Is Cowling's anti-dynamo theorem valid. near a rotating black hole? Axel Brandenburg, Nordita, Blegdamsvej 17, DK-2100 Copenhagen. (January 17, 1996)

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1 Is Cowling's anti-dynamo theorem valid near a rotating black hole? Axel Brandenburg, Nordita, Blegdamsvej 17, DK-2100 Copenhagen (January 17, 1996) Abstract The kinematic evolution of axisymmetric magnetic and electric elds near a rotating black hole is investigated numerically in Kerr geometry. In the cases investigated it is found that a magnetic eld cannot be sustained against Ohmic dissipation. In at space, this result is known as Cowling's antidynamo theorem. No support is found for the possibility that a gravitomagnetic dynamo eect could lead to self-excited axisymmetric solutions. In practice, therefore, Cowling's anti-dynamo theorem, may still hold in Kerr geometry, although here the original proof can no longer be applied. 1 Introduction Axisymmetric magnetic elds can generally be decomposed into a poloidal eld and an azimuthal component. Dierential rotation can stretch poloidal eld and convert it into azimuthal eld. This gives an ecient amplication of the magnetic eld, but it requires the poloidal eld as a source. However, in strictly two-dimensional geometry, there is no corresponding source for the poloidal eld, which must eventually decay. The ow can only advect the poloidal eld around, but, because of axisymmetry, it cannot stretch or amplify the poloidal eld. This is known as Cowling's (1934) anti-dynamo theorem (see also Parker 1979). In Kerr geometry, near a rotating black hole, the situation is dierent. The hole's rotation drags all physical objects near it into orbital motion in the same direction as the hole rotates. Physical quantities, such as electric and magnetic elds, are measured by ducial observers (FIDOs), who are orbiting with angular velocity! relative to a rigid coordinate grid. This frame dragging eect acts in a similar way as ordinary dierential rotation in nonrelativistic hydromagnetics and leads to eld line stretching. However, the frame dragging eect stretches not only magnetic elds, but also electric elds. Khanna & Camenzind (1994, 1996; hereafter referred to as KC94 and KC96, respectively) pointed out that this could be important for Cowling's anti-dynamo theorem, because it leads to a back-reaction on the poloidal 1

2 magnetic eld: the poloidal electric eld that is induced by the azimuthal magnetic eld is stretched and converted into an azimuthal electric eld, which could then induce a new poloidal magnetic eld. Now, this new poloidal magnetic eld could either enhance or diminish the original poloidal magnetic eld. A simple sketch suggests that the original poloidal magnetic eld is in fact enhanced by this sequence of events; see Fig. 1. a b B p B t d c E t E p Figure 1: a: A poloidal magnetic eld pointing outwards in the northern hemisphere becomes stretched into a negative azimuthal magnetic eld. b: This azimuthal magnetic eld induces a poloidal electric eld that goes from low to high latitudes. c: This poloidal electric eld becomes stretched into a negative azimuthal electric eld at low latitudes and a positive electric eld at high latitudes. d: This toroidal electric eld induces a new poloidal magnetic eld that points outwards at mid-latitudes. If strong enough, this would enhance the original poloidal magnetic eld. In the chain of processes depicted above only changes in the azimuthal electric and magnetic elds by stretching are considered, but they also change directly by induction. In particular, the poloidal magnetic eld would induce a positive azimuthal electric eld that tends to oppose the azimuthal electric eld generated by the relativistic 3-step process of stretch{induce{stretch. Thus, it is not clear whether the relativistic stretching eects would be able to dominate over the ordinary induction eects to produce a self-excited dynamo (especially in the presence of additional eld line stretching by dierential rotation in a surrounding accretion disk). Based on numerical simulations, KC94 and KC96 suggested that a self-excited gravitomagnetic dynamo is indeed possible. However they presented only two dierent cases, and it remains therefore unclear for which parameters (e.g., electrical conductivity, angular momentum of the black hole, geometry of the accretion ow, etc.) this process would work. In addition, certain features of their model might give rise to concern: neglect of the Faraday displacement current, insucient numerical resolution near the horizon, and an initial magnetic eld that is (almost) frozen into a highly conducting corona. The purpose of this Letter is to investigate the possibility of dynamo action using an independent 2

3 numerical method. A highly nonuniform mesh is adopted to resolve the accumulation of elds near the horizon, the Faraday displacement current is retained in most cases, and dierent conductivity proles are examined. 2 The model and numerical method There are many papers on what is now called black hole electrodynamics (e.g., Znajek 1977, Thorne & Macdonald 1982, Macdonald & Suen 1986, Park & Vishniac 1989), a topic that has recently also reached the text book level (see Thorne, Price, & Macdonald 1986; Novikov & Frolov 1989). The standard procedure is to rewrite the covariant Maxwell equations in the form of evolution equations with respect to a universal time t. These equations have the familiar = $e ^(B r)!? cr (E); = $e ^(E r)! + cr (B)? 4j; (2) together with rb = 0 and re = 4 e, where e is the charge density. Axisymmetry has been assumed and Boyer-Lindquist coordinates are employed, r is the radial coordinate, is colatitude, is longitude, e ^ is the unit vector in the -direction, and the hats indicate the physically measurable components of a vector. The dierential operators are written in curvilinear coordinates with metric coecients g 1=2 rr, g 1=2 and g 1=2, that are functions of r and, which can be found in the publications mentioned above. E and B are the electric and magnetic elds, respectively, = (?g tt )?1=2 is the gravitational redshift factor,! = g t =g tt the angular velocity caused by the frame dragging eect, $ = g 1=2 is the circumference of a circle around the axis (divided by 2), and c is the speed of light. (From now on c = G = 1 is assumed.) The current density j is governed by Ohm's law, j = (E + v B) + 0 ev; with 0 e = e? v E; (3) where is the conductivity, v is the bulk velocity of the plasma, = (1? v 2 )?1=2 is the relativistic Lorentz factor, and 0 and are the charge densities in the rest frames of the plasma and the FIDO, respectively. In order to be consistent with KC94 and KC96, however, 0 will be omitted (except in one case, see Fig. 4). The elds E and B are split into poloidal and toroidal elds, E = E p + E t and B = B p + B t, where E t = E ^e ^, B t = B ^e ^, and B p = r (A ^e ^) guarantees that B is solenoidal. Equations (1) and (2) are then written in the ^ =?E ^; (4) = $B p r!? e ^ (r E p )? e ^ r E p ; (5) 3

4 @E ^ = $E p r! + e ^ (r B p )? D 2 A ^? 4j ^; p = r (B ^ e ^)? 4j p ; (7) where D 2 A ^ =?e ^ [r r (A ^e ^)] is the Stokes operator (in Boyer-Lindquist coordinates), and Ohm's law is written as h j p = E p + (v ^ e ^) B p + v p (B ^ i e ^) + 0 ev p ; (8) j ^ h = E ^ i + e ^ (v p B p ) + 0 e v ^: (9) (Note that the redshift factor has nothing to do with the alpha in mean eld dynamo theory.) Following the model assumptions made by KC96, the velocities v ^ and v p are calculated such that the angular momentum is 99% of its Keplerian value outside the marginally stable orbit r ms, and equal to the value at r ms for r < r ms, where free fall is assumed. The relevant formulae can be found in KC96 and Shapiro & Teukolsky (1983). The angular velocity is constant on spherical shells, and v p is conned to the equatorial plane using a Gaussian prole with a scale height of H = 0:5 M, simulating an accretion ow in a disk. For the conductivity the prescription of KC96 is adopted: (4)?1 = 0:2H 2 v ^R?1 e?z2 =H 2, where z = r cos and R = r sin. For comparison, models with uniform conductivity and no accretion ow are also computed. Equations (4){(9) are solved using a 3rd order timestep and 2nd order centered dierences for most of the terms. However, for stability reasons (to ensure that waves on the scale of the mesh can still propagate) the curl operations in Eqs. (5) and (7) had to be performed on a staggered mesh for E p. Furthermore, it is essential to use a nonuniform mesh that is linear at large radia, but logarithmic near the horizon. In the present scheme a tortoise coordinate has been used, x = r + r H ln(r=r H? 1); (10) where r H = M + (M 2? a 2 ) 1=2 is the outer horizon, M is the mass of the black hole, and a is its angular momentum per unit mass in terms of c 3 =G = 1. All r-derivatives are evaluated on a uniform grid in x and then multiplied by dx=dr = (1? r H =r)?1. The original mesh coordinate r(x) is obtained iteratively, r! r 0, with r 0 = x? r H ln(r=r H? 1) for x > 2r H ; r 0 = r H f1 + exp[(x? r)=r H ]g for x < 2r H : (11) Tortoise coordinates were used earlier for a Schwarzschild black hole (r H = 2M), in which case the Maxwell equations simplify considerably (Macdonald & Suen 1985). Nevertheless, even for a Kerr black hole these coordinates have not only the obvious advantage of resolving the region near the horizon properly, but they also relax the timestep criterion considerably, because near the horizon, where the 4

5 eld evolution is sluggish, the dierence in the group velocity between neighboring points is similar to the dierences further away, even though r changes by a factor 10 5 :::10 8. The boundary conditions for B and E are formulated at the \stretched" horizon (Macdonald & Suen 1985), i.e. a very small distance above the actual horizon where is still nite (in the present case, x min is varied between?20 and?12, and so min = 10?5 :::4 10?4 ). On the boundary electromagnetic waves travel into the horizon, so the Poynting vector points into the hole, i.e. E ^ = B ^; B ^ =?E ^; (12) where, in turn, B ^ (or rather A ^) and E ^ satisfy the condition for an outgoing ^ = (g rr g g A ^ ; ^ = (g rr g 1=2 ^ E : (14) No explicit condition for E ^r is needed. On the outer boundary B ^ = B ^ = 0 is assumed (except in those cases where a perfect conductor is assumed). In some cases the eld parity is xed by suitable boundary conditions at the equator so that B has even or odd symmetry. The code was tested for = 0 by reproducing the relaxation of an initially radial magnetic eld to a steady vertical eld solution, that is known analytically (Wald 1974). For nite conductivity and with Keplerian dierential rotation the code was tested by computing dynamo action in Schwarzschild geometry giving results in qualitative agreement with expectations. The eective DC resistivity of the black hole of 377 ohms was veried numerically by applying an electric eld and measuring the resulting current (see Fig. 15 of Thorne et al. 1986). 1 For a parameter survey a resolution of meshpoints in the r and directions proved to be sucient; comparison with meshpoints gave very similar results. For those calculations where the Faraday displacement is neglected the E-eld is obtained from (8) and (9). 3 Results The results presented here are for M = 1, a = 0:998 (corresponding to maximal rotation, cf. Thorne 1974), and r max = 5. Case A. In order to compare with the model of KC96, an initially vertical magnetic eld is adopted is neglected. The purely dipole-like symmetry was broken by imposing an initial toroidal eld in the upper disk plane only, but this died out almost immediately. There is a strong increase of the magnetic eld due to the shear between the disk and the corona, see Fig. 2. Note that at all times the 1 Note that KC96 erroneously adopted an enhanced magnetic diusion on the stretched horizon to mimic the 377 ohms. 5

6 eld remains smooth and there are no ripples of the sort found in KC96. As time goes on, the eld in the disk plane is advected into the hole and does not accumulate near the horizon; this is unlike the behavior in KC96. Figure 2: The poloidal eld (contours of $A ^) for three dierent times [in diusion times t d t=(4 0 ), where 4 0 = 42:6 min(4)]. The shaded areas indicate the extent of the hole (r H = 1:06) and the disk (r ms = 1:237, H = 0:5) = 0. Case B. In the previous case the eld was frozen into a perfectly conducting corona and acted like an imposed eld. This is unsuitable for the study of dynamo action. Therefore, the initial eld is now conned to the disk. In order to allow for mixed parity, the initial eld is zero in the lower disk plane. The eld increased initially due to shear, but at later times it decayed due to dissipation or because it was simply absorbed by the black hole; see Fig. 3. Thus, there is no self-excited dynamo. Also, the eld 6

7 parity became quadrupole like, i.e. the toroidal eld has the same sign above and below the disk plane, thus minimizing Ohmic diusion. Figure 3: The poloidal eld at dierent times in the case without imposed magnetic eld. Dotted contours indicate opposite orientation. The eld dies out and is absorbed by the black hole = 0. Case C. In order to understand why the gravitomagnetic dynamo eect does not operate in the present case, the angular velocity! of the frame dragging was increased articially above its physical value! phys. This computational device helps to locate the dynamo eect in parameter space. In the absence of a radial accretion ow and for uniform conductivity there is a self-excited solution for! crit =! phys 2:2; see Fig. 4a. Radial accretion impedes dynamo action (! crit =! phys 3:1) but the value and prole of has very little eect on the threshold. Even for a very high angular momentum of the black hole 7

8 (a = 0: )! crit =! phys is well above unity ( 1:9); see Fig. 4b. This suggests that the threshold for dynamo action cannot be attained for any physically reasonable input parameters. Figure 4: a: Growth rate versus!=! phys. The angular momentum density L of the disk is +99%, 0, and?99% of its Keplerian value, and 4 = 20. Comparison is made with 4 = 50 and 100 (L > 0), is either included or neglected (for 4 = 50). The dierence between dipole-like and quadrupolelike solutions is negligible, and so is the inclusion of the 0 e term. b: Inverse threshold (! crit =! phys )?1 for dierent values of a meshpoints. In Kerr geometry, Cowling's proof cannot be applied: the toroidal current induced at the neutral O-type line could in principle be supported by the E p r! term. In order that je ^j does not vanish one has to require ^) 2 = 0, and, from (6), E ^E p r! > 0. Inspection of solutions with! =! phys showed, however, that this is not the case. This is because the negative toroidal electric eld in Fig. 1d is dominated by the positive electric eld induced by B p. Therefore E ^E p > 0 in the equatorial plane and, because r r! < 0, E ^E p r! < 0. (On the other hand, for!! crit the sign of E ^E p r! is indeed positive, as expected.) 8

9 4 Conclusions The present calculations suggest that the gravitomagnetic dynamo eect does not lead to self-excited solutions in the presence of an accretion ow with Keplerian rotation. These results are at variance with those of KC94 and KC96. This could possibly be explained by an incorrect boundary condition on the stretched horizon used by KC96 (see footnote in Sect. 2). The neglect of the Faraday displacement current and the charge density proves to be a good approximation in the cases considered here. Although axisymmetric dynamo action seems not to occur, it is quite clear however that accretion disks will always be magnetized by three-dimensional dynamo action (cf. Brandenburg et al. 1995). Acknowledgements AB thanks Ramon Khanna, David Moss and Ulf Torkelsson for illuminating discussions and comments on the manuscript. References Brandenburg, A., Nordlund, A., Stein, R. F., & Torkelsson, U Dynamo generated turbulence and large scale magnetic elds in a Keplerian shear ow. Astrophys. J. 446, Cowling, T. G The magnetic eld of sunspots. Monthly Notices Roy. Astron. Soc. 94, Khanna, R. & Camenzind, M The gravitomagnetic dynamo eect in accretion disks of rotating black holes. Astrophys. J. Letters 435, L129-L132. (KC94) Khanna, R. & Camenzind, M The! dynamo in accretion disks of rotating black holes. Astron. Astrophys. (in press). (KC96) Macdonald, D. A. & Suen, W.-M Membrane viewpoint on black holes: dynamical electromagnetic elds near the horizon. Phys. Rev. D 32, Macdonald, D. A. & Thorne, K. S Black-hole electrodynamics: an absolute-space/universal-time formulation. Monthly Notices Roy. Astron. Soc. 198, Novikov, I. D. & Frolov, V. P Physics of black holes. Dordrecht: Kluwer Academic Publishers. Park, S. J. & Vishniac, E. T An axisymmetric, nonstationary model of the central engine in an active galactic nucleus: I. Black hole electrodynamics. Astrophys. J. 337, Parker, E. N Cosmical Magnetic Fields. Clarendon Press, Oxford. Shapiro, S. A. & Teukolsky, S. L Black holes, White Dwarfs and Neutron Stars. Wiley & Sones, New York. Thorne, K. S Disk-accretion onto a black hole. II. Evolution of the hole. Astrophys. J. 191, Thorne, K. S., Price, R. H., & Macdonald, D. A Black holes: the membrane paradigm. New Haven: Yale University Press. 9

10 Wald, R. M Black hole in a uniform magnetic eld. Phys. Rev. D 10, Znajek, R. L Black hole electrodynamics and the Carter tretrad. Monthly Notices Roy. Astron. Soc. 179,