Gravitational electromotive force in magnetic reconnection around Schwarzschild black holes
GGravitational electromotive force in magnetic reconnection around Schwarzschildblack holes
Felipe A. Asenjo ∗ and Luca Comisso † Facultad de Ingenier´ıa y Ciencias, Universidad Adolfo Ib´a˜nez, Santiago 7941169, Chile. Department of Astronomy and Columbia Astrophysics Laboratory,Columbia University, New York, NY 10027, USA
We analytically explore the effects of the gravitational electromotive force on magnetic reconnec-tion around Schwarzschild black holes through a generalized general-relativistic magnetohydrody-namic model that retains two-fluid effects. It is shown that the gravitational electromotive forcecan couple to collisionless two-fluid effects and drive magnetic reconnection. This is allowed by thedeparture from quasi-neutrality in curved spacetime, which is explicitly manifested as the emergenceof an effective resistivity in Ohm’s law. The departure from quasi-neutrality is owed to differentgravitational pulls experienced by separate parts of the current layer. This produces an enhancementof the reconnecion rate due to purely gravitational effects.
PACS numbers: 52.27.Ny; 52.30.Cv; 52.35.Vd, 04.20.-qKeywords: Magnetic reconnection; General relativity; Relativistic plasmas
Magnetic fields are ubiquitous in the universe and theyplay a major role in a variety of astrophysical systems.At large scales, the behavior of highly conducting mag-netized plasmas is well described by the equations ofideal magnetohydrodynamics (MHD), which impose sig-nificant constraints on the plasma dynamics. Indeed, anideal MHD evolution implies the frozen-in condition andtherefore the preservation of field line connectivity amongfluid elements. This is a remarkably general result, whichis valid in non-relativistic [1], special relativistic [1–3], aswell as general relativistic [4] plasmas.On the other hand, at small spatial scales, physical ef-fects beyond ideal MHD can break the frozen-in conditionand allow for a topological rearrangement of the magneticfield configuration that occurs on time scales much fasterthan the global magnetic diffusion time. This process,known as magnetic reconnection [5], enables a rapid con-version of magnetic energy into plasma particle energy,and is generally believed to be the underlying mechanismthat powers some of the most energetic astrophysical phe-nomena in the universe, such as solar and stellar flares[6, 7], nonthermal signatures of pulsar wind nebulae [8, 9],and gamma-ray flares in blazar jets [10, 11].Electrical resistivity due to Coulomb interactions be-tween charged particles is the prototypical effect that canbreak the frozen-in condition and allow for the reconnec-tion of magnetic field lines. This was indeed employedin many models of magnetic reconnection, from the pio-neering Sweet-Parker model [12, 13] to the more recentmodels of fast magnetic reconnection mediated by theplasmoid instability [14–19]. Anomalous resistivity dueto wave-particle interactions and scatterings off the tur-bulent fluctuations can also enable magnetic reconnec-tion, and they have been considered as a possible agent ∗ Electronic address: [email protected] † Electronic address: [email protected] of fast reconnection [20–22]. Depending on the value ofthe classical/anomalous resistivity, other non-ideal effectscan be even more important. For example, electron in-ertia effects are indeed known to permit nondissipativemagnetic reconnection [23–25], and in an analogous fash-ion, nongyrotropic electron pressure tensor effects canbreak the frozen-in constraint and sustain most of thereconnection electric field required for fast reconnection[26–28].In relativistic plasmas, thermal effects proportional tothe relativistic enthalpy density couple to the inertial ef-fects, leading to an increase of the magnetic reconnectionrate [29, 30]. Furthermore, the Hall terms, that cannotcause magnetic reconnection per se in the nonrelativisticcase, do allow for a change in the magnetic field line con-nectivity if there is a significant difference between theenthalpy density of the positively and negatively chargedfluids constituting the plasma [31]. The situation is ren-dered even more complex in the presence of a strong grav-itational field, as in the vicinity of compact objects likeblack holes. Several studies have predicted the forma-tion of reconnection layers in the vicinity of black holes[32–38], and the theoretical investigation of magnetic re-connection in curved spacetime has just started [30, 39].With this manuscript we intend to explore the effectsof the gravitational electromotive force on magnetic re-connection in a curved spacetime around a black hole.In previous works [30, 39] the role of the radial gravi-tational force due to the black hole was not studied, asit requires a correct definition of the gravitational elec-tromotive forces as well as understanding the influenceof the charge density in curved spacetimes (see below).That the gravitational electromotive force contributes tomagnetic reconnection was suggested by Koide [40], with-out working out explicitly its quantitative effects on thereconnection rate. Here we focus on the simplest formof the gravitational field created by a black hole, i.e., aSchwarzschild black hole, and we calculate the reconnec-tion rate due to the gravitational electromotive force. a r X i v : . [ phy s i c s . p l a s m - ph ] M a r In order to show that the gravitational field of aSchwarzschild black hole introduces new effects thatare relevant for reconnection, we adopt a generalizedversion of the general-relativistic magnetohydrodynamic(GRMHD) equations [30, 40] which retain two-fluid ef-fects that are neglected in the simpler single-fluid descrip-tions. In particular, we employ a set of equations [30]that describes electron-ion plasmas in the thermal-inertiaregime [41, 42]. This is the regime in which the thermal-inertial terms are larger than the Hall terms. Therefore,by taking into account the proper mass ratio between thepositively and negatively charged particles, the same setof equations describes also pair plasmas, where the Hallterms vanish identically.The considered spacetime x µ = ( t, x , x , x ) is char-acterized by a metric g µν , where the line element is givenby ds = g µν dx µ dx ν . Note that we choose units in whichthe speed of light c is unity. The GRMHD equationsdeal with a single-fluid plasma model with proper en-thalpy density h = n ( h + /n + h − /n − ), where n ± in-dicate the proper particle number density for the posi-tively (+) and negatively ( − ) charged components flu-ids. Similarly, the enthalpy density h ± of each chargedfluid is specified with the corresponding subscript, and n = n + + n − . Furthermore, it is assumed that ∆ h (cid:28) h ,where ∆ h = mn ( h + /m + n − h − /m − n − ) / m = m + + m − , and m ± indicating the mass of the corre-sponding charged particle). It is also assumed the equa-tion of state h ± = m ± n ± K ( m ± /k B T ± ) /K ( m ± /k B T ± )[43, 44], where K and K are the modified Bessel func-tions of the second kind of orders two and three, T ± arethe temperatures of each fluid, and k B is the Boltzmannconstant.In this model the momentum equation that retainsthermal-inertia effects is [30, 40] ∇ ν (cid:20) h (cid:18) U µ U ν + ξ n e J µ J ν (cid:19)(cid:21) = −∇ µ p + J ν F µν , (1)where ∇ ν denotes the covariant derivative associatedwith the spacetime metric g µν , U µ is the plasma four-velocity, J µ is the four-current density, and F µν isthe electromagnetic field tensor. Furthermore, p = p + + p − indicates the proper plasma pressure, e isthe electron charge, and ξ = 1 − (∆ µ ) , with ∆ µ =( m + − m − ) / ( m + + m − ). Observe that ξ ≈ m − /m + for an electron-ion plasma, while ξ = 1 for a pair plasma.Furthermore, the generalized Ohm’s law in thethermal-inertial regime is [30] U ν F µν = η [ J µ − ρ (cid:48) e U µ ]+ ξ e n ∇ ν (cid:20) hn (cid:18) U µ J ν + J µ U ν − ∆ µne J µ J ν (cid:19)(cid:21) , (2)where ρ (cid:48) e = − U ν J ν is the charge density observed by thelocal center-of-mass frame, and η is the electrical resistiv-ity, which is considered as a phenomenological parameter.Notice that, in comparison with the model equations of Ref. [30], we are considering a plasma where the thermalenergy excange rate between the two fluids is negligible[40], i.e., the redistribution coefficient of the thermalizedenergy to the positively and negatively charged fluids iszero.The plasma dynamics is completed by the continuityequation ∇ ν ( nU ν ) = 0 , (3)and Maxwell’s equations ∇ ν F µν = J µ , ∇ ν F ∗ µν = 0 , (4)where F ∗ µν is the dual of the electromagnetic field tensor.To explicitly display the gravitational effects in theabove plasma model in a familiar fashion, we write theprevious equations in the 3+1 formalism [45–47]. In suchform, that spacetime curvature effects become aparent ina set of vectorial equations. For a Schwarzschild back-ground, with spherical geometry, the line element be-comes ds = − α dt + h dr + h dθ + h dφ , (5)with α = (cid:112) − r s /r , h = 1 /α , h = r , and h = r sin θ . Here, α is known as the lapse function, r isthe radial distance to the black hole, r s is the halfof the Schwarzschild radius (hereafter G = 1 = c ),0 ≤ θ ≤ π , and 0 ≤ φ ≤ π . In order to properlydescribe the plasma dynamics, it is also useful to re-write the plasma vectorial equations by introducing a lo-cally nonrotating frame called “zero-angular-momentum-observer” (ZAMO) frame [30, 39, 40, 48], which intro-duces a locally Minkowskian spacetime in where the lineelement (5) can be written as ds = − d ˆ t + (cid:80) i =1 ( d ˆ x i ) ,where d ˆ t = αdt and d ˆ x i = h i dx i . In the following, quan-tities observed in the ZAMO frame are denoted with hats.We first consider the continuity equation (3), whichcan be rewritten in the ZAMO frame as [30, 40] ∂ ( γn ) ∂t + αr sin θ (cid:88) j ∂∂x j (cid:18) r sin θh j γn ˆ v j (cid:19) = 0 , (6)where ˆ v is the velocity in the ZAMO frame, and γ =(1 − ˆ v ) − / is the Lorentz factor (we use latin indices forspace components). We also consider the spatial compo-nents of the generalized momentum equation (1), whichlead to the dynamical equation ∂ ˆ P i ∂t = − αr sin θ (cid:88) j ∂∂x j (cid:18) r sin θh j ˆ T ij (cid:19) − ( (cid:15) + γρ ) 1 h i ∂α∂x i + (cid:88) j α (cid:104) G ij ˆ T ij − G ji ˆ T jj (cid:105) , (7)where ˆ P i = hγ ˆ v i + hξ n e ˆ J i ˆ J + (cid:88) j,k ε ijk ˆ E j ˆ B k , (8) (cid:15) = hγ + hξ e n ( ˆ J ) − p − ργ + 12 (cid:16) ˆ B + ˆ E (cid:17) , (9)and ˆ T ij = pδ ij + hγ ˆ v i ˆ v j + hξ e n ˆ J i ˆ J j + 12 (cid:16) ˆ B + ˆ E (cid:17) δ ij − ˆ B i ˆ B j − ˆ E i ˆ E j . (10)Here, ˆ J is the separation of charge density while ˆ J i isthe current density, both observed in the ZAMO frame.It is the main goal of this work to show (below) that ˆ J affects the magnetic reconnection process by the gravi-tational electromotive force. Besides, it is important tonotice that ˆ J is related to the invariant ρ (cid:48) e = − U µ J µ .We also specify that ˆ E j and ˆ B j are the electric andmagnetic fields measured in the ZAMO frame, G ij = − (1 /h i h j )( ∂h i /∂x j ), and ε ijk is the Levi-Civita symbol.For the spatial components of the generalized Ohm’slaw (2), in the ZAMO frame we have ξen ∂∂t (cid:20) hγ en (cid:16) ˆ J i + ˆ J ˆ v i (cid:17)(cid:21) = − hξγ ˆ J e n h i ∂α∂x i − αenr sin θ (cid:88) j ∂∂x j (cid:18) r sin θh j ˆ K ij (cid:19) + αen (cid:88) j (cid:16) G ij ˆ K ij − G ji ˆ K jj (cid:17) + αγ ˆ F i + αγ ˆ v j ˆ F ij − αη (cid:16) ˆ J i − ρ (cid:48) e γ ˆ v i (cid:17) , (11)where ˆ K ij = ( hξγ/ en )(ˆ v i ˆ J j + ˆ v j ˆ J i ). Similarly, the tem-poral component of Eq. (2) becomes [40] ξ en ∂∂t (cid:32) hγ ˆ J en (cid:33) = − hξγ e n (cid:88) j h j ∂α∂x j (cid:16) ˆ J j + ˆ J ˆ v j (cid:17) − αenr sin θ (cid:88) j ∂∂x j (cid:18) r sin θh j hξγ en (cid:104) ˆ J j + ˆ J ˆ v j (cid:105)(cid:19) + αγ ˆ v j ˆ F j − αη (cid:16) ˆ J − ρ (cid:48) e γ (cid:17) . (12)Finally, we rewrite Maxwell’s equations (4) in theZAMO frame. These are (cid:88) j ∂∂x j (cid:18) r sin θαh j ˆ B j (cid:19) = 0 , (13) αr sin θ (cid:88) j ∂∂x j (cid:18) r sin θαh j ˆ E j (cid:19) = ˆ J , (14) α ˆ J i + ∂ ˆ E i ∂t = αh i r sin θ (cid:88) j,k ε ijk ∂∂x j (cid:16) αh k ˆ B k (cid:17) , (15) ∂ ˆ B i ∂t = − αh i r sin θ (cid:88) j,k ε ijk ∂∂x j (cid:16) αh k ˆ E k (cid:17) . (16) The gravitational field of a Schwarzschild black holeintroduces effects in the generalized Ohm’s law (11)that can be seen as effective electric fields. In par-ticular, terms with the form G ij ˆ K ij and G ji ˆ K jj inEq. (11) can introduce effective resistivities of the or-der ( hξ/ en )( ∂ j h i /h i h j ), in where both the gravitationalfield and the thermal-inertial effects are important. How-ever, as we will see below, in the simplest possible geom-etry for the reconnection layer, these both terms vanish.On the other hand, as noticed by Koide in Ref. [40], theterm proportional to ˆ J ( ∂ i α/h i ) in Eq. (11) producesa radial contribution to the generalized Ohm’s law thatcan be interpreted as an effective electric field, as longas ˆ J does not vanish. Therefore, in this work we an-alyze this possibility, showing that a reconnection layeraround a Schwarzschild black hole allows a solution inwhich the separation of charge ˆ J is finite, and that inthis case the electromotive force due to gravity can drivemagnetic reconnection.Without loss of generality, let us assume that the re-connection layer is at θ = π/ r . We consider a quasi–two–dimensional reconnectionlayer having characteristic length L and width δ suchthat δ (cid:28) L . The length L is in the φ -direction, whilethe width δ is in the θ -direction, as depicted in Fig. 1.We also assume that the layer is not close to the blackhole, δ (cid:28) L (cid:28) r . This model allow us to study mag-netic reconnection using a Sweet-Parker-like approach fora plasma that is supported against the black hole gravity[49–51], as for the model we investigated in Kerr curvedspacetime [30, 39]. We also assume that the radial plasmavelocity is null or negligible, i.e. ˆ v r = 0, and that in thediffusion region ˆ J θ = 0 and ˆ J φ = 0. Then, it is impor-tant to observe that ρ (cid:48) e = − U µ J µ = γ ˆ J (cid:54) = 0, in general[40]. Furthermore, the reconnecting magnetic field hasmagnitude ˆ B in in φ –direction (with no radial componentin the reconnection layer), while the electric field is inthe radial direction (see Fig. 1).Under the above assumptions, we can readily calculatethe outflow velocity of the plasma accelerated throughthe reconnection channel. This plasma outflow is in φ -direction along the neutral line. By using the momentumequation (7) we find (cid:88) j ∂∂x j (cid:18) r sin θh j ˆ T φj (cid:19) = 0 , (17)as other terms indentically vanish along the φ –direction.The solution for this equation is T φφ = 0. Takingthe tensor (10) along the neutral line, and using that p ≈ ˆ B / ≈ h/ γ out ˆ v out ≈ / √ FIG. 1: Sketch of a magnetic reconnection layer showing thestudied configuration. The shaded gray area represents themagnetic diffusion region. θ -direction is ˆ B θ (cid:12)(cid:12)(cid:12) out ≈ δL ˆ B in . (18)On the other hand, using the continuity equation (6) forthe Schwarzschild geometry, the inflow plasma velocitycan be written as γ in ˆ v in ≈ δL γ out ˆ v out . (19)Besides, from Eq. (15) we obtain that the radial currentdensity at the X point is simplyˆ J r (cid:12)(cid:12)(cid:12) X ≈ ˆ B in δ . (20)The results (18), (19) and (20) are equivalent to thosepertaining relativistic plasmas in flat spacetimes [29, 52].The explanation for this is the chosen configurationaround the Schwarzschild black hole. The simple geom-etry studied here, with the invoked assumptions, impliesthat no gravitational effects appear in the momentumequation or Maxwell’s equations when they are evalu-ated in the reconnecion layer. As we shall see now, allthe gravitational effects appear in the generalized Ohm’slaw.We focus on the spatial part of the generalized Ohm’slaw (11) along r -direction. For our geometry, in the cur-rent sheet this equation becomes αenr sin θ (cid:88) j ∂∂x j (cid:18) r sin θh j hξγ en ˆ v j ˆ J r (cid:19) + hξαγ ˆ J e n ∂α∂r =+ αγ ˆ E r − αγ ˆ v θ ˆ B φ + αγ ˆ v φ ˆ B θ − αη ˆ J r . (21)We evaluate this equation in the inflow point, where theinflow plasma velocity is in the θ -direction and the term proportional to the resistivity is negligible. Thus, we getˆ E r (cid:12)(cid:12)(cid:12) in ≈ ˆ v in ˆ B in + hξ ˆ J e n r s αr (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) in , (22)where we have used that ∂ r α = r s /αr . Here, we haveneglected the non-linear terms and considered γ in ≈ X -point (where the plasmavelocity vanishes), obtainingˆ E r (cid:12)(cid:12)(cid:12) X ≈ ( η + Λ) ˆ J r + hξ ˆ J e n r s αr (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) X , (23)where we have introduced the effective relativitistic col-lisionless resistivity [29]Λ = hξ e n L . (24)Both results (22) and (23) reduce to those of Ref. [29] inthe flat spacetime limit r s → J different fromzero, the electric fields ˆ E r | in and ˆ E r | X are not equal. Thisis due to the presence of different gravitational gradientsat the inflow and X points. The radial distance of theinflow point r | in is related to the radial distance r of the X -point by r | in ≈ r + δ / (8 r ), where r | X ≡ r , and therebythe two points experience slightly different gravitationalpulls. We can obtain the difference between the electricfield at the inflow and X points by using Eq. (14). Byintegration among these two points in the current layer,and the radial distance of the inflow point, we getˆ E r (cid:12)(cid:12)(cid:12) in − ˆ E r (cid:12)(cid:12)(cid:12) X ≈ δ αr ˆ J , (25)where the lapse function must be evaluated at the dis-tance r of the X -point.What remains to be done is to obtain a relation be-tween the current density and ˆ J . This can be achievedthrough the temporal part of the generalized Ohm’s law,namely Eq. (12). We can use that ρ (cid:48) e = γ ˆ J is an in-variant to calculate ˆ J by evaluation of Eq. (27) at theoutflow point. Thereby, assuming that the variations ofthe current density are neglegible in this geometry com-pared to the gravitational gradient, i.e., neglecting thedivergence of the current density with respect to the gra-dient of the lapse function projected along the current αr ∂∂r (cid:16) αr ˆ J r (cid:17) (cid:28) α ∂α∂r ˆ J r , (26)then from Eq. (12) evaluated in the outflow point weobtain0 ≈ − γ Λ Lr s r ˆ J r (cid:12)(cid:12)(cid:12)(cid:12) out − αγ Λ Lr ∂∂φ (cid:16) ˆ v φ ˆ J (cid:17)(cid:12)(cid:12)(cid:12)(cid:12) out + αηγ ˆ v ˆ J (cid:12)(cid:12)(cid:12) out , (27)where we have used that 1 − γ = − γ ˆ v . As ˆ J decreasesto the X -point, the previous equation can be solved forˆ J to finally get ˆ J ≈ Lχr s αr ( η + Λ) ˆ J r , (28)where χ = 1 − L / (4 r ) − r s L / (8 α r ), and we haveused the radial distance of the outflow point r | out ≈ r + L / (8 r ) in terms of the radial distance r of the X -point.Notice that the separation of charge ˆ J is only relevantin curved spacetimes, as it vanishes when r s → r s (cid:28) r . Inthis case, the reconnection rate becomes simplyˆ v in ≈ (cid:18) S + Λ L (cid:19) / (cid:20) L r s α r ( η + Λ) (cid:21) , (29)where S = L/η (cid:29) v in ≈ S − / (whenΛ = 0). In the flat spacetime limit, r s →
0, we recoverthe reconnetion rates for a special relativistic pair plas-mas ˆ v in ≈ (1 /S + Λ /L ) / studied in Ref. [29].The physical mechanism for the increase of the recon-nection rate due to gravity is straightforward to under-stand. The gravitational force (due to gradients of α )at the inflow point is along the radial direction at anangle θ ≈ π/ − δ/ (2 r ). This is the force proportionalto ˆ J ( r s /αr ) | in that appears in Eq. (22). On the otherhand, the gravitational force that the plasma experiencesat the X –point is also along the radial direction but nowat an angle θ = π/
2. Anew, this force is proportionalto the term ˆ J ( r s /αr ) | X in Eq. (23). These two gra-dient forces point in radial direction at different angles,implying the existence of a net force antiparallel to the θ –direction, along the plane of the reconnection layer.Therefore, the net force pushes the plasma toward the X –point, producing an increase of the reconnection rate.In case in which the difference of gravitational forcesbetween the inflow and X points is neglected, then theplasma can be considered as quasi–neutral, with ˆ J = 0.This is the case of the analyses presented Refs. [30, 39],where quasi–neutral plasma were studied around Kerrblack holes, and only the curvature due to spacetime ro-tation was considered. However, if the most general case for the simplest gravitational effect produced by any com-pact object is considered into the study of magnetic re-connection in the surrounding plasma, a deviation fromquasi–neutrality is expected.Finally, the reconnection rate (29) explicity display theimportance of taking into consideration the collisionlesseffects. Those effects are the ones coupled to gravity. Inparticular, the difference of the reconnection rate (29)in the limit S → ∞ and its flat spacetime counterpartˆ v in ≈ (cid:112) Λ /L , is proportional toˆ v in (cid:112) Λ /L − (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) S →∞ ∝ (cid:18) hm − n (cid:19) / (cid:18) d e (cid:19) (cid:18) Lr (cid:19) (cid:18) r s r (cid:19) , (30)for pair plasmas, and proportional toˆ v in (cid:112) Λ /L − (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) S →∞ ∝ (cid:18) hm + n (cid:19) / (cid:18) d e (cid:19) (cid:18) Lr (cid:19) (cid:18) r s r (cid:19) , (31)for ion–electron plasmas (here d e = λ e /L is the dimen-sionless electron inertial length, with λ e indicating theelectron skin depth). Both results show that reconnectionrates are larger in plasmas around Schwarzschild blackholes, depending on the size of the black hole ∝ r s /r ,and on the geometry of the current sheet ∝ L/r . Nev-ertheless, the reconnection rate for pair plasmas is largeraccording to the fact that positrons contribute as theelectrons to the effective relativitistic collisionless resis-tivity Λ.The presented results complete the theoretical analysisof magnetic reconnection in curved spacetime initiated inRefs. [30, 39]. In this way, we have shown that spacetimecurvature effects (gravitational pull or rotation) form anintrinsic part of magnetic reconnection processes in astro-physical plasmas around compact objects. Future high-resolution numerical simulations with general relativisticcodes should be able to extend the predictions of theanalytic theory to more complex scenarios, as asymmet-ric reconnection layers, strong field inhomogeneities inall three spatial directions, and non-steady reconnectionprocesses.
Acknowledgments
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