dM − Ω H dJ − Φ H dQ e = 0 holdsif and only if χ µ ( ∂ µ + ieA µ ) ψ = 0 everywhere on the future part of the event horizon.It is easy to see that in this case the charge, energy and angular momentum fluxes dQdτ , dEdτ and dLdτ into the black hole are zero, thus dM = dJ = dQ e = 0. Furthermore,since χ µ A µ is a real constant on the event horizon, the equation χ µ ( ∂ µ + ieA µ ) ψ = 0implies that either ψ = 0 everywhere on the future part of the event horizon, or ψ is11f the form ψ = ψ e iατ , where α ∈ R and ψ = 0, along some integral curves of χ µ .The second possibility is excluded if we assume lim τ →−∞ ψ = 0 at the event horizon.We close this section with two remarks. Although the expressions E µ = ˆ T µ τ + A τ j µ and J µ = ˆ T µ ϕ + ( A ϕ − Q m C ) j µ appear to be gauge dependent because of the explicitpresence of A τ and A ϕ in them, it is important to note that the vector potential A that is used in these expressions is invariant under time translations and rotations,which is a property that is also used when Noether’s theorem is applied, and whichfixes A τ and A ϕ uniquely (up to additive constants), as mentioned in section 2.Furthermore, if A is replaced by some gauge transformed vector potential A + d Φin the Lagrangian (26), then the quantity K µ appearing in the invariance condition(A.3) also has to be modified as K µ → K µ + j µ ∂ τ Φ or K µ → K µ + j µ ∂ ϕ Φ (fortime translations and rotations, respectively), where j µ is the electric current. Theeffect of this modification is that the A τ and A ϕ appearing explicitly in the formulas E µ = ˆ T µ τ + A τ j µ and J µ = ˆ T µ ϕ + ( A ϕ − Q m C ) j µ remain unchanged. Of course,the vector potential in the expressions for ˆ T µ τ and j µ will be the gauge transformedone. The expressions (21) and (22) for the conserved energy and angular momentumin the case of the pointlike test particle can also be derived using Noether’s theorem,and an argument analogous to the one above shows that the A τ and A ϕ appearing inthese expressions are also well defined.The tensor ˆ T µν coincides with the Einstein-Hilbert energy-momentum tensor − δ L δg µν + g µν L obtained from the Lagrangian (26), and the inequality ˆ T µν χ µ χ ν ≥ E µ = ˆ T µ τ and J µ = ˆ T µ ϕ ,however, these currents are not conserved, which can be seen by considering that theirconservation would imply the conservation of A τ j µ and A ϕ j µ . The non-conservationof these currents also means that ∇ µ ˆ T µ ν = 0. By looking at the derivation of theconservation of the Einstein-Hilbert energy-momentum tensor (see e.g. section E.1 of[2] around equation (E.1.27)) it can be seen that the obstacle to the conservation ofˆ T µν is the presence of the fixed electromagnetic field. The Lagrangian density of the scalar and electromagnetic fields in Kerr-Newmanspacetime is L = − g µν ( ∂ µ − ie ˜ A µ ) ψ ∗ ( ∂ ν + ie ˜ A ν ) ψ − m ψ ∗ ψ − π ˜ F µν ˜ F µν , (40)where ˜ F µν = ∂ µ ˜ A ν − ∂ ν ˜ A µ and the tilde is used to distinguish the vector potential ofthe full electromagnetic field from the vector potential of the electromagnetic field of12he dyonic Kerr-Newman black hole introduced in section 2. The electric current is j µ = ie [ ψ ∗ ( ∂ µ + ie ˜ A µ ) ψ − ψ ( ∂ µ − ie ˜ A µ ) ψ ∗ ] . (41)In the present setting the Einstein-Hilbert energy-momentum tensor T µν = − δ L δg µν + g µν L = ( ∂ µ − ie ˜ A µ ) ψ ∗ ( ∂ ν + ie ˜ A ν ) ψ + ( ∂ µ + ie ˜ A µ ) ψ ( ∂ ν − ie ˜ A ν ) ψ ∗ + 14 π ˜ F µλ ˜ F λν + g µν L (42)is conserved (i.e. ∇ µ T µν = 0) and is suitable for defining the energy and angularmomentum currents as E µ = T µτ , J µ = T µϕ . (43)These currents are conserved (i.e. ∇ µ E µ = 0 and ∇ µ J µ = 0) because ∂/∂τ and ∂/∂ϕ are Killing vectors and ∇ µ T µν = 0. E µ and J µ are also clearly gauge invariant. Thesame definition is taken for the energy and angular momentum currents in [7].The charge, energy and angular momentum fluxes through the event horizon aregiven by dQdτ = − Z H √− g j r dθdϕ (44) dEdτ = Z H √− g T rτ dθdϕ (45) dLdτ = − Z H √− g T rϕ dθdϕ . (46)We assume that the field ψ goes to zero as τ → ∞ , in accordance with the fundamentalassumption that the final state of the physical process under consideration is a dyonicKerr-Newman state. The vector potential, on the other hand, will become A + dQ e A e (up to gauge transformation) as τ → ∞ due to the change dQ e of the charge of theblack hole. This change of the electromagnetic field implies that the energy and theangular momentum of the electromagnetic field around the black hole also changes,which has to be taken into account in the calculation of dM and dJ . Thus dM and dJ are given by dM = Z ∞−∞ dEdτ dτ − Z ∞ r + dr Z √− g T τ τ | ˜ A = A + dQ e A e , ψ =0 dθdϕ + Z ∞ r + dr Z √− g T τ τ | ˜ A = A, ψ =0 dθdϕ (47)13nd dJ = Z ∞−∞ dLdτ dτ + Z ∞ r + dr Z √− g T τ ϕ | ˜ A = A + dQ e A e , ψ =0 dθdϕ − Z ∞ r + dr Z √− g T τ ϕ | ˜ A = A, ψ =0 dθdϕ . (48)The second term on the right hand side of (47) and (48) gives the energy and angularmomentum, respectively, of the electromagnetic field around the black hole at τ → ∞ ,whereas the third terms give the energy and angular momentum of the electromag-netic field around the black hole in the initial state. dQ e is given by dQ e = R ∞−∞ dQdτ dτ ,as in section 4.1. We note that in [7] the change of the energy and angular momen-tum of the electromagnetic field around the black hole is included in dM and dJ byconsidering fluxes through spherical surfaces of radius r → ∞ rather than r = r + .Aiming to derive an equation similar to (38), we consider now the quantity dM − Ω H dJ . From (45), (46) and (15) it is easy to see that the contribution of the first termson the right hand side of (47) and (48) to dM − Ω H dJ is R ∞−∞ dτ R H √− g T µν ω µ χ ν dθdϕ .Since A e and A m are known explicitly, the contribution of the second and third termson the right hand side of (47) and (48) can also be evaluated. This task can besimplified by partial integrations and by using the properties of A e , A m and of thecorresponding electromagnetic fields. In addition, those terms that are higher thanfirst order in dQ e should be neglected. One finds that all integrals can be evaluatedtrivially except for one integral over θ , and the final result is that the contribution ofthe terms in question is Φ H dQ e . Thus Z ∞−∞ dτ Z H √− g T µν ω µ χ ν dθdϕ = dM − Ω H dJ − Φ H dQ e , (49)which is analogous to (38) in section 4.1. From the fact that χ µ is a null vector at theevent horizon and from the form (42) of T µν it is obvious that at the event horizon T µν satisfies the inequality T µν χ µ χ ν ≥ A µ and ψ functions. Taking into consideration (18), this implies that T µν ω µ χ ν ≥ dM − Ω H dJ − Φ H dQ e ≥ . (50)This inequality, which has the same form as (39), implies dη ≥ E µ = T µτ − π √− g ∂ ρ ( √− g ˜ A τ ˜ F ρµ ) (51) J µ = T µϕ − π √− g ∂ ρ ( √− g ( ˜ A ϕ − Q m C ) ˜ F ρµ ) . (52)14he additional terms − π √− g ∂ ρ ( √− g ˜ A τ ˜ F ρµ ) and − π √− g ∂ ρ ( √− g ( ˜ A ϕ − Q m C ) ˜ F ρµ ) are ofthe form ∇ ν f µν , where f µν is antisymmetric. Currents of this form are automaticallyconserved regardless of the value of f µν . It is not difficult to verify using partialintegration that these terms do not give any contribution to dM and dJ , thereforethe definitions (51) and (52) also lead to the results (49) and (50).We also note finally that the presence of the scalar field is not essential in thederivation above. If it is omitted, then the case of a purely electromagnetic test fieldis obtained. Appendix. Noether’s theorem
Let the action of a physical system described by a collection of real fields Φ i ( x a ) onan n -dimensional spacetime be S = Z dx dx . . . dx n L (Φ i ( x a ) , ∂ b Φ i ( x a ) , x a ) , (A.1)with Lagrangian L (Φ i ( x a ) , ∂ b Φ i ( x a ) , x a ). The equations of motions are the Euler-Lagrange equations ∂ L ∂ Φ i − D µ ∂ L ∂ ( ∂ µ Φ i ) = 0 . (A.2)The notation D µ is used for the total derivative with respect to x µ . If, for example, f is a function of x a , then D µ f = ∂ µ f , whereas for a function f (Φ i , x a ) we have D µ f = ∂f∂ Φ i ∂ µ Φ i + ∂f∂x µ .Assume that Φ i satisfy the Euler-Lagrange equations, and the invariance condition ∂ L ∂ Φ i ∆Φ i + ∂ L ∂ ( ∂ µ Φ i ) D µ (∆Φ i ) = D µ K µ (A.3)holds with some functions ∆Φ i and K µ . ∆Φ i denote the change of the fields underan infinitesimal transformation Φ i → Φ i + ǫ ∆Φ i . The expression on the left hand sideis the change of L under this transformation. Now it is straightforward to see, using(A.2) and (A.3), that the current j µ = ∂ L ∂ ( ∂ µ Φ i ) ∆Φ i − K µ (A.4)is conserved, i.e. D µ j µ = 0 . (A.5)This theorem is independent of any metric structure on the spacetime manifold.In section 4 we have L = √− g L ; for time translations∆ ψ = − ∂ τ ψ, ∆ ψ ∗ = − ∂ τ ψ ∗ , ∆ ˜ A µ = − ∂ τ ˜ A µ , K µ = − δ µτ √− g L ; (A.6)15or rotations∆ ψ = − ∂ ϕ ψ, ∆ ψ ∗ = − ∂ ϕ ψ ∗ , ∆ ˜ A µ = − ∂ ϕ ˜ A µ , K µ = − δ µϕ √− g L . (A.7)For global U (1) gauge transformations we have∆ ψ = iψ, ∆ ψ ∗ = − iψ ∗ , K µ = 0 . (A.8)The invariance condition is satisfied for any fields in these cases, not only for thesolutions of the Euler-Lagrange equations. Acknowledgment
The author would like to thank Istv´an R´acz for useful discussions.
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