e − e + pair creation by vacuum polarization around electromagnetic black holes
aa r X i v : . [ a s t r o - ph . H E ] M a y e − − e + pair creation by vacuum polarization around electromagnetic black holes C. Cherubini,
1, 2
A. Geralico,
1, 3
J. A. Rueda H.,
1, 3 and R. Ruffini
1, 3, ∗ ICRANet, I-65100 Pescara, Italy Nonlinear Physics and Mathematical Modeling Lab, C.I.R.,University Campus Bio-Medico, I-00128 Rome, Italy Physics Department and ICRA, University of Rome “La Sapienza,” I-00185 Rome, Italy (Dated: November 15, 2018)The concept of “dyadotorus” was recently introduced to identify in the Kerr-Newman geometrythe region where vacuum polarization processes may occur, leading to the creation of e − − e + pairs.This concept generalizes the original concept of “dyadosphere” initially introduced for Reissner-Nordstr¨om geometries. The topology of the axially symmetric dyadotorus is studied for selectedvalues of the electric field and its electromagnetic energy is estimated by using three differentmethods all giving the same result. It is shown by a specific example the difference between adyadotorus and a dyadosphere. The comparison is made for a Kerr-Newman black hole with thesame total mass energy and the same charge to mass ratio of a Reissner-Nordstr¨om black hole.It turns out that the Kerr-Newman black hole leads to larger values of the electromagnetic fieldand energy when compared to the electric field and energy of the Reissner-Nordstr¨om one. Thesignificance of these theoretical results for the realistic description of the process of gravitationalcollapse leading to black hole formation as well as the energy source of gamma ray bursts are alsodiscussed. PACS numbers: 04.20.CvKeywords: Black hole physics; vacuum polarization
I. INTRODUCTION
Relativistic astrophysics differs from the otherbranches of physics and astronomy by exploring new fun-damental processes unprecedented for the spectacularlylarge scales of the involved observables and for their ex-tremely short time variability. Following the well knowncase of supernova with energies . ergs on time scalesof months, gamma-ray bursts (GRBs) have offered anextreme example of the most energetic ( E . ergs)and the fastest transient (∆ t . − − s) phenomenaever observed in the universe [1]. The dynamics of GRBsis dominated by an electron-positron plasma [2]. Thetheoretical model based on the vacuum polarization pro-cesses [3] occurring in a Kerr-Newman geometry [4] canindeed explain such enormous energetics and the sharptime variability. What is most important is that sucha model is based on explicit analytic solutions of well-tested ultrarelativistic field theories. The formation ofsuch black holes in a process of gravitational collapse isexpected from a large variety of binary mergers composedof neutron stars, white dwarfs and massive stars at theend point of their thermonuclear evolution [5] in all pos-sible combinations.In particular, in the merging of two neutron stars andin the final process of gravitational collapse to a blackhole is expected the occurrence of electromagnetic fieldswith strength larger than the critical value of vacuum ∗ Electronic address: ruffi[email protected] polarization E c = m e c ~ e , (1)where m e and e are the electron mass and charge respec-tively [1]. We are currently reexamining the electrody-namical processes of a neutron star via an ultrarelativisticThomas-Fermi equation to identify the possible physicalorigin of this overcritical electric field [6, 7, 8].The time evolution of the gravitational collapse (oc-curring on characteristic gravitational time scales τ = GM/c ≃ × − M/M ⊙ s) and the associated elec-trodynamical process are too complex for a direct de-scription. We address here a more confined problem:the polarization process around an already formed Kerr-Newman black hole. This is a well defined theoreticalproblem which deserves attention. It represents a physi-cal state asymptotically reachable in the process of gravi-tational collapse. We expect such an asymptotic configu-ration be reached when all the multipoles departing fromthe Kerr-Newman geometry have been radiated away ei-ther by process of vacuum polarization or electromag-netic and gravitational waves. What it is most impor-tant is that by performing this theoretical analysis wecan gain a direct evaluation of the energetics of the spec-tra and dynamics of the e − − e + plasma created on theextremely short time scales due to the quantum phenom-ena of ∆ t = ~ / ( m e c ) ≃ − s. This entire transientphenomena, starting from an initial neutral condition,undergoes the formation of the Kerr-Newman black holeby the collective effects of gravitation, strong, weak, elec-tromagnetic interactions during a fraction of the abovementioned gravitational characteristic time scale of col-lapse.The aim of this article is to explore the initial condi-tion for such a process to occur using the recently intro-duced concept of “dyadotorus” [4] which generalizes tothe Kerr-Newman geometry the concept of the “dyado-sphere” previously introduced in the case of the spheri-cally symmetric Reissner-Nordstr¨om geometry [9, 10].Damour and Ruffini [3] showed that vacuum polar-ization processes `a la Sauter-Heisenberg-Euler-Schwinger[11, 12, 13] can occur in the field of a Kerr-Newman blackhole endowed with a mass ranging from the maximumcritical mass for neutron stars (3 . M ⊙ ) all the way upto 7 . × M ⊙ . It is an almost perfectly reversible pro-cess in the sense defined by Christodoulou and Ruffini[14], leading to a very efficient mechanism of extractingenergy from the black hole.In the case of absence of rotation in spacetime, we havea Reissner-Nordstr¨om black hole as the background ge-ometry. The region where vacuum polarization processestake place is a sphere centered about the hole, and hasbeen called dyadosphere [9, 10]. Its main properties arerecalled in Section II.We investigate in Section III how the presence of rota-tion in spacetime modifies the shape of the surface con-taining the region where electron-positron pairs are cre-ated. Due to the axial symmetry we call that region asdyadotorus and we give the conditions for its existence.We then provide some pictorial representations of theboundary surface of the dyadotorus by using the Boyer-Lindquist radial and angular coordinates as polar coor-dinates in flat space as well as by employing Kerr-Schildcoordinates. We show in Section IV the dyadotorus onthe corresponding embedding diagrams, which reveal theintrinsic structure of the spacetime geometry. In SectionV we provide an estimate of the electromagnetic energycontained in the dyadotorus by using three different defi-nitions commonly adopted in the literature, i.e. the stan-dard definition in terms of the timelike Killing vector (seee.g. [15]), the one recently suggested by Katz, Lynden-Bell and Biˇc´ak [16, 17] for axially symmetric asymptoti-cally flat spacetimes, which is an observer dependent def-inition of energy, and the last one involving the theoryof pseudotensors (see e.g. [18]). All these approaches areshown to give the same results. Finally, a comparison ismade between the electromagnetic energy of an extremeKerr-Newman black hole and the corresponding one of aReissner-Nordstr¨om black hole with the same total massand charge to mass ratio. In addition to the topologicaldifferences between the dyadotorus and the dyadosphere,it is shown how larger field strengths are allowed in thecase of a Kerr-Newman geometry close to the horizon,when compared with a Reissner-Nordstr¨om black hole ofthe same mass energy and charge to mass ratio.We finally draw some general conclusions. II. THE DYADOSPHERE
In this section we recall the definition of dyado-sphere and its main properties in the field of a Reissner-Nordstr¨om black hole as derived in [9, 10]. In standardSchwarzschild-like coordinates the Reissner-Nordstr¨omblack hole metric is given by ds = − (cid:18) − Mr + Q r (cid:19) dt + (cid:18) − Mr + Q r (cid:19) − dr + r ( dθ + sin θdφ ) , (2)where geometric units G = c = 1 have been adopted.The associated electromagnetic field is given by F = − Qr dt ∧ dr . (3)The horizons are located at r ± = M ± p M − Q ; weconsider the case | Q | ≤ M and the region r > r + outsidethe outer horizon. For an extremely charged hole we have | Q | = M and the two horizons coalesce.Let us introduce an orthonormal frame adapted to thestatic observers e ˆ t = (cid:18) − Mr + Q r (cid:19) − / ∂ t ,e ˆ r = (cid:18) − Mr + Q r (cid:19) / ∂ r ,e ˆ θ = 1 r ∂ θ , e ˆ φ = 1 r sin θ ∂ φ . (4)The electric field as measured by static observers withfour-velocity U = e ˆ t is purely radial E ( U ) = Qr e ˆ r . (5)The radius r ds at which the electric field strength | E | = | E ˆ r | reaches the critical value E c has been defined in [9,10] as the outer radius of the dyadosphere, which extendsdown to the horizon and within which the electric fieldstrength exceeds the critical value r ds ≃ . × p λµ cm , (6)where the dimensionless quantities λ = Q/M and µ = M/M ⊙ have been introduced. The critical electric field(1) in geometric units is given by E c ≈ . × − cm − .The electromagnetic energy contained inside the dya-dosphere has been evaluated by Vitagliano and Ruffini[15] E ( ξ ) ( r + ,r ds ) = Q r + (cid:18) − r + r ds (cid:19) , (7)by using a “truncated version” of the definition of energyin terms of the conserved Killing integral E ( ξ ) = Z Σ T (em) µν ξ µ d Σ ν , (8) E em µ FIG. 1: The behavior of the electromagnetic energy (7) insolar mass units is shown as a function of the mass parameter µ for selected values of the charge parameter λ = [0 . , . , Q / (2 r + ). where ξ = ∂ t is the timelike Killing vector. We refer toSection V for a detailed discussion on this point. Fig. 1shows the behaviour of the electromagnetic energy (7) asa function of the mass parameter µ for fixed values of thecharge parameter λ .Ruffini and collaborators estimated also the total en-ergy of pairs converted from the “static electric energy”(7) and deposited within the dyadosphere E pairs = Q r + (cid:18) − r + r ds (cid:19) " − (cid:18) r + r ds (cid:19) . (9)Its behaviour as a function of the charge and mass pa-rameters λ and µ is shown in Fig. 2.The rate of pair creation per unit four-volume is givenby the Schwinger formula [13]2Im L = 14 π (cid:18) | E | eπ ~ (cid:19) ∞ X n =1 n e − nπE c / | E | . (10)The leading term n = 1 agrees with the WKB resultsobtained by Sauter [11] and Heisenberg-Euler [12]2Im L = 14 π (cid:18) | E | eπ ~ (cid:19) e − πE c / | E | . (11) λ µ λ µ FIG. 2: The total energy of pairs (9) is plotted as a functionof the two mass and charge parameters µ and λ . The differentcurves correspond to selected values of the the energy (inergs). Only the solutions below the solid line are physicallyrelevant. The configurations above the solid line correspondinstead to unphysical solutions with r ds < r + . The plot isreproduced from [19] with the kind permission of the authors. The dyadosphere has been defined by Ruffini and col-laborators [9, 10] by the condition | E | = E c . One mightbetter define it by requiring the electric field strength tobe such that the rate of pair creation is suppressed ex-actly by a factor 1 /e , leading to the condition | E | = πE c .However, from Eq. (10) it is clear that no sharp thresh-old exists for electron-positron pair creation, so that thedefinition | E | = kE c (12)appears to be more appropriate and should be exploredfor different values of the constant parameter k , even for k <
1. Consequently, we shall define in the followingboth dyadosphere and dyadotorus as the locus of pointswhere the electric field satisfies the condition (12).
III. THE DYADOTORUS
The Kerr-Newman metric in standard Boyer-Linquisttype coordinates writes as [20] ds = − (cid:18) − M r − Q Σ (cid:19) dt − a sin θ Σ (cid:0) M r − Q (cid:1) dtdφ + Σ∆ dr + Σ dθ + (cid:20) r + a + a sin θ Σ (2
M r − Q ) (cid:21) sin θdφ , (13)with associated electromagnetic field F = Q Σ ( r − a cos θ ) dr ∧ [ dt − a sin θdφ ]+2 Q Σ ar sin θ cos θdθ ∧ [( r + a ) dφ − adt ] , (14)where Σ = r + a cos θ and ∆ = r − M r + a + Q .Here M , Q and a are the total mass, total charge andspecific angular momentum respectively characterizingthe spacetime. The (outer) event horizon is located at r + = M + p M − a − Q .Let us introduce the Carter’s observer family [21],whose four-velocity is given by U car = r + a √ ∆ Σ (cid:20) ∂ t + ar + a ∂ φ (cid:21) . (15)An observer adapted frame to U car is then easily con-structed with the triad e ˆ r = 1 √ g rr ∂ r , e ˆ θ = 1 √ g θθ ∂ θ , ¯ U car = a sin θ √ Σ (cid:20) ∂ t + 1 a sin θ ∂ φ (cid:21) . (16)The Carter observers measure parallel electric and mag-netic fields E and B [3], with components E ( U car ) α = F αβ U β car , B ( U car ) α = ∗ F αβ U β car , (17)where ∗ F is the dual of the electromagnetic field (14).Both E and B are directed along e ˆ r and assuming asusual Q >
0, the strength of electric and magnetic fieldsare given by | E | = | E ˆ r | = Q Σ ( r − a cos θ ) , | B | = | B ˆ r | = (cid:12)(cid:12)(cid:12)(cid:12) Q Σ ar cos θ (cid:12)(cid:12)(cid:12)(cid:12) . (18)It is worth noting that the Carter orthonormal frame isthe unique frame in which the flat spacetime Schwingerdiscussion can be locally applied. This is due both to themeaning of the Carter orthonormal frame and its rela-tion to the geometry of the Weyl curvature tensor andthe spacetime itself, as well as to the fact that the invari-antly described Schwinger process demands this uniqueframe for its application. An alternative but equiva-lent derivation of this result is presented in Appendix A,where the electric and magnetic field strengths are ob-tained in terms of the electromagnetic invariants by us-ing the Newman-Penrose formalism, hence showing moreclearly the invariant character of the dyadotorus.The Schwinger formula generalized to include bothelectric and magnetic fields, i.e.2Im L = 14 π (cid:18) | E | eπ ~ (cid:19) ∞ X n =1 n (cid:18) nπ | B || E | (cid:19) × coth (cid:18) nπ | B || E | (cid:19) e − nπE c / | E | , (19) has been used by Damour and Ruffini [3] for the case ofa Kerr-Newman geometry.We are interested in the region exterior to the outerhorizon r ≥ r + . Solving Eq. (12) for r and introducingthe dimensionless quantities λ = Q/M , α = a/M , µ = M/M ⊙ and ǫ = kE c M ⊙ ≈ . k × − (with M ⊙ ≈ . × cm) we get (cid:18) r d ± M (cid:19) = 12 λµǫ − α cos θ ± (cid:20) λ µ ǫ − λµǫ α cos θ (cid:21) / (20)where the ± signs correspond to the two different parts ofthe surface. They join at the particular values θ ∗ and π − θ ∗ of the polar angle given by the condition of vanishingargument of the square root in Eq. (20) θ ∗ = arccos √ α s λµǫ ! . (21)The requirement that cos θ ∗ ≤ k , giving the range ofallowed values for which the dyadotorus appears indeedas a torus-like surface (see Figs. 4 (b), (c) and (d)) k ≥ λ E c M ⊙ µα ≈ . × λµα ; (22)for lower values of k the dyadotorus consists instead oftwo disjoint parts, one of them (corresponding to thebranch r d + ) always external to the other (correspondingto the branch r d − ), and has rather the shape of an el-lipsoid (see Fig. 4 (a)). Therefore, the use of the termdyadoregion should be more appropriate in this case.In terms of the dimensionless quantities λ and α thehorizon radius is then given by r + M = 1 + p − λ − α . (23)The condition for the existence of the dyadotorus is givenby r d ± ≥ r + . The allowed region for the pairs ( λ, µ ) (withfixed values of the rotation parameter α and the polarangle θ ) satisfying this condition is shown in Fig. 3.Figure 4 shows the shape of the projection of thedyadotorus on a plane containing the rotation axis for anextreme Kerr-Newman black hole with fixed µ and λ anddifferent values of the parameter k using Cartesian-likecoordinates X = r sin θ , Z = r cos θ , built up simply bytaking the Boyer-Lindquist coordinates r and θ as polarcoordinates in flat space.A “dynamical” view of topology change in the shapeof the dyadoregion is shown in Fig. 5, where the case of aReissner-Nordstr¨om black hole with the same total massand charge is also shown for comparison. We point outsome interesting qualitative differences between dyado-torus and dyadosphere which can be seen clearly fromthese plots. In particular, the dyadotorus appear to leadto larger values of the electric field than the correspond-ing dyadosphere close to the horizon. A key point here is α = 0α = 0.4α = 0.6α = 0.8α = 0.9α = 0.99 .1e3.1e4.1e51e+05 µ λ FIG. 3: The space of parameters ( λ, µ ) is shown for dif-ferent values of the rotation parameter α = a/M =[0 , . , . , . , . , .
99] and fixed value of the polar angle θ = π/
3. The region below each curve represents the allowedregion for the existence of the dyadoregion with fixed α . Theconfigurations above each line correspond to unphysical solu-tions where r d ± < r + for the selected set of parameters. Thevalue of the parameter k has been set equal to one. the size of the horizon, which in the limit of small chargeto mass ratio λ ≪ r + ∼ M , while in the case of a Reissner-Nordstr¨om black hole goes to r + ∼ M . This fact iscrucial because it leads to the presence of stronger elec-tric fields for the Kerr-Newman black hole in contrastwith the Reissner-Nordstr¨om one. We can compare forinstance the maximum electric field E max = Q/r ofan extreme Kerr-Newman black hole and of a Reissner-Nordstr¨om black hole, which is obtained for r = r + , θ = π/ r = r + in the lattercase, in the limit of small charge to mass ratio E KNmax = QM = 4 E RNmax . (24)We will turn to the energetics of the dyadoregion in Sec-tion V.Three-dimensional images of the dyadotorus can begenerated also in terms of Kerr-Schild coordinates(˜ t, x, y, z ), which are related to the standard Boyer- Lindquist ones ( t, r, θ, φ ) by the equations (see e.g. [20]) d ˜ t = dt − M r − Q ∆ dr ,dψ = dφ − ar + a M r − Q ∆ dr ,x = p r + a sin θ cos ψ ,y = p r + a sin θ sin ψ ,z = r cos θ . (25)Note that the spatial coordinates ( x, y, z ) satisfy the re-lation x + y r + a + z r = 1 , (26)and the auxiliary angular coordinate ψ is a function of r ,as from the second relation of Eq. (25) ψ = φ − Z r ar + a M r − Q ∆ dr . (27)The shape of the dyadotorus using Kerr-Schild coordi-nates is shown in Fig. 6 for the same choice of parametersas in Fig. 4. IV. EMBEDDING DIAGRAM
The plots of Figs. 4 and 6 actually shows a dis-torted view of the shape of the dyadotorus; we shouldrather look at the corresponding embedding diagram,which gives the correct geometry allowing to visual-ize the spacetime curvature. Because of our familiarthree-dimensional intuition, the most useful and easilyunderstood embedding diagrams are those which takea Riemannian two-surface from the original geometry,then reconstructing it as a distorted surface in a three-dimensional Euclidean space.The dyadotorus implicitly defined by Eq. (20) can bevisualized as a 2-dimensional surface of revolution aroundthe rotation axis embedded in the usual Euclidean 3-space by suppressing the temporal and azimuthal depen-dence. Using Boyer-Lindquist coordinates, form Eq. (13)we get the following induced metric of the constant timeslice ( dt = 0) of the world sheet r = r d ± given by Eq. (20) (2) ds = h ηη dη + g φφ dφ ,h ηη = g rr (cid:18) dr d ± dη (cid:19) + g θθ − η , (28)where η ≡ cos θ and all the metric coefficients are evalu-ated at r = r d ± , which is indeed a function of the polarangle θ (so that dr has been related to dη ).Following a standard procedure [22, 23], consider theflat-space line element written in spherical-like coordi-nates (3) ds = dX + dY ± dZ , (29) - - - X - - - Z - - - X - - - Z (a) (b) - - - X - - - Z - - - X - - - Z (c) (d)FIG. 4: The projection of the dyadotorus on the X − Z plane ( X = r sin θ , Z = r cos θ are Cartesian-like coordinates builtup simply using the Boyer-Lindquist radial and angular coordinates) is shown for an extreme Kerr-Newman black hole with µ = 10, λ = 1 . × − and different values of the parameter k : (a) k = 0 . k = 1 . k = 1 . k = 1 . k ≈ . where the plus sign refers to the Euclidean case and theminus sign to the Minkowskian case. For the embeddingsurface in the parametric form X = F ( η ) cos φ , Y = F ( η ) sin φ , Z = G ( η ) , (30)the corresponding line element becomes (2) ds = "(cid:18) dFdη (cid:19) ± (cid:18) dGdη (cid:19) dη + F dφ . (31) Comparison with (28) implies (cid:18) dFdη (cid:19) ± (cid:18) dGdη (cid:19) = h ηη , F = √ g φφ . (32)The relation F = F ( η ) is already given by the secondequation and one can then numerically integrate the firstequation to get the function G ( η ) G ± ( η ) = Z ηη vuut ± h ηη − (cid:20) ddη ( √ g φφ ) (cid:21) ! dη , (33)with the initial condition G ( η ) = 0. Note that the (a)(b)FIG. 5: The projections of the dyadotorus on the X − Z planecorresponding to different values of the ratio | E | /E c ≡ k areshown in Fig. (a) for µ = 10 and λ = 1 . × − . Thecorresponding plot for the dyadosphere with the same massenergy and charge to mass ratio is shown in Fig. (b) forcomparison. dyadotorus is embeddable entirely in the Euclidean 3-space, whereas the embedding of the outer horizon maybecome Minkowskian depending on the values of thecharge and rotation parameters of the black hole [22]. Inthe latter case the embedding cross section has a horizon-tal tangent line when the signature switch of sign in Eq.(33) takes place at a certain value of η given by η = η (ss) ,where dG/dη = 0. The integration must be performedwith the plus sign (into the Euclidean part of the embed-ding) or with the minus sign (into the Minkowskian part of the embedding) starting from such a signature-switchpoint with the initial condition G ( η (ss) ) = 0.Fig. 7 shows the embedding diagram of the dyado-torus for the same choice of parameters as in Figs. 4 and6 as concerns Figs. (a), (c) and (d). Fig. (b) corre-sponds instead to a slightly different choice of the chargeparameter, satisfying Eq. (22) with the equality sign(implying θ ∗ = 0), i.e. to the limiting value of k suchthat the dyadotorus still appears as a torus-like surface,the two branches r d ± still joining (at θ = 0 , π ). Despitethe appearance the cusps on the axis do not correspondto conical singularities at the axis ( θ = 0 , π ), as it oc-curs in contrast in the case of the ergosphere [23, 24].In fact, expanding the induced metric (28) about θ = 0(or equivalently θ = π ) to the second order we get theapproximate metric (up to an ignorable constant factor) (2) ds ≃ dθ + θ dφ (34)with φ ∈ [0 , π ], which is the intrinsic metric of a rightcone with no deficit angle.The projections on the X − Z plane of the embeddingdiagrams of Fig. 7 are shown in Fig. 8. V. ON THE ENERGY OF THE DYADOREGION
The total electromagnetic energy distributed in a sta-tionary spacetime can be determined by evaluating theconserved Killing integral (see e.g. [15]) E ( ξ ) = Z Σ T (em) µν ξ µ d Σ ν , (35)where ξ = ∂ t is the timelike Killing vector, T (em) µν is theelectromagnetic energy-momentum tensor of the source, d Σ ν = n ν d Σ is the surface element vector with n the unittimelike normal to the smooth compact spacelike hyper-surface Σ. The integration is meant to be performedthrough the whole spacetime occupied by the electro-magnetic field, i.e. by allowing Σ to extend up to thespatial infinity. Evaluating the electromagnetic energystored inside a finite region with boundary r = const ofspacetime would require instead the introduction of theconcept of “quasilocal energy.” However, it is interestingto compare the results of the quasilocal treatment withthe expression of the electromagnetic energy containedin the portion of spacetime with boundary r = const ob-tained simply by truncating the integration over r at agiven R in Eq. (35) E ( ξ ) ( r + ,R ) = Z Rr + Z π Z π E ( ξ ) p h n drdθdφ = Q r + (cid:16) − r + R (cid:17) + Q r + (cid:20)(cid:18) a r (cid:19) × arctan( a/r + ) a/r + − r + R (cid:18) a R (cid:19) × arctan( a/R ) a/R (cid:21) , (36) –3 –2 –1 0 1 2 3 x –2–10123 y –3–2–10123 z –3 –2 –1 0 1 2 3 x –2–10123 y –3–2–10123 z (a) (b) –3 –2 –1 0 1 2 3 x –2–10123 y –3–2–10123 z –3 –2 –1 0 1 2 3 x –2–10123 y –3–2–10123 z (c) (d)FIG. 6: Three-dimensional images of the dyadotorus are shown using Kerr-Schild coordinates. The parameter choice is thesame as in Fig. 4. The surfaces have been cut in half for a better view of the interior. The horizon instead has been shownentirely (black region). where E ( ξ ) = T (em) µν ξ µ n ν = Q π Σ / √ ∆ r − a cos θ + 2 a q ( r + a ) − ∆ a sin θ (37)can be interpreted as the electromagnetic energy density, n is the unit normal to the time coordinate hypersurfacesand h n = (Σ / ∆)[( r + a ) − ∆ a sin θ ] sin θ is the de-terminant of the induced metric. It is interesting to notethat the same results can be obtained by using the the-ory of pseudotensors [18] (see Appendix B). In the limitof vanishing rotation parameter Eq. (36) becomes E ( ξ ) ( r + ,R ) = Q r + (cid:16) − r + R (cid:17) , (38) which is just the expression for the electromagnetic en-ergy obtained by Vitagliano and Ruffini [15] for theReissner-Nordstr¨om geometry. Eq. (35) can be actu-ally considered as a possible quasilocal definition of en-ergy [25], although it strongly depends on the existenceof certain spacetime symmetries, i.e. the existence ofa timelike Killing vector, which characterizes stationaryspacetimes. In addition, we can see that since the cur-rent J µ ( ξ ) = T µ (em) ν ξ ν is a conserved vector, the result-ing energy does not depend on the chosen cut throughspacetime. In contrast, in any given spacetime one canalways introduce a physically motivated congruence ofobservers U measuring the energy irrespective of space-time symmetries. But the current J µ ( U ) = T µ (em) ν U ν isnot a conserved vector in general. Therefore, in this case –3 –2 –1 0 1 2 3 X –2–10123 Y –3–2–10123 Z –3 –2 –1 0 1 2 3 X –2–10123 Y –3–2–10123 Z (a) (b) –3 –2 –1 0 1 2 3 X –2–10123 Y –3–2–10123 Z –3 –2 –1 0 1 2 3 X –2–10123 Y –3–2–10123 Z (c) (d)FIG. 7: The dyadotorus is shown on an embedding diagram. The choice of the parameters is the same as in Fig. 4 as concernsFigs. (a), (c) and (d). In the case of Fig. (b) the value of the parameter k has been changed to the critical value k ≈ .
998 inorder to satisfy the condition (22) with the equality sign, so representing the limiting case in which the dyadotorus still appearsas a torus-like surface (as in Figs. (c) and (d)) for the chosen values of parameters µ and α . The surfaces have been cut inhalf for a better view of the interior, where the embedding of the horizon is also shown (the black shaded region is Euclidean,whereas the white regions are Minkowskian). Note that in this case the coordinates ( X, Y, Z ) are given by Eq. (30). the energy has an observer dependent meaning; in addi-tion, the results of the measurement could be differentfor different cuts through spacetime.Such an approach consists of using the definition [16,17] E Σ ( U ) = Z Σ T (em) µν U µ d Σ ν , (39)where Σ is now a bounded hypersurface containing only afinite portion of spacetime, and U is the 4-velocity of theobserver measuring the energy. In general the flux inte-gral of the current J µ ( U ) = T µ (em) ν U ν depends on the hy-persurface, because this is not connected with the space-time symmetries. In particular, the vector field U can be chosen to be the unit timelike normal n of Σ. Therefore,generally we may always evaluate E Σ ( U ) with respect toany preferred observer U , but should not expect to getan answer independent of the chosen cut. In the caseof axially symmetric spacetimes in practice there is nor-mally a good time coordinate such as Boyer-Lindquist inKerr and cuts are chosen to be at constant time. Thecurrent J µ ( U ) will be conserved both for static observersand ZAMOs (Zero Angular Momentum Observers), sincetheir 4-velocities are aligned with Killing vectors.Due to the spacetime symmetries it is indeed quitenatural to consider in the Kerr-Newman spacetime twofamilies of observers which are described by two geomet-rically motivated congruences of curves: 1) static ob-0 (a) (b)(c) (d)FIG. 8: The projections on the X − Z plane of the embedding diagrams of Fig. 7 are shown. Dashed lines correspond to theMinkowskian part of the embedding of the outer horizon. servers, at rest at a given point in the spacetime, whose4-velocity m = 1 / √ g tt ∂ t is aligned with the Killing tem-poral direction; 2) ZAMOs, a family of locally nonrotat-ing observers with 4-velocity n = N − ( ∂ t − N φ ∂ φ ), where N = ( − g tt ) − / and N φ = g tφ /g φφ are the lapse and shiftfunctions respectively, characterized as that normalizedlinear combination of the two given Killing vectors whichis orthogonal to ∂ φ and future-pointing, and it is the unitnormal to the time coordinate hypersurfaces. Since thestatic observers do not exist inside the ergosphere, theZAMOs seem to be the best candidates to construct the energy (39). However, their 4-velocity diverges at thehorizon, since the lapse function goes to zero there.In order to obtain a finite energy at the horizon one canthen chose a family of infalling observers as the Painlev´e-Gullstrand observers, which move radially with respectto ZAMOs and form a congruence of geodesic and ir-rotational orbits, whose 4-velocity is given by U P G = N − ( n −√ − N e ˆ r ). Since they do not follow the space-time symmetries the current J µ ( U P G ) = T µ (em) ν U νP G isnot conserved, so the corresponding energy E Σ ( U P G ) de-pends on the hypersurface. The result is that the expres-1sion (36) of the electromagnetic energy contained in thedyadoregion constructed by means of the (not normal-ized) timelike Killing vector agrees with the electromag-netic energy assessed by the Painlev´e-Gullstrand geodesicfamily of infalling observers through the T = const cutof the Kerr-Newman spacetime, where T denotes thePainlev´e-Gullstrand time coordinate, i.e. E Σ ( ξ ) ≡ Z Σ T (em) µν ξ µ d Σ ν | {z } BL coordinates, Killing vector, t = const cut = Z Σ T (em) µν N µ d Σ ν | {z } PG coordinates, PG 4-velocity, T = const cut ≡ E Σ ( N ) , (40)with N the timelike normal to the chosen cut. Detailscan be found in Appendix B.From Eq. (36), a rough estimate of the electromag-netic energy stored inside the “dyadoregion” turns outto be given by E ( ξ ) ( r + ,R ) ≈ . × − cm ≈ . × ergs by assuming R = 2 r + with the same parameters asin Fig. 4 (d), and E ( ξ ) ( r + ,R ) ≈ . × − cm ≈ . × ergs if R = 3 r + with the same choice of parameters asin Fig. 4 (a). We note that an exact analytic expressionfor the electromagnetic energy can also be obtained bytaking the actual shape r = r d ± given by Eq. (20) in-stead of the approximate expression r = R = const inthe evaluation of the integral (36). However, this onlycomplicates matters by introducing a nontrivial depen-dence on the polar angle θ which makes the integrationprocedure more involved, even if it can be analyticallyperformed (not shown here for the sake of brevity). Fur-thermore, the numerical values of the energy correspond-ing to the above choice of parameters agree with previousestimates.It is interesting to compare the electromagnetic energy(36) of an extreme Kerr-Newman black hole contained inthe portion of spacetime with boundary R = const andthat of a Reissner-Nordstr¨om black hole (38) with thesame total mass and charge in the limit of small chargeto mass ratio. In this limit we have E RN ≃ Q M (cid:18) − MR (cid:19) ,E KN ≃ Q M (cid:18) − MR (cid:19) (41)+ Q M (cid:20) π MR − (cid:18) M R (cid:19) arctan( M/R ) (cid:21) . A comparison between energies is meaningful only at in-finity, where the radial coordinates of a Kerr-Newmanand a Reissner-Nordstr¨om geometry can be identified(both with an ordinary radial coordinate in flat space).For R → ∞ we thus have E KN − E RN → Q M π > . (42) VI. CONCLUSIONS
Vacuum polarization processes can occur in the fieldof a Kerr-Newman black hole inside a region we havecalled dyadotorus, whose properties have been investi-gated here. Such a region has an invariant character, i.e.its existence does not depend on the observer measuringthe electromagnetic field: therefore, it is a true physicalregion.Some pictorial representations of the boundary surfacesimilar to those commonly used in the literature havebeen shown employing Cartesian-like coordinates (i.e. or-dinary spherical coordinates built up simply using theBoyer-Lindquist radial and angular coordinates) as wellas Kerr-Schild coordinates. The dyadotorus has been alsoshown on the corresponding embedding diagram, whichgives the correct geometry allowing to visualize the space-time curvature.We have then estimated the electromagnetic energycontained in the dyadotorus by using three different ap-proaches, which give rise to the same final expressionfor the energy. The first one follows the standard ap-proach consisting of using the (not normalized) timelikeKilling vector through the Boyer-Lindquist constant timecut of the Kerr-Newman spacetime (see e.g. [15]), thesecond one follows a recent observer dependent defini-tion by Katz, Lynden-Bell and Biˇc´ak [16, 17] for axi-ally symmetric asymptotically flat spacetimes, for whichwe have used the Painlev´e-Gullstrand geodesic family ofinfalling observers through the Painlev´e-Gullstrand con-stant time cut, and the last one adopts the pseudotensortheory (see e.g. [18]). We have found by rough estimatesthat the extreme Kerr-Newman black hole leads to largervalues of the electromagnetic energy as compared with aReissner-Nordstr¨om black hole with the same total massand charge.It is appropriate to recall that the release of energy viathe electron-positron pairs in the dyadotorus is the mostpowerful way to extract energy from black holes and inall sense corresponds to a new form of energy: the “black-holic” energy [4]. This is a new form of energy differentfrom the traditional ones known in astrophysics. Thethermonuclear energy has been recognized to be energysource of main sequence stars lasting for 10 years [32],the gravitational energy released by accretion processesin neutron stars and black holes has explained the energyobserved in binary X-ray sources on time scales 10 − years [33]. The “blackholic” energy appears to be energysource for the most transient and most energetic eventsin the universe, the GRBs [4]. Acknowledgments
The authors thank Dr. C. L. Bianco for the technicalassistance plotting Fig. 5.2
APPENDIX A: NEWMAN-PENROSEQUANTITIES AND INVARIANT DEFINITIONOF THE DYADOTORUS
The existence of the dyadotorus has an invariant char-acter. This fact appears more evident if the electric andmagnetic field strengths are expressed in terms of theelectromagnetic invariants. Let us adopt here the met-ric signature (+ , − , − , − ) in order to use the Newman-Penrose formalism in its original form and then easilyget the necessary physical quantities [26, 27]. The Kerr-Newman metric is thus given by ds = (cid:18) − M r − Q Σ (cid:19) dt + 2 a sin θ Σ (cid:0) M r − Q (cid:1) dtdφ − Σ∆ dr − Σ dθ − (cid:20) r + a + a sin θ Σ (2
M r − Q ) (cid:21) sin θdφ , (A1)with associated electromagnetic field F = Q Σ ( r − a cos θ ) dr ∧ [ dt − a sin θdφ ]+2 Q Σ ar sin θ cos θdθ ∧ [( r + a ) dφ − adφ ] . (A2)Introduce the standard Kinnersley principal tetrad [28] l µ = 1∆ [ r + a , ∆ , , a ] ,n µ = 12Σ [ r + a , − ∆ , , a ] ,m µ = 1 √ r + ia cos θ ) (cid:20) ia sin θ, , , i sin θ (cid:21) , (A3)which gives nonvanishing spin coefficients ρ = − r − ia cos θ , τ = − ia √ ρρ ∗ sin θ ,β = − ρ ∗ √ θ , π = ia √ ρ sin θ ,µ = 12 ρ ρ ∗ ∆ , γ = µ + 12 ρρ ∗ ( r − M ) ,α = π − β ∗ , (A4)and the only nonvanishing Weyl scalar ψ = M ρ + Q ρ ∗ ρ , (A5)showing clearly the Petrov type D nature of the Kerr-Newman spacetime, whereas the Maxwell scalars are φ = φ = 0 , φ = Q ρ . (A6)The electromagnetic invariants are given by F ≡ F µν F µν = 12 ( B − E ) = 2Re( φ φ − φ ) , G ≡ F µν ∗ F µν = E · B = − φ φ − φ ) , (A7) where E and B are the electric and magnetic fields. Re-quiring parallel electric and magnetic fields [3] as mea-sured by the Carter observer [21], the previous relationsbecome | B | − | E | = − φ ) , | E | | B | = 2Im( φ ) , (A8)taking into account Eq. (A6). This system can thenbe easily solved for the magnitudes of E and B in theKerr-Newman background, which turn out to be givenby | E | = (cid:12)(cid:12)(cid:12)(cid:12) Q Σ ( r − a cos θ ) (cid:12)(cid:12)(cid:12)(cid:12) , | B | = (cid:12)(cid:12)(cid:12)(cid:12) Q Σ ar cos θ (cid:12)(cid:12)(cid:12)(cid:12) , (A9)which coincide with those of Eq. (18). We have thus re-covered the results by Damour and Ruffini [3], but usinga different faster derivation using the Newman-Penroseformalism.Finally, the Schwinger formula for the rate of pair cre-ation per unit four-volume in terms of the electromag-netic invariants (A7) is given by [13]2Im L = e |G| π ~ ∞ X n =1 n coth ( nπ (cid:20) ( F + G ) / + F ( F + G ) / − F (cid:21) / ) × e − nπE c / [( F + G ) / −F ] / . (A10)After introducing the Carter frame (15)–(16) with re-spect to which electric and magnetic fields are parallel,the previous formula reduces to Eq. (19), since[( F + G ) / + F ] / = | B | , [( F + G ) / − F ] / = | E | , |G| = | E | | B | . (A11) APPENDIX B: ELECTROMAGNETIC ENERGYUSING PAINLEV´E-GULLSTRAND OBSERVERSAND PSEUDOTENSOR THEORY
In order to evaluate the energy E Σ ( U P G ) it is usefulto transform the Kerr-Newman metric (13) from Boyer-Lindquist coordinates ( t, r, θ, φ ) to Painlev´e-Gullstrandcoordinates (
T, R, Θ , Φ) [29, 30], which are related bythe transformation T = t − Z r f ( r ) dr , R = r , Θ = θ ,
Φ = φ − Z r ar + a f ( r ) dr , (B1)where f ( r ) = − p (2 M r − Q )( r + a )∆ . (B2)Let us notice that r and R are identified. This is alsotrue for their differential dr = dR but it is no more truefor the associated differentiations ∂ r = ∂ R . Hereafter we3will always use r in place of R , except for the differentia-tion operations. In differential form, this transformationwrites as dT = dt − f ( r ) dr , dR = dr , d Θ = dθ ,d
Φ = dφ − ar + a f ( r ) dr . (B3)Finally, the Kerr-Newman metric in the Painlev´e-Gullstrand coordinates is given by ds = − (cid:18) − M r − Q Σ (cid:19) dT + 2 r M r − Q r + a dT dr − a (2 M r − Q )Σ sin θdT d Φ+ sin θ (cid:20) r + a + a (2 M r − Q )Σ sin θ (cid:21) d Φ − a sin θ r M r − Q r + a drd Φ+ Σ r + a dr + Σ dθ , (B4)with associated electromagnetic field F = Q Σ ( r − a cos θ ) dr ∧ [ dT − a sin θd Φ]+2 Q Σ ar sin θ cos θdθ ∧ [( r + a ) d Φ − adT ] , (B5)which has the same form as (14) with dt → dT and dφ → d Φ.The limit of vanishing rotation parameter a = 0 of theprevious equations (B4)–(B5) gives rise to the Reissner-Nordstr¨om solution in Painlev´e-Gullstrand coordinates ds = − (cid:18) − Mr + Q r (cid:19) dT + 2 p M r − Q r dT dr + dr + r ( dθ + sin θdφ ) ,F = Qr dr ∧ dT . (B6)In the Painlev´e-Gullstrand coordinates the slicing ob-servers ( T − slicing hereafter) have 4-velocity N = ∂ T − p (2 M r − Q )( r + a )Σ ∂ R (B7)and associated 1-form N ♭ = − dT . This family of T − slicing-adapted observers does not coincide with the t − slicing-adapted observers in Boyer-Lindquist coordi-nates once the coordinate transformation is performed.In fact, when expressed in Boyer-Lindquist coordinatesthe T − slicing-adapted observers move with respect tothe t − slicing-adapted observers in the radial direction,as already pointed out in Section V.We are now ready to evaluate the energy (39) througha T = const hypersurface as measured by Painlev´e-Gullstrand observers with 4-velocity (B7). The energy density turns out to be E ( N ) = T (em) µν N µ N ν = Q π Σ ( r − a cos θ + 2 a ) , (B8)where T (em) µν is the Kerr-Newman electromagnetic energy-momentum tensor expressed in Painlev´e-Gullstrand co-ordinates. Let us assume that the boundary S of Σ bethe 2-surface r = R = const for simplicity. Therefore theenergy (39) turns out to be given by E ( N ) ( r + ,R ) = 2 π Z Rr + Z π E ( N ) p h N drdθ = − Q a (cid:20) ar + r + a r arctan ar − π (cid:21) Rr + = Q r + − Q R + 14 Q ar ( r + a ) arctan ar + − Q aR ( R + a ) arctan aR , (B9)where h N = Σ sin θ is the determinant of the inducedmetric. The total electromagnetic energy contained inthe whole spacetime is obtained by taking the limit R →∞ in the previous equation E ( N ) ( r + , ∞ ) = Q r + + 14 Q ar ( r + a ) arctan ar + , (B10)which in the limiting case a = 0 reduces to E RN ( N ) ( r + , ∞ ) = Q r + . (B11)It is interesting to note that the same result (B9) forthe energy assessed by Painlev´e-Gullstrand observer isachieved simply by using the Killing vector ξ = ∂ T , since E ( N ) = E ( ξ ). But it is quite surprising that the sameresult is again obtained by taking a t = const hypersur-face in Boyer-Linquist coordinates with unit normal theZAMO 4-velocity n with respect to “Killing observers” ξ = ∂ t (see Eq. (36)), since E ( N ) p h N = E p h n = Q π Σ ( r − a cos θ + 2 a ) sin θ . (B12)For completeness we list here similar results presentedin Ref. [18] by using the standard definition of symmetricenergy-momentum pseudotensor as given by Landau andLifshitz [31] (LL), although we stress that the physicalinterpretation of these quantities are controversial in theliterature, due to their strict relation with specific coor-dinate sets. This fact is clearly not in the spirit of gen-eral relativity. The LL prescription for the pseudotensoris given by 16 πL αβ = λ αβγδ,γδ , where comma denotespartial derivative and λ αβγδ = − g (cid:0) g αβ g γδ − g αγ g βδ (cid:1) .The conservation law L αβ,β = 0 implies that the to-tal energy is given by E = R R R L dx dx dx . By4computing the pseudotensor in the quasi-Cartesian Kerr-Schild coordinates previously introduced in Eq. 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