kerr geodesic

ar X iv :g r-q c/ 07 02 12 3v 2 1 1 Ju n 20 09 Kerr Geodesics, the Penrose Process and Jet Collimation by a Black Hole J. Gariel1, M.A.H. MacCallum2, G. Marcilhacy1 and N.O. Santos1,2,3 1LERMA-PVI, Universite´ Paris 06, Observatoire de Paris, CNRS, 3 rue Galile´e, Ivry sur Seine 94200, France. 2School of Mathematical Sciences, Queen Mary, University of London, Mile End Road, London E1 4NS, U.K. 3Laborato´rio Nacional de Computac¸a˜o Cient´ıfica, 25651-070 Petro´polis RJ, Brazil. (Dated: June 11, 2009) We re-examine the possibility that astrophysical jet collimation may arise from the geometry of rotating black holes and the presence of high-energy particles resulting from a Penrose (or similar) process, without the help of magnetic fields. Our analysis uses Weyl coordinates, which are better adapted to the desired shape of the jets. It shows that such a collimated jet would consist of particles with (almost) no conserved axial angular momentum. We therefore give a detailed study of these geodesics. Among them are a set of perfectly collimated geodesics whose cross-sectional radius is of order a/M . We discuss the significance of the results for jet collimation. PACS numbers: 04.20.-q, 04.25.-g, 04.40.Dg, 95.30.Sf Keywords: black hole physics; relativity; galaxies: jets; accretion; accretion discs I. INTRODUCTION There is increasing evidence [1] that a single mech- anism is at work in the production and collimation of various very energetic observed jets, such as those in gamma ray bursts (GRB) [2, 3, 4], and jets ejected from active galactic nuclei (AGN) [5] and from microquasars [6, 7]. Here we limit ourselves to jets produced by a black hole (BH) type core. The most often invoked pro- cess is the Blandford-Znajek [8] or some closely similar mechanism (e.g. [9, 10, 11]) in the framework of mag- netohydrodynamics, always requiring a magnetic field. However such mechanisms are limited to charged parti- cles, and would be inefficient for neutral particles (neu- trons, neutrinos and photons), which are currently the presumed antecedents of very thin and long duration GRB [4]. Moreover, even for charged particles, some questions persist (see for instance the conclusion of [12]). Finally, while the observations of synchrotron radiation prove the presence of magnetic fields, they do not prove that those fields alone cause the collimation: magnetic mechanisms may be only a part of a more unified mecha- nism for explaining the origin and collimation of powerful jets, see [13] p. 234 and section 5, and, in particular, for collimation of jets from AGN to subparsec scales, see [14]. Considering this background, it is worthwhile looking for other types of model to explain the origin and struc- ture of jets. Other models based on a purely general relativistic origin for jets have been considered. A sim- ple model was obtained by [15] by assuming the centres of galaxies are described by a cylindrical rotating dust. That paper showed that confinement occurs in the radial motion of test particles while the particles are acceler- ated in the axial direction thus producing jets. Another relativistic model was put forward in [16]. This showed that the sign of the proper acceleration of test particles near the axis of symmetry of quasi-spherical objects and close to the horizon can change. Such an outward accel- eration, that can be very big, might cause the production of jets. However, these models show a powerful gravitational effect of repulsion only near the axis, and are built in the framework of axisymmetric stationary metrics which do not have an asymptotic behaviour compatible with possible far away observations. So we want to explore the more realistic rotating black hole, i.e. Kerr, metrics instead. We thus address here the issue of whether it is possible, at least in principle (i.e. theoretically) to obtain a very energetic and perfectly collimated jet in a Kerr black hole spacetime without making use of magnetic fields. Other authors (see [12, 17, 18, 19] and references therein) have made related studies to which we refer below. Most such authors agree that the strong gravitational field gener- ated by rotating BHs is essential to understanding the origin of jets, or more precisely that the jet originates from a Penrose-like process [12, 20] in the ergosphere of the BH; collimation may also arise from the gravitational field and that is the main topic in this paper. Our work can therefore be considered as covering the whole class of models in which particles coming from the ergosphere form a jet collimated by the geometry. Al- though a complete model of an individual jet would re- quire use of detailed models of particle interactions inside the ergosphere, such as that given by [12], we show that thin and very long and energetic jets, with some generic features, can be produced in this way. In particular the presence of a characteristic radius, of the size of the ergo- sphere, around which one would find the most energetic particles, might be observationally testable. From a strictly general relativistic point of view, test particles in vacuum (here, a Kerr spacetime) follow geodesics; this applies to both charged and uncharged particles, although, of course, in an electrovacuum space- time, such as Kerr-Newman, charged particles would fol- low accelerated trajectories, not geodesics. Thus, in Kerr fields, what produces an eventual collimation for test par- ticles, or not, is the form of the resulting geodesics. Hence we discuss here the possibilities of forming an outgoing jet of collimated geodesics followed by particles arising from 2 a Penrose-like process inside the ergosphere of a Kerr BH. We show that it is possible in principle to obtain such a jet from a purely gravitational model, but it would re- quire the “Penrose process” to produce a suitable, and rather special, distribution of outgoing particles. The model is based on the following considerations. Most studies of geodesics, e.g. [21], employ generalized spherical, i.e. Boyer-Lindquist, coordinates. We trans- form to Weyl coordinates, which are generalized cylin- drical coordinates, and are more appropriate, as we shall see, for interpreting the collimated jets. We consider test particles moving in the axisymmet- ric stationary gravitational field produced by the Kerr spacetime, whose geodesic equations, as projected into a meridional plane, are known [21]. Our study is restricted to massive test particles, moving on timelike geodesics, but of course massless test particles on null geodesics could be the subject of a similar study. (Incidentally the compendium of [22] shows that analytic studies of gen- eral timelike geodesics have been much less frequent than detailed studies of more restricted problems.) For particles outgoing from the ergosphere of the Kerr BH we examine their asymptotic behaviour. Among the geodesic particles incoming to the ergosphere, we discuss only the ones coming from infinity parallel to the equa- torial plane, because these are in practice the particles stemming from the accretion disk. We show that only those with a small impact parameter are of high enough energy to provide energetic outgoing particles. In the ergosphere, a Penrose-like process can occur. In the original Penrose process, an incoming particle decays into two parts inside the ergosphere. It could also decay into more than two parts, or undergo a collision with an- other particle in this region, or give rise to pair creation (e−, e+) from incident photons which would follow null geodesics. The different possible cases do not affect our considerations, and that is why we do not study them here, although the distribution function of outgoing par- ticles would be required in a more detailed model of the type discussed, in particular to explain why only parti- cles with low angular momentum and not diverging from the rotation axis are produced. For detailed studies see [12, 18, 23]. After a decay, one (or more) of the parti- cles produced crosses the event horizon and irreversibly plunges into the BH, while a second particle arising from the decay can be ejected out of the ergosphere following a geodesic towards infinity. This outgoing particle could be ejected so that asymptotically it runs parallel to the axis of symmetry, but we do not discuss only such particles. In our model there is no appeal to electromagnetic forces to explain the ejection or the collimation of jets, though the particles therein may themselves be charged. The gravitational field suffices, in the case of strong fields in general relativity, which is the case near the Kerr BH, provided the ergosphere produces particles of appropri- ate energy and initial velocity. The gravitomagnetic part of the gravitational field then provides the collimation. Hence, our model is, in this respect, simpler than the standard model of [8], and is in accordance with the anal- ysis given in [12]. The paper starts with a study of Kerr geodesics in Weyl coordinates in section II; the next section studies the asymptotic behaviour of geodesics of outgoing parti- cles with Lz = 0; section IV analyses incoming particles stemming from the accretion; a sample Penrose process and the plotting of geodesics are presented in section V; and finally we discuss in section VI the significance of our results for jets. In the conclusion, we succinctly summa- rize our main results and evoke some perspectives. II. KERR GEODESICS We start from the projection in a meridional plane φ = constant of the Kerr geodesics in Boyer-Lindquist spher- ical coordinates r¯, θ and φ. The metric is ds2 = (r¯2 + a2 cos2 θ) ( dr¯2 r¯2 − 2Mr¯ + a2 + dθ 2 ) + sin2 θ (r¯2 + a2 cos2 θ) ( adt− (r¯2 + a2)dϕ)2 (1) − (r¯ 2 − 2Mr¯ + a2) (r¯2 + a2 cos2 θ) ( dt− a sin2 θdϕ)2 . where M and Ma are, respectively, the mass and the angular momentum of the source, and we have taken units such that c = 1 = G where G is Newton’s con- stant of gravitation. The ‘radial’ coordinate in (1) has been named r¯ because it is more convenient for us to use the rescaled coordinate r = r¯/M [24]. The projected timelike geodesic equations are then M2r˙2 = (a4r 4 + a3r 3 + a2r 2 + a1r + a0)[ r2 + ( a M )2 cos2 θ ]2 , (2) M2θ˙2 = b4 cos 4 θ + b2 cos 2 θ + b0 (1− cos2 θ) [ r2 + ( a M )2 cos2 θ ]2 , (3) with coefficients a0 = −a 2Q M4 , (4) a1 = 2 M2 [ (aE − Lz)2 +Q ] , (5) a2 = 1 M2 [ a2(E2 − 1)− L2z −Q ] , (6) a3 = 2, (7) a4 = E 2 − 1, (8) and b0 = Q M2 , (9) b2 = 1 M2 [ a2(E2 − 1)− L2z −Q ] = a2, (10) b4 = − ( a M )2 (E2 − 1); (11) 3 where the dot stands for differentiation with respect to an affine parameter and E, Lz and Q are constants. Here Chandrasekhar’s δ1 has been set to 1, its value for time- like curves. Assuming that the affine parameter is proper time τ along the geodesics, then these equations implic- itly assume a unit mass for the test particle [44], so that E and Lz have the usual significance of total energy and angular momentum about the z-axis, and Q is the cor- responding Carter constant (which, as described in [25] for example, arises from a Killing tensor of the met- ric, while E and Lz arise from Killing vectors). With this understanding, E, Lz, and Q have the dimensions of Mass, Mass2 and Mass4 respectively, in geometrized units, while δ1, though 1 numerically, has dimensions Mass2, as do all the ai and bi. In this paper we con- sider only particles on unbound geodesics with E ≥ 1. (For the conditions for existence of a turning point, giv- ing bound geodesics, which are related to the parameter values for associated circular orbits, see [12, 18, 21].) The dimensionless Weyl cylindrical coordinates, in multiples of geometrical units of mass M , are given by ρ = [ (r − 1)2 −A]1/2 sin θ, z = (r − 1) cos θ, (12) where A = 1− ( a M )2 . (13) From (12) we have the inverse transformation r = α+ 1, (14) sin θ = ρ (α2 −A)1/2 , cos θ = z α , (15) with α = 1 2 ( [ρ2 + (z + √ A)2]1/2 + [ρ2 + (z − √ A)2]1/2 ) . (16) Here we have assumed A ≥ 0, and taken the root of the second degree equation obtained from (15) for the function α2(ρ, z) that allows the extreme black hole limit A = 0. The other root, in this limit, is the constant α = 0. The equation (16) shows that in the (ρ, z) plane the curves of constant α (constant r) are ellipses with semi- major axis α and eccentricity e = √ A/α: for large α, these approximate circles. Note that ρ = 0 consists of the rotation axis θ = 0 or π together with the ergosphere surface. Now, with (14) and (15) we can write the geodesics (2) and (3) in terms of ρ and z coordinates, producing the following autonomous system of first order equations Mρ˙ = Pα3ρ α2 −A + S(α2 −A)z αρ (α+ 1)2α2 + ( a M )2 z2 , (17) Mz˙ = (Pz − S)α [ (α+ 1)2α2 + ( a M )2 z2 ] −1 ,(18) where P = ǫ1 [ a4(α + 1) 4 + a3(α+ 1) 3 (19) + a2(α+ 1) 2 + a1(α+ 1) + a0 ]1/2 , S = ǫ1ǫ2(b4z 4 + b2α 2z2 + b0α 4)1/2, (20) and ǫi = ±1 for i = 1, 2: ǫ1 indicates whether the geodesic is incoming or outgoing in r (i.e. the sign of r˙), while ǫ1ǫ2 indicates whether θ is increasing or decreasing. Note we always mean the non-negative square roots to be taken. The ratio between the first order differential equations (17) and (18) yields the special characteristic equation of this system of equations dz dρ = (|P |z − ǫ2|S|)(α2 −A)α2ρ |P |α4ρ2 + ǫ2|S|(α2 −A)2z . (21) We restrict our study to the quadrant ρ > 0 and z > 0 in the projected meridional plane (orbits in fact spiral round in φ in general: this information is contained in the conserved Lz). The results for the other three quad- rants will follow by symmetry, although this symmetry does not imply that individual geodesics are symmet- ric with respect to the equatorial plane. From numeri- cal solutions of the geodesics we obtained asymmetrical geodesics, confirming the analysis in [12, 18]. Geodesics can also cross the polar axis, which would be represented by a reflection from ρ = 0 back into the quadrant. Geodesics going to or coming from the expected ac- cretion disk would, if the disk were thin, go to or from values of ρ much larger than z. In this limit (ρ≫ z and ρ≫ √ A) , we have α = ρ(1 +O(ρ−2)), (22) |P | = √a4ρ2(1 + k/ρ+O(ρ−2)), (23) where k = 2E2 − 1 E2 − 1 , (24) |S| = √ b0ρ 2(1 +O(ρ−2)), (25) and thus dz dρ ≈ z − ǫ2z1 + zk/ρ ρ(1 + k/ρ) +O(ρ−3), (26) where z1 = ( b0 a4 )1/2 = 1 M ( Q E2 − 1 )1/2 , (27) and we have to assume Q ≥ 0 to obtain a real z1 (the form of (3) makes it obvious that geodesics with Q < 0 are bounded away from the equatorial plane cos θ = 0, though those with very small |Q| could still satisfy ρ ≫ z). The truncated series development of dz/dρ now yields dz dρ ≈ 1 ρ ( z − ǫ2z1 + ǫ2kz1 ρ ) +O(ρ−3). (28) 4 If ǫ2 = −1 the curve crosses the equatorial plane and gives similar asymptotic behaviour in the second quad- rant, so we can take ǫ2 = 1. A thicker accretion disk would absorb or release parti- cles on geodesics with larger values of z/ρ, which might include particles with Q < 0. Geodesics in an axial jet would have z ≫ ρ. For this limit, we first observe that from (20) we have S ≈ ǫ2 [ (b0 + b2 + b4)z 4 + (2b0 + b2)ρ 2z2 ]1/2 +O(z−1), (29) where b0 + b2 + b4 = − ( Lz M )2 ≤ 0, (30) 2b0 + b2 = 1 M2 [ a2(E2 − 1)− L2z +Q ] . (31) Hence in this limit S is well defined and real for indef- initely small ρ/z only for Lz = 0. The geodesics obey- ing this restriction, imposed after similar reasoning, were studied by [17], but in BL or Kerr-Schild coordinates. Here we re-examine these geodesics in the more reveal- ing cylindrical coordinates. Before doing so, we may note that in contrast to geodesics with Lz 6= 0, geodesics with E2 > 1 and Lz = 0 may lie arbitrarily close to the polar axis [26]. For Lz 6= 0, the value of S2 at the axis is −z2L2z < 0 which is not al- lowed and thus there is some upper bound θ0 on θ. The value of S2 at θ = π/2 is b0α 2 so if Q < 0 there is also a lower bound θ1 on θ. III. GEODESICS WITH Lz = 0 We shall discuss unbounded (E2 > 1) outgoing geodesics. Corresponding incoming geodesics will follow the same curves in the opposite direction. For Lz = 0, S 2 factorizes as S2 = (α2 − z2)(Qα2 + a2(E2 − 1)z2)/M2. (32) Hence S can only be zero at the symmetry axis, where cos θ = 1, α = z and ρ = 0, or, if Q < 0, at some z/α = (|Q|/a2[E2−1])1/2 = cos θ1, say. Correspondingly θ˙ = 0 only at the axis, at θ = θ1 if Q < 0, and at α→∞. Thus for Lz = 0 and Q < 0, geodesics which initially have θ˙ < 0 will become asymptotic to θ = θ1. The angle may be narrow if a2(E2 − 1)− |Q| ≪ |Q|, (33) and then θ1 ≪ 1. Such geodesics may provide a conical jet, as discussed later. Our other polynomial, P 2, can be written as P 2 = (E2 − 1) ( r4 + a2 M2 r2 + 2a2 M2 r ) (34) +2r3 + 2 a2 M2 r − Q M2 ( r2 − 2r + a 2 M2 ) . From this form it easily follows that any unbound geodesic (E2 > 1) with Lz = 0 has at most one turn- ing point in r (i.e. value such that r˙ = 0) and this, if it exists, lies inside the horizon (and a fortiori inside the ergosphere) [27]. The argument is very simple. If E2 > 1 and Q ≤ 0, then P 2 is strictly positive for all r > 0. If Q > 0, P 2 is negative at r = 0 but positive at the outer black hole horizon (where r2−2r+a2/M2 = 0), so its one zero lies inside the black hole. This implies that unbounded outgoing geodesics followed by particles with Lz = 0 must come from the ergosphere. Correspond- ingly, geodesics incoming from infinity with Lz = 0 will fall into the ergosphere. Although there are no turning points of r, one can have turning points of ρ, if ǫ2 = −1. Such turning points are solutions of the equation D(ρ, z) ≡ |P |α4ρ2 − |S|(α2 −A)2z = 0 (35) where D(ρ, z) is the denominator of (21) with ǫ2 = −1. At each of these turning points (ρ2, z2), dz/dρ → ∞, which means that the geodesics have a vertical tangent (parallel to the z-axis). Before reaching the turning point, these geodesics have dρ/dz > 0, and, at any z, dz/dρ > dz/dρ|D, where dz/dρ|D is the slope of the curve (35), and afterwards they have dρ/dz < 0, imply- ing that they subsequently cross the axis. They will then cross the curve (35) again but from above in the (ρ, z) plane and hence with dz/dρ < dz/dρ|D, and afterwards stay in the region outside (35). For outgoing geodesics outside (35) which reach points at large z and ρ (≫ √ A), then unless the ratio of z to ρ is very large (the case which we discuss next) or very small, approximating (21) gives dz/dρ ≈ z/ρ, so all such geodesics approximate ρ = Cz for suitable C, regardless of the sign of ǫ2. In the limit z ≫ ρ and z ≫ √A, α = z(1 +O(z−2)), (36) |P | = √a4 z2(1 + k/z +O(z−2)), (37) |S| = √ 2b0 + b2 ρz(1 +O(z −2)), (38) so the equation (21) can be approximated by dz dρ = z(1 + k/z) ρ(1 + k/z) + ǫ2ρ1 + O ( 1 ρz ) , (39) where ρ1 = ( 2b0 + b2 a4 )1/2 = ρe [ 1 + Q a2(E2 − 1) ]1/2 ,(40) and ρe ≡ a/M . Here ρ1 is real if a2(E2− 1)+Q > 0 but we see from (32) that for S to be real near the axis, this condition must be satisfied. In Figure 1 we show a plot of the values of dz/dρ, using (21), for ε2 = −1, with the parameters M = 1, a = 1/2, E = 104, Q = −9×103. The only asymptotes are parallel to the z axis at ρ = ρ1 as expected from (40). 5 0.2 0.4 0.6 0.8 · 0 250000 500000 750000 1·106 z -1·107 0 1·107 dzd· 250000 500000 750000 1·106 FIG. 1: Plot of the surface dz/dρ = f(ρ, z) given by equation (21) for an outgoing particle with the parameters E = 104, Lz = 0 and Q = −9 × 10 3 for a black hole with parameters M = 1 and a = 0.5, in the case where ǫ2 = −1. The points where dz/dρ → ±∞ correspond to the asymptotes given by the equation (40), ρ = ρ1 = 0.49991. We also plot in Figure 2 a set of such outgoing geodesics obeying (21), for the same values of the pa- rameters of the BH (a = 1/2, M = 1) and of the particle (Lz = 0, Q = −2.2 × 105, E = 2.103, so