Magnetic neutral points and electric lines of force in strong gravity of a rotating black hole
aa r X i v : . [ a s t r o - ph . H E ] M a r Magnetic neutral points and electric lines offorce in strong gravity of a rotating black hole ∗ V. Karas, O. Kop´aˇcek, and D. KunneriathAstronomical Institute, Academy of Sciences,Boˇcn´ı II 1401, CZ-14100 Prague, Czech RepublicE-mail: [email protected] 1, 2013
Abstract
Magnetic field can be amplified and twisted near a supermassiveblack hole residing in a galactic nucleus. At the same time mag-netic null points develop near the horizon. We examine a large-scaleoblique magnetic field near a rotating (Kerr) black hole as an originof magnetic layers, where the field direction changes abruptly in theergosphere region. In consequence of this, magnetic null points candevelop by purely geometrical effects of the strong gravitational fieldand the frame-dragging mechanism. We identify magnetic nulls aspossible sites of magnetic reconnection and suggest that particles maybe accelerated efficiently by the electric component. The situation wediscuss is relevant for starving nuclei of some galaxies which exhibitepisodic accretion events, namely, Sagittarius A* black hole in ourGalaxy.
Most galaxies including the Milky Way are believed to host a supermassiveblack hole in the centre [13, 31]. The black hole is embedded in a surrounding ∗ To appear in International Journal of Astronomy and Astrophysics × solar masses resides [16, 34]. Given the compact size of the blackhole horizon, the magnetic field generated by external sources appears to beeffectively uniform on the length-scale a several gravitational radii. However,the field intensity is uncertain. On large scales (i.e., greatly exceeding thegravitational radius) the field should not go beyond a few milligauss [35, 36],while on medium scales ( ≃ r g ) it might be amplified to tens of gauss[11, 14, 30].Magnetic reconnection occurs when the magnetic field lines change theirconnectivity [40, 41]. This happens as topologically distinct regions approacheach other. The standard setup involves the violation of the ideal MHDapproximation just on the boundary between neighbouring magnetic domainswhere the field direction changes rapidly.One can ask if the BH proximity creates conditions favourable to incitereconnection, leading to plasma heating and particle acceleration. This couldgenerate the flaring activity [2, 17, 39]. The typical rise time of Sgr A* flareslasts several minutes, i.e. a fraction of the orbital period near the innermoststable circular orbit (ISCO). Variety of processes have been considered forSgr A* radiation. For example, Liu et al. [27, 28] propose that stochasticacceleration of electrons in the turbulent magnetic field is responsible for thesubmillimeter emission within ≃ r g . Also, Ballantyne et al. [3] elaborate onthe idea that Sgr A* may be an important site for particle acceleration. We2ill discuss a complementary scheme of a magnetically dominated system.We show that antiparallel field lines are brought into mutual contact,within the low-density conditions, by the frame-dragging (gravito-magnetic)action of the rotating BH. One expects that a dissipation region developswhere the magnetic field structure changes abruptly across a separatrix curve,so these spots, occurring just above the ISCO, can act as places where par-ticles are energised [11]. Ingredients necessary for this scenario to work – i.e.an ordered magnetic field due to external sources plus the diluted plasmaenvironment of disturbed stars – are naturally present near Sgr A* blackhole. In addition to our previous work [19] we show also the electric fieldthreading the magnetic nulls. Thereby the electric component is capable ofaccelerating the charged matter once it is injected in the area of the magneticnull. In underluminous galactic nuclei, it is likely that plasma is only episodicallyinjected into the central region, perhaps by passing stars gradually sinkingdown to the BH. We model the gravitational field by Kerr metric [33]; wewill use geometrical units with c = G = 1 and scale all quantities by the BHmass, M . The gravitational radius is r g = c − GM ≈ . × − M pc, andthe corresponding light-crossing time-scale t g = c − GM ≈ M sec, where M ≡ M/ (10 M ⊙ ).Hereafter we use spheroidal coordinates with µ ≡ cos θ , σ ≡ sin θ , andthe metric line element in the form d s = − ∆Σ A − d t + Σ∆ − d r + Σ d θ + A Σ − σ (d φ − ω d t ) , where a denotes the specific angular momentum, | a | ≤ a = 0 is for a non-rotating BH, while a = ± r − r + a , Σ = r + a µ , A = ( r + a ) − ∆ a σ , and ω = 2 ar/A . The outer horizon, r ≡ r + ( a ), is where ∆ = 0,whereas the ISCO ranges between r + ( a ) ≤ r isco ( a ) ≤
6. The presence ofterms ∝ ω in Kerr metric indicates that frame dragging operates and affectsthe motion of particles and the structure of fields [20, 32].The influence of general relativistic frame-dragging on accreted particlesresembles the effect of a rotating viscous medium: it forces the particles toshare the rotational motion of the central body. Similarly, it also affects the3 y −3 −2 −1 0 1 2 3−2.5−2−1.5−1−0.500.511.522.5 x y Figure 1: Magnetic field lines lying in the equatorial plane xy , perpendicularto the rotation axis of an extremely rotating BH (units are scale with ofgravitational radius r g ). Left panel: an asymptotically uniform magneticfield is directed along the x -axis at large radii and plotted with respect tothe physical frame of an orbiting star. The case of a co-rotating frame isshown ( a = 1). Right panel: example of the projection of the rotationinduced electric field, corresponding to the magnetic field in the left panel.magnetic field lines and generates the electric component. For the magneticfield, we might liken the impact of frame-dragging to the Parker spiral, de-scribing the shape of the (rotating) Sun’s magnetic field. In a way, the actionof the frame-dragging is reminiscent of neutral points arising due to the in-teraction of the interplanetary magnetic field with the Earth dipole [12], buthere the strong gravity plays a major role. Besides other contrasts, the influ-ence of the rotating BH grows overwhelmingly as one enters the ergosphericregion below r = 2 r g .The models of electromagnetic acceleration and collimation of jets havebeen greatly advanced during the last decade [15, 22, 23, 25]. These worksdemonstrate the astrophysical importance of BH rotation, but their setup issomewhat different from this paper. In particular, the field is typically as-sumed to be frozen into an equatorial accretion disc; the magnetic field linesare twisted by differential rotation of the disc plasma. On the other hand, thecase of Sgr A* is distinct, with only a tiny accretion rate, ˙ M ≤ − M ⊙ yr − [29], although any firm estimates are currently not possible because of un-certainties in the accretion efficiency [13]. The standard disc is not presenthere; the gas chunks arrive episodically [9, 10]. In these circumstances one4 y * −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8−0.6−0.4−0.200.20.40.6 x* y * Figure 2: The field structure as in the left panel of Fig. 1, now using the newradial coordinate r ∗ . The magnetic field (left panel) and the correspondingelectric field (right panel) are shown.cannot expect the BH and the ordered magnetic field to have a common sym-metry axis (the Bardeen-Petterson effect does not operate due to the lack ofa steady accretion flow) neither that the black hole is resting in the centre.We assume that the electromagnetic field does not contribute to the sys-tem gravity, which is correct for every astrophysically realistic situation.Within a limited volume around the BH, typically of size ≃ (10 r g ) , themagnetic field lines have a structure resembling the asymptotically uniformfield. The electromagnetic field is a potential one and can be written as a su-perposition of two parts: the aligned component [21, 43], plus the asymptot-ically perpendicular field [6, 7]. The four-potential has been given explicitlyin terms of functions ψ ≡ φ + aδ − ln [( r − r + ) / ( r − r − )] is the Kerr ingoingangular coordinate, Ψ = r cos ψ − a sin ψ , δ = r + − r − , and r ± = 1 ± √ − a [7, 19]. It is exactly the variable ψ that determines the growing twist of mag-netic lines, which eventually leads to the formation of magnetic null pointsnear a rotating black hole. We can thus employ these expressions to drawlines of force.A particle can be accelerated by the equipartition field, acting along thedistance ℓ , to energy γ max ≃ q e ( B/ G ) ( ℓ/r g ) eV, where q e is in units ofthe elementary charge. Naturally, this rough estimate can be exceeded if anon-stationary field governs the acceleration process.The aligned vacuum field is gradually expelled out of the BH as its rota-tion increases [21]. Non-vacuum fields are more complicated and have been5 .4 0.5 0.6 0.7 0.8 0.90.911.11.21.31.4 x y Figure 3: Two examples of detailed magnetic structure surrounding a nullpoint, where the magnetic field vanishes in the black hole equatorial plane.The separatrix line is shown (solid line by red colour) in the right panel.studied in detail; e.g. Komissarov & McKinney [24] notice that high conduc-tivity of the medium changes the field expulsion properties of the horizon.On the other hand, only few aspects of the misaligned BH magnetic fieldshave been explored so far [37, 42]. This is because of an extra complexitycaused by the frame-dragging [1, 18]. What we put forward here is that theframe-dragging can actually contribute to the acceleration mechanisms.To obtain the physical components of the electromagnetic tensor, F =2 dA , we project it onto the local observer tetrad, e ( a ) [33]. The appropriatechoice of the projection tetrad is the one attached to a frame in Keplerianorbital motion, which exists for r ≥ r isco . Free circular motion can be stableat r ≥ r isco , otherwise the star has to spiral downwards while maintainingconstant angular momentum of l = l ( r isco ). The corresponding frame andthe field lines can be derived in a straightforward manner [19]. We interpretthe frame, quite naturally, as the one connected with an orbiting star. In theextremely rotating case, the circular motion of a star is possible all the waydown to r isco = r g . The dependence of the electromagnetic components onthe ψ angle then indicates the ever increasing effect of the frame draggingnear the horizon.The electric and magnetic intensities, measured by the physical observer,are: E ( a ) = e µ ( a ) F µν u ν , B ( a ) = e µ ( a ) F ∗ µν u ν , where u ν ≡ e ν ( t ) is the observer’sfour-velocity (the remaining three basis vectors can be conveniently chosenas space-like, mutually perpendicular vectors).6igure 1 shows the typical structure of the field lines representing anasymptotically transverse magnetic field ( B k = 0). Formation of the layersin the ergospheric region just above ISCO is an interesting and generic featureof the rotating spacetime. Two critical points can be seen occurring at radiiup to ≃ . r g for a = 1. Naturally, the co-rotating case ( a >
0) is notequivalent with the counter-rotating ( a <
0) one, but in both, a site ofoppositely directed field lines arises.An example of electric lines of force is plotted in the right panel of Fig. 1.We notice that the electric field along the magnetic lines does not vanish andis capable of accelerating charged particles. Figure 2 then reveals the shape ofthe near-horizon field lines in terms of the new radial variable, r ∗ ≡ − r g /r ,which we introduce to better resolve the complicated structure (Cartesian-type coordinates are used, x ∗ + y ∗ = r ∗ ).So far we have assumed zero translational motion of the BH with respectto the magnetic field. However, our approach can be generalised and theuniform motion can be taken into account by Lorentz boost of the fieldintensities E ( a ) , B ( a ) . To this end we notice that the Lorentz transform,when applied in the asymptotically distant region (i) changes the directionof the uniform magnetic field by an ϑ , and (ii) generates a new (uniform)electric component. The desired form of the electromagnetic test field near amoving black hole is thus written in a symbolic way, F ′ r →∞ = Λ( β ) T F Λ( β ),and obtained as superposition of the two parts combined together in the rightratio, i.e. the asymptotically uniform magnetic field, rotated into the desireddirection, and the solution for the asymptotically uniform electric field. Thelatter one can be found by applying the dual transform to Wald’s field.This way we can explore also the magnetic fields near a drifting BH,in which case the additional electric component arises. As an example, inthe right panel of Fig. 2 we assumed constant velocity β y = − .
99 directedalong the y -axis. A null point (again shown in the orbiting frame) arisesjust above ISCO, located at x + y = 1 for a = 1. Finally, Figure 3 showsan enlarged detail of the magnetic structure around the region where themagnetic null point develops. The two examples differ by the magnitude ofthe translatory boost of the black hole which affects the exact location of thenull point, although it always appears very close to the horizon and requiresrapid rotation of the black hole.Finally, we remind the reader that the shape of lines of force obviouslydepends on the observer’s frame with respect to which the lines are plot-ted. Figure 4 compares three distinct examples. The there panels of the7 x [M] y [ M ] −3 −2 −1 0 1 2 3 4−4−3−2−10123 x [M] y [ M ] −3 −2 −1 0 1 2 3−3−2−10123 x [M] y [ M ] −5 0 5−5−4−3−2−1012345 x [M] y [ M ] −3 −2 −1 0 1 2 3 4−4−3−2−101234 x [M] y [ M ] −2 −1 0 1 2−2.5−2−1.5−1−0.500.511.522.5 x [M] y [ M ] −5 0 5−5−4−3−2−1012345 x [M] y [ M ] −3 −2 −1 0 1 2 3 4−4−3−2−101234 x [M] y [ M ] −2 −1 0 1 2−2.5−2−1.5−1−0.500.511.522.5 x [M] y [ M ] −0.6 −0.4 −0.2 0 0.2 0.4 0.6−0.6−0.4−0.200.20.40.6 X Y −0.6 −0.4 −0.2 0 0.2 0.4 0.6−0.6−0.4−0.200.20.40.6 X Y −0.6 −0.4 −0.2 0 0.2 0.4 0.6−0.6−0.4−0.200.20.40.6 X Y Figure 4: Equatorial behaviour of the magnetic field with the purely per-pendicular magnetic asymptotes ( B x = 0, B y = B z = 0) for three distincttetrads (see the main text for details).8 X Y −0.5 0 0.5−0.8−0.6−0.4−0.200.20.40.60.8 X −0.5 0 0.5−0.8−0.6−0.4−0.200.20.40.60.8 X −0.5 0 0.5−0.8−0.6−0.4−0.200.20.40.60.8 X Y −0.5 0 0.5−0.8−0.6−0.4−0.200.20.40.60.8 X −0.5 0 0.5−0.8−0.6−0.4−0.200.20.40.60.8 X −0.5 0 0.5−0.8−0.6−0.4−0.200.20.40.60.8 X Y −0.5 0 0.5−0.8−0.6−0.4−0.200.20.40.60.8 X −0.5 0 0.5−0.8−0.6−0.4−0.200.20.40.60.8 X −0.5 0 0.5−0.8−0.6−0.4−0.200.20.40.60.8 X Y −0.5 0 0.5−0.8−0.6−0.4−0.200.20.40.60.8 X −0.5 0 0.5−0.8−0.6−0.4−0.200.20.40.60.8 X Figure 5: Equatorial behaviour of the electric field projection. Case of theextreme Kerr black hole drifting with respect to the aligned magnetic field.9op row present the lines of magnetic force with respect to the free-fallingtetrad (observer’s angular momentum vanishes). Further down, the secondrow corresponds to the case of a free-falling observer (below the innermoststable circular orbit) in combination with a freely orbiting observer (Kep-lerian prograde rotation) above it; likewise in the third row (for retrograderotation). The Schwarzschild limit a = 0 is shown in the first column, middlecolumn corresponds to spin a = 0 .
9, and the last one represents the extremecase, a = 1. In the bottom row the rescaled dimensionless radial coordinate R ≡ r − r + r is used in order to stretch the region close to the horizon (case ofextreme rotation).Complementary to magnetic field lines, Figure 5 shows the projection ofelectric lines of force induced by the presence of the magnetic componentby rotation and the linear motion of the black hole. The first (left) columnpresents the non-drifting case (zero boost velocity v i ); in the second we set v x = 0 . v y = 0; and in the third v x = 0 . v y = 0 .
5. Four distinct cases arecompared in the rows (top to bottom): zero-angular momentum observers,free-falling observers, co-rotating and free falling, and counter-rotating andfree-falling observers, respectively. We observe that in all these cases theneutral electric points develop as the drift is introduced. We stress that asthe original field is aligned ( B x = 0) we measure the latitudinal component E ( θ ) = 0 in the equatorial plane provided that v z = 0, so actually these plotspresent true shape of field lines.We notice that non-vanishing electric component passes through the mag-netic null, thereby accelerating electrically charged particles in this region.One expects reconnection to occur intermittently, as the plasma is injectedinto the dissipation region where the differently directed field lines approacheach other due to their interaction with a highly curved spacetime. Will thegravito-magnetic effect produce the same layered structure of the magneticfield also in the presence of non-negligible amount of plasma, or will thefield structure change entirely? Numerical simulations will be necessary tosee whether this mechanism can be part of a broader picture in astrophys-ically realistic situations, which vary wildly under different circumstances,and to determine the actual speed at which the process operates. We remarkthat, on the other end of analytical approximations, the solution for non-axisymmetric accretion of stiff adiabatic gas onto a rotating black hole alsoexhibits critical points near the horizon [38], so the conditions for magneticreconnection of the frozen-in magnetic field will be again fulfilled.10 Conclusions
We considered the influence of the black hole rotation acting onto the or-dered magnetic field in the physical frame of a star orbiting a black hole,or plunging down to it. If rotation is fast enough, the magnetic layers andthe corresponding null points exist just above the innermost stable circularorbit. Although we prescribed a special configuration of the electro-vacuummagnetic field and we considered only the test-field approximation, the pro-cess of warping the field lines is a general feature that should operate also inmore complicated settings: the frame-dragging is expected to take over anddetermine the field structure near the horizon. The layered structure of themagnetic field lines with neutral points suggest this should become a site ofparticle acceleration.The essential ingredients of the scheme described here are the rotatingblack hole and the oblique magnetic field into which the black hole is embed-ded. The interaction region is very near the horizon, representing, to ourknowledge, the acceleration site nearest to the black hole horizon among thevariety of mechanisms proposed so far. Even if we treated the problem ina very simplified scheme, the idea of geometrical effects of frame draggingcausing the acceleration of matter is very promising in the context of rapidlyrotating black hole inside nuclei of galaxis.We concentrated on the equatorial plane in which the transverse magneticfield lines reside, but this constrain was imposed only to keep graphs asclean as possible. Otherwise, the lack of symmetry complicates the situation.Plasma motions in the close vicinity of the Sgr A* black hole are currentlyinaccessible to direct observation. However, there are chances that the regionwill be resolved with future interferometers, such as GRAVITY in the near-infrared spectral band and the Event Horizon Telescope VLBI project insubmillimeter wavelengths. The resulting signatures in the light-curve areexpected to occur synchronous to the X-ray signal, with time-delays specificto the radiation mechanism.
Acknowledgments
We acknowledge the Czech Science Foundation (GA ˇCR 13-00070J) and Ger-man Forschungsgemeinschaft (DFG) collaboration project for support. Wethank an anonymous referee for helpful suggestions.11 eferenceseferences