Kerr Geodesics, the Penrose Process and Jet Collimation by a Black Hole

A&A 515, A15 (2010) Astronomy DOI: 10.1051/0004-6361/200913678 & c ESO 2010 Astrophysics

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

1 LERMA-UPMC, Université Pierre et Marie Curie, Observatoire de Paris, CNRS, UMR 8112, 3 rue Galilée, 94200 Ivry-sur-Seine, France e-mail: [email protected]; [email protected] 2 School of Mathematical Sciences, Queen Mary, University of London, Mile End Road, London E1 4NS, UK e-mail: [email protected] 3 Laboratório Nacional de Computação Cientíﬁca, 25651-070 Petrópolis RJ, Brazil e-mail: [email protected] Received 16 November 2009 / Accepted 10 February 2010

ABSTRACT

Aims. 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 process, without the help of magnetic ﬁelds. Methods. Our analysis uses the Weyl coordinates, which are revealed better adapted to the desired shape of the jets. We numerically integrate the 2D-geodesics equations. Results. We give a detailed study of these geodesics and give several numerical examples. Among them are a set of perfectly colli- ρ = ρ ρ Q a mated geodesics with asymptotes 1 parallel to the z-axis, with 1 only depending on the ratios E2−1 and M ,wherea and M are the parameters of the Kerr black hole, E the particle energy and Q the Carter’s constant. Key words. black hole physics – acceleration of particules – relativistic processes

1. Introduction a cylindrical rotating dust. That paper showed that confinement occurs in the radial motion of test particles while the particles are It has long been speculated that a single mechanism might be accelerated in the axial direction thus producing jets. Another at work in the production and collimation of various very ener- relativistic model was put forward in Herrera & Santos (2007). getic observed jets, such as those in gamma ray bursts (GRB, This showed that the sign of the proper acceleration of test par- Piran et al. 2001; Sheth et al. 2003; Fargion 2003), and jets ticles near the axis of symmetry of quasi-spherical objects and ejected from active galactic nuclei (AGN) (Sauty et al. 2002) close to the horizon can change. Such an outward acceleration, and from microquasars (Mirabel & Rodriguez 1994, 1999). Here that can be very big, might cause the production of jets. we limit ourselves to jets produced by a black hole (BH) type However, these models show a powerful gravitational effect core. The most often invoked process is the Blandford-Znajek of repulsion only near the axis, and are built in the framework of (Blandford & Znajek 1977) or some closely similar mechanism axisymmetric stationary metrics which do not have an asymp- (e.g. Punsly & Coroniti 1990a,b; Punsly 2001) in the framework totic behaviour compatible with possible far away observations. of magnetohydrodynamics, always requiring a magnetic field. So we want to explore the more realistic rotating black hole, i.e. However such mechanisms are limited to charged particles, and Kerr, metrics instead. ffi would be ine cient for neutral particles (neutrons, neutrinos and We thus address here the issue of whether it is possible, at photons), which are currently the presumed antecedents of very least in principle (i.e. theoretically) to obtain a very energetic thin and long duration GRB (Fargion 2003). Moreover, even for and perfectly collimated jet in a Kerr black hole spacetime with- charged particles, some questions persist (see for instance the out making use of magnetic fields. Other authors (see Bicák˘ conclusion of Williams 2004). Finally, while the observations of et al. 1993; Williams 1995, 2004; de Felice & Carlotto 1997, synchrotron radiation prove the presence of magnetic fields, they and references therein) have made related studies to which we do not prove that those fields alone cause the collimation: mag- refer below. Most such authors agree that the strong gravitational netic mechanisms may be only a part of a more unified mecha- field generated by rotating BHs is essential to understanding the nism for explaining the origin and collimation of powerful jets origin of jets, or more precisely that the jet originates from a (see Livio 1999, p. 234, and Sect. 5), and, in particular, for col- Penrose-like process (Penrose 1969; Williams 2004) in the ergo- limation of jets from AGN to subparsec scales (see de Felice & sphere of the BH; collimation may also arise from the gravita- Zanotti 2000). tional field and that is the main topic in this paper. Considering this background, it is worthwhile looking for Our work can therefore be considered as covering the whole other types of model to explain the origin and structure of jets. class of models in which particles coming from the ergosphere Other models based on a purely general relativistic origin for jets form a jet collimated by the geometry. Although a complete have been considered. A simple model was obtained by Opher model of an individual jet would require use of detailed models et al. (1996) by assuming the centres of galaxies are described by of particle interactions inside the ergosphere, such as that given Article published by EDP Sciences Page 1 of 9 A&A 515, A15 (2010) by Williams (2004), we show that thin and very long and ener- suffices, in the case of strong fields in general relativity, which getic jets, with some generic features, can be produced in this is the case near the Kerr BH, provided the ergosphere produces way. In particular the presence of a characteristic radius, of the particles of appropriate energy and initial velocity. The gravit- size of the ergosphere, around which one would find the most omagnetic part of the gravitational field then provides the col- energetic particles, might be observationally testable. limation. Hence, our model is, in this respect, simpler than the From a strictly general relativistic point of view, test particles standard model of Blandford & Znajek (1977), and is in accor- in vacuum (here, a Kerr spacetime) follow geodesics; this applies dance with the analysis given in Williams (2004). to both charged and uncharged particles, although, of course, The paper starts with a study of Kerr geodesics in Weyl co- in an electrovacuum spacetime, such as Kerr-Newman, charged ordinates in Sect. 2; the next section studies the asymptotic be- particles would follow accelerated trajectories, not geodesics. haviour of geodesics of outgoing particles with Lz = 0; Sect. 4 Thus, in Kerr fields, what produces an eventual collimation for analyses incoming particles stemming from the accretion; a sam- test particles, or not, is the form of the resulting geodesics. ple Penrose process and the plotting of geodesics are presented Hence we discuss here the possibilities of forming an outgoing in Sect. 5; and finally we discuss in Sect. 6 the significance of jet of collimated geodesics followed by particles arising from our results for jets. In the conclusion, we succinctly summarize a Penrose-like process inside the ergosphere of a Kerr BH. We our main results and evoke some perspectives. show that it is possible in principle to obtain such a jet from a purely gravitational model, but it would require the “Penrose process” to produce a suitable, and rather special, distribution of 2. Kerr geodesics outgoing particles. We start from the projection in a meridional plane φ = constant The model is based on the following considerations. of the Kerr geodesics in Boyer-Lindquist spherical coordinatesr ¯, Most studies of geodesics, (e.g. Chandrasekhar 1983), em- θ and φ. The metric is ploy generalized spherical, i.e. Boyer-Lindquist, coordinates. We transform to Weyl coordinates, which are generalized cylin- d¯r2 ds2 = r¯2 + a2 cos2 θ + dθ2 drical coordinates, and are more appropriate, as we shall see, for r¯2 − 2Mr¯ + a2 interpreting the collimated jets. 2 sin θ 2 We consider test particles moving in the axisymmetric sta- + adt − r¯2 + a2 dϕ (1) tionary gravitational field produced by the Kerr spacetime, r¯2 + a2 cos2 θ whose geodesic equations, as projected into a meridional plane, (¯r2 − 2Mr¯ + a2) 2 − dt − a sin2 θdϕ , are known (Chandrasekhar 1983). Our study is restricted to mas- r¯2 + a2 cos2 θ sive test particles, moving on timelike geodesics, but of course massless test particles on null geodesics could be the subject where M and Ma are, respectively, the mass and the angular of a similar study (Incidentally the compendium of Sharp 1979 momentum of the source, and we have taken units such that shows that analytic studies of general timelike geodesics have c = 1 = G where G is Newton’s constant of gravitation. The been much less frequent than detailed studies of more restricted “radial” coordinate in Eq. (1) has been namedr ¯ because it is problems). more convenient for us to use the rescaled coordinate r = r¯/M For particles outgoing from the ergosphere of the Kerr BH (O’Neill 1995). The projected timelike geodesic equations are we examine their asymptotic behaviour. Among the geodesic then particles incoming to the ergosphere, we discuss only the ones 4 + 3 + 2 + + 2 2 (a4r a3r a2r a1r a0) coming from infinity parallel to the equatorial plane, because M r˙ = , (2) 2 2 these are in practice the particles stemming from the accretion r2 + a cos2 θ disk. We show that only those with a small impact parameter are M of high enough energy to provide energetic outgoing particles. 4 θ + 2 θ + 2 2 b4 cos b2 cos b0 M θ˙ = , (3) In the ergosphere, a Penrose-like process can occur. In the 2 2 − 2 θ 2 + a 2 θ original Penrose process, an incoming particle decays into two (1 cos ) r M cos parts inside the ergosphere. It could also decay into more than two parts, or undergo a collision with another particle in this with coefficients region, or give rise to pair creation (e−, e+) from incident pho- a2Q tons which would follow null geodesics. The different possible a = − , (4) ff 0 M4 cases do not a ect our considerations, and that is why we do not study them here, although the distribution function of outgo- 2 a = (aE − L )2 + Q , (5) ing particles would be required in a more detailed model of the 1 M2 z type discussed, in particular to explain why only particles with 1 a = a2(E2 − 1) − L2 −Q , (6) low angular momentum and not diverging from the rotation axis 2 M2 z are produced. For detailed studies see Williams (1995, 2004) a3 = 2, (7) and Piran & Shaham (1977). After a decay, one (or more) of = 2 − , the particles produced crosses the event horizon and irreversibly a4 E 1 (8) plunges into the BH, while a second particle arising from the and decay can be ejected out of the ergosphere following a geodesic Q towards infinity. This outgoing particle could be ejected so that b0 = , (9) asymptotically it runs parallel to the axis of symmetry, but we M2 do not discuss only such particles. 1 b = a2(E2 − 1) − L2 −Q = a , (10) In our model there is no appeal to electromagnetic forces to 2 2 z 2 M explain the ejection or the collimation of jets, though the parti- a 2 b = − (E2 − 1); (11) cles therein may themselves be charged. The gravitational field 4 M Page 2 of 9 J. Gariel et al.: Kerr geodesics, the Penrose process and jet collimation by a black hole where the dot stands for differentiation with respect to an affine and i = ±1fori = 1, 2: 1 indicates whether the geodesic is parameter and E, Lz and Q are constants. Here Chandrasekhar’s incoming or outgoing in r (i.e. the sign ofr ˙), while 1 2 indicates δ1 has been set to 1, its value for timelike curves. Assuming whether θ is increasing or decreasing. Note we always mean the that the affine parameter is proper time τ along the geodesics, non-negative square roots to be taken. then these equations implicitly assume a unit mass for the test The ratio between the first order differential Eqs. (17) 1 particle ,sothatE and Lz have the usual significance of total and (18) yields the special characteristic equation of this system energy and angular momentum about the z-axis, and Q is the of equations corresponding Carter constant (which, as described in Hughson dz (|P|z − |S |)(α2 − A)α2ρ et al. 1972, for example, arises from a Killing tensor of the met- = 2 · (21) ρ | |α4ρ2 + | | α2 − 2 ric, while E and Lz arise from Killing vectors). With this un- d P 2 S ( A) z Q 2 derstanding, E, Lz,and have the dimensions of Mass, Mass We restrict our study to the quadrant ρ>0andz > 0in 4 δ and Mass respectively, in geometrized units, while 1, though the projected meridional plane (orbits in fact spiral round in φ 1 numerically, has dimensions Mass2, as do all the a and b .In i i in general: this information is contained in the conserved Lz). this paper we consider only particles on unbound geodesics with The results for the other three quadrants will follow by sym- ≥ E 1 (for the conditions for existence of a turning point, giv- metry, although this symmetry does not imply that individual ing bound geodesics, which are related to the parameter values geodesics are symmetric with respect to the equatorial plane. for associated circular orbits, see Chandrasekhar 1983; Williams From numerical solutions of the geodesics we obtained asym- 1995, 2004). metrical geodesics, confirming the analysis in Williams (1995, The dimensionless Weyl cylindrical coordinates, in multiples 2004). Geodesics can also cross the polar axis, which would be of geometrical units of mass M,aregivenby ρ = represented by a reflection from 0 back into the quadrant. 1/2 Geodesics going to or coming from the expected accretion ρ = (r − 1)2 − A sin θ, z = (r − 1) cos θ, (12) disk would, if the disk were thin, go to or from√ values of ρ much where ρ ρ larger than z. In this limit ( z and A), we have a 2 A = 1 − · (13) α = ρ 1 + O ρ−2 , (22) M √ 2 −2 From (12) we have the inverse transformation |P| = a4ρ 1 + k/ρ + O ρ , (23) r = α + 1, (14) 2E2 − 1 ρ z where k = , (24) θ = , θ = , E2 − 1 sin 2 1/2 cos (15) (α − A) α 2 −2 |S | = b0ρ (1 + O(ρ )), (25) with

√ / √ / and thus α = 1 ρ2 + + 2 1 2 + ρ2 + − 2 1 2 . (z A) (z A) (16) dz z − z + zk/ρ 2 ≈ 2 1 + O(ρ−3), (26) Here we have assumed A ≥ 0, and taken the root of the second dρ ρ(1 + k/ρ) 2 degree equation obtained from (15) for the function α (ρ, z)that where = allows the extreme black hole limit A 0. The other root, in this / / α = b 1 2 1 Q 1 2 limit, is the constant 0. z = 0 = , 1 2 (27) The Eq. (16) shows that in the (ρ, z) plane the curves of a4 M E − 1 constant α (constant√ r) are ellipses with semi-major axis α and and we have to assume Q≥0toobtainarealz1 (the form of eccentricity e = A/α:forlargeα, these approximate circles. Q < ρ = θ = π Eq. (3) makes it obvious that geodesics with 0 are bounded Note that 0 consists of the rotation axis 0or together away from the equatorial plane cos θ = 0, though those with very with the ergosphere surface. small |Q| could still satisfy ρ z). Now, with Eqs. (14)and(15) we can write the geodesics The truncated series development of dz/dρ now yields Eqs. (2)and(3)intermsofρ and z coordinates, producing the following autonomous system of first order equations dz 1 2kz1 −3 ≈ z − 2z1 + + O ρ . (28) α3ρ α2 − dρ ρ ρ P + S ( A)z α2 − A αρ If = −1 the curve crosses the equatorial plane and gives simi- Mρ˙ = , (17) 2 2 lar asymptotic behaviour in the second quadrant, so we can take α + 2α2 + a 2 ( 1) z 2 = 1. M −1 A thicker accretion disk would absorb or release particles on a 2 /ρ Mz˙ = (Pz − S )α (α + 1)2α2 + z2 , (18) geodesics with larger values of z , which might include parti- M cles with Q < 0. ρ where Geodesics in an axial jet would have z . For this limit, we first observe that from Eq. (20)wehave = α + 4 + α + 3 P 1 a4( 1) a3( 1) (19) 1/2 − / ≈ + + 4 + + ρ2 2 + 1 , 2 1 2 S 2 (b0 b2 b4)z (2b0 b2) z O(z ) (29) + a2(α + 1) + a1(α + 1) + a0 , 1/2 where S = b z4 + b α2z2 + b α4 , (20) 1 2 4 2 0 L 2 b + b + b = − z ≤ 0, (30) 1 An alternative interpretation is to assume that for a particle of 0 2 4 M mass m,theaffine parameter τ/m has been used (Wilkins 1972; 1 2b + b = a2(E2 − 1) − L2 + Q . (31) Williams 1995). 0 2 M2 z Page 3 of 9 A&A 515, A15 (2010)

6 Hence in this limit S is well defined and real for indefinitely 110 small ρ/z only for Lz = 0. The geodesics obeying this restric- tion, imposed after similar reasoning, were studied by Bicák˘ 7500007500 et al. (1993), but in BL or Kerr-Schild coordinates. Here we re- examine these geodesics in the more revealing cylindrical coor- 500000500 z dinates. Before doing so, we may note that in contrast to geodesics 2 with Lz 0, geodesics with E > 1andLz = 0 may lie arbitrarily 25000025 2 close to the polar axis (Carter 1968). For Lz 0, the value of S − 2 2 < at the axis is z Lz 0 which is not allowed and thus there is 2 2 some upper bound θ0 on θ.ThevalueofS at θ = π/2isb0α so 0 Q < θ θ if 0 there is also a lower bound 1 on . 1107 0 dzd -1107 3. Geodesics with Lz = 0 0.2 0.4 0.6 0.8 We shall discuss unbounded (E2 > 1) outgoing geodesics. Corresponding incoming geodesics will follow the same curves in the opposite direction. Fig. 1. Plot of the surface dz/dρ = f (ρ, z)givenbyEq.(21)foran 2 4 3 For Lz = 0, S factorizes as outgoing particle with the parameters E = 10 , Lz = 0andQ = −9×10 for a black hole with parameters M = 1anda = 0.5, in the case where 2 = α2 − 2 Qα2 + 2 2 − 2 / 2. S ( z )( a (E 1)z ) M (32) 2 = −1. The points where dz/dρ →±∞correspond to the asymptotes givenbytheEq.(40), ρ = ρ1 = 0.49991. 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 θ , say. Correspondingly θ˙ = 0 only at the axis, at θ = θ if cross the curve (35) again but from above in the (ρ, z)planeand 1 1 / ρ< / ρ| Q < 0, and at α →∞. hence with dz d dz d D, and afterwards stay in the region Thus for Lz = 0andQ < 0, geodesics which initially have outside (35). θ<˙ 0 will become asymptotic to θ = θ1. The angle may be For outgoing√ geodesics outside (35) which reach points at narrow if 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 2 2 a (E − 1) − |Q| |Q|, (33) Eq. (21)givesdz/dρ ≈ z/ρ, so all such geodesics approximate ρ = θ Cz for suitable C, regardless√ of the sign of 2. and then 1 1. Such geodesics may provide a conical jet, as ρ discussed later. In the limit z and z A, Our other polynomial, P2, can be written as α = + −2 , z√(1 O(z )) (36) 2 −2 2 2 |P| = a4 z (1 + k/z + O(z )), (37) 2 2 4 a 2 2a P = (E − 1) r + r + r (34) − M2 M2 |S | = 2b + b ρz(1 + O(z 2)), (38) 0 2 a2 Q a2 +2r3 + 2 r − r2 − 2r + · so the Eq. (21) can be approximated by M2 M2 M2 dz z(1 + k/z) 1 = + O , (39) From this form it easily follows that any unbound geodesic dρ ρ(1 + k/z) + 2ρ1 ρz 2 (E > 1) with Lz = 0 has at most one turning point in r (i.e. value such thatr ˙ = 0) and this, if it exists, lies inside the horizon (and where / / a fortiori inside the ergosphere, Stewart & Walker 1974). The 2b + b 1 2 Q 1 2 argument is very simple. If E2 > 1andQ≤0, then P2 is strictly ρ = 0 2 = ρ 1 + , (40) 1 e 2 2 − positive for all r > 0. If Q > 0, P2 is negative at r = 0 but positive a4 a (E 1) at the outer black hole horizon (where r2 −2r+a2/M2 = 0), so its 2 2 and ρe ≡ a/M.Hereρ1 is real if a (E − 1) + Q > 0butwesee one zero lies inside the black hole. This implies that unbounded from Eq. (32) that for S to be real near the axis, this condition = outgoing geodesics followed by particles with Lz 0 must come must be satisfied. from the ergosphere. Correspondingly, geodesics incoming from In Fig. 1 we show a plot of the values of dz/dρ,using(21), infinity with Lz = 0 will fall into the ergosphere. 4 for ε2 = −1, with the parameters M = 1, a = 1/2, E = 10 , Although there are no turning points of r, one can have turn- Q = −9 × 103. The only asymptotes are parallel to the z axis at ing points of ρ,if 2 = −1. Such turning points are solutions of ρ = ρ1 as expected from Eq. (40). the equation We also plot in Fig. 2 a set of such outgoing geodesics ρ, ≡| |α4ρ2 −| | α2 − 2 = obeying (21), for the same values of the parameters of the BH D( z) P S ( A) z 0 (35) 5 (a = 1/2, M = 1) and of the particle (Lz = 0, Q = −2.2 × 10 , = × 3 ρ = . ff where D(ρ, z) is the denominator of (21) with 2 = −1. At E 2 10 ,so 1 0 441588), but with di erent initial values each of these turning points (ρ2, z2), dz/dρ →∞,which of the position. The set of turning points of these geodesics is the means that the geodesics have a vertical tangent (parallel to the curve defined by Eq. (35). For the rightmost of these geodesics, z-axis). Before reaching the turning point, these geodesics have the numerical integration was also continued back√ towards the −4 dρ/dz > 0, and, at any z,dz/dρ>dz/dρ|D,wheredz/dρ|D is the ergosphere as far as ρ = 10 , z = 0.843407 < A = 0.866025. slope of the curve (35), and afterwards they have dρ/dz < 0, To confirm the picture obtained from these numerical exper- implying that they subsequently cross the axis. They will then iments, one can show, without assuming z ρ, the existence of

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large ρ which gives z = z1ρ ln(1 + ρ/k)/k + cρ with similar interpretation). From Eq. (45), either (a) ρ/z is approximately constant or (b) ρ → ρ1. In case (a), we note that for consistency of the approximation z ρ, C must be small, although the conclusion is the same as was reached√ above merely with the assumption that both z and ρ are A. In case (b), we have a limit-outgoing geodesic for which ρ<ρ1 at all points and as z →∞, ρ → ρ1. This limit is obtained since the turning point for ρ has z2 →∞ when ρ2 → ρ1. We can see from Eq. (44) that the coordinate z2 of this turning point tends to infinity like z3 = −kρ1/(ρ2 − ρ1). The geodesics asymptotic to ρ1 would provide a perfectly collimated jet parallel to z. One might think (and we initially thought) that there also existed geodesics eventually tending to the same asymptote but approaching it from the right in the (ρ, z) plane (for example, Fig. 2. Plots of geodesics obeying Eq. (21), showing the turning points. directly from the accretion disk, or coming from the ergosphere From left to right these curves start at ρ = 0.07 and z = 0.98, 0.95, 0 0 but with a turning point ρ2 >ρ1). However, such geodesics do 0.93, 0.92, 0.91, 0.9 and 0.89935501. not exist, since they require that dz/dρ<0 in the limit z ρ and for ρ>ρ1, contradicting Eq. (39) which implies dz/dρ>0. This exactly one zero of D on any curve r = constant, 0 <θ<π/2, is entirely in agreement with the results of Stewart & Walker so that the conclusion that a geodesic has at most one turning (1974). ρ>ρ /ρ point in ρ is not an artefact of the approximation at large z.The The geodesics in 1 may asymptote to any ratio z , ρ argument is as follows. from Eq. (39). Moreover, geodesics which do turn in then cross = Along an r = constant curve, |P| and α are constant, ρ = the axis, cannot cross the curve D 0 from below again, and so ρ sin θ and z = α cos θ,whereρ is a constant (related to r). cross it from above and also asymptotically have some fixed ratio 0 0 /ρ Then z . For astrophysical applications, it may be important to write = | |α4ρ2 2 θ −| | α2 − 2α θ D P 0 sin S ( A) cos (41) the results in the normal units of length and time. We have, from Eq. (17), that asymptotically for outgoing particles in z ρ, where, defining ρ˙ → 0, t˙ → E,˙z > 0 (for − 2 = 1 = 1)andEq.(18)isgivenby √ 2 1/2 F ≡ (b0 + |b4| cos θ) Mz˙ ≈ a4; (46) = [(Q + a2(E2 − 1) cos2 θ)]1/2/M, (42) hence, restoring normal units of length and time and taking a par- v from Eq. (32)wehave|S | = α2F sin θ. For real S we need F ≥ 0 ticle of mass m, the asymptotic value of the speed of outgoing and as θ decreases, F increases. particles is given by At θ = π/2, D > 0, while at small θ, D < 0. Hence there is at 1/2 θ = θ E2 v least one zero of D. Let the largest one be at 0 say. On the mMz˙ ≈ − m2 = mγ , (47) r = constant curve, we will then have c4 c θ 3 2 2 sin where we have used (8)and D = α (α − A) (F0 sin θ cos θ0 − F cos θ sin θ0). (43) sin θ − / 0 v 2 1 2 E = θ = θ γ = 1 − = , (48) Here we have used D 0at 0 to substitute for the constant c mc2 | |α4ρ2 2 θ α θ θ θ θ P 0 sin 0 in terms of and 0.As decreases from 0,sin decreases, cos θ increases and F increases. Thus the combination is the Lorentz factor. Hence, asymptotically, the speed of the (F0 sin θ cos θ0 − F cos θ sin θ0) becomes and stays negative for particle is all θ<θ0. Thus D < 0forallθ<θ0, though D approaches 0, / θ θ → Mz˙ m2c4 1 2 due to the further factor sin ,as 0. This implies that the v = c = c 1 − , (49) only points on r = constant such that D = 0areatθ = 0and t˙ E2 θ = θ0. √ For large z we see from Eq. (39) that the turning points lie which is ultrarelativistic if E mc2.FromEq.(47)wehave approximately on a curve ρ = ρ1z/(z + k)orz = −kρ/(ρ − ρ1). that asymptotically limit-outgoing particles have an uniform mo- Actually, the differential equation for large z,ifwedropthe1/z2 tion parallel to the z axis. terms, has an analytic solution

Ckz = kρ + ρ1z ln[z/(z + k)], (44) 4. Incoming particles where C is a constant of integration, so We describe as “incoming particles” the particles, with param- Q eters E , Lz,and , coming into the ergosphere following un- ρ = ρ1z/k ln(1 + k/z) + Cz → ρ1 + Cz... (45) bound geodesics and having a turning point in z (i.e. such that z˙ = 0). Such turning points {ρ , z } are deﬁned as solutions of →∞ / 4 4 as z . Keeping the next order terms in 1 z would be incon- the equation sistent with the terms dropped during the derivation (Similarly, there is an analytic solution for the approximate Eq. (26)at N2(ρ4, z4) = 0 (50)

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z 5. Penrose process and plotting of geodesics 1 To make a jet using the geodesics discussed above, we would have to assume that incoming particles arrive in the ergosphere 0.8 and undergo a Penrose process. As mentioned earlier, in its original version (Penrose 1969), each particle may be decomposed 0.6 into two subparticles and one of them may cross the horizon and fall irreversibly into the BH, while the other is ejected to 0.4 the exterior of the ergosphere; or the incoming particle may collide with another particle resulting in one plunging into the 0.2 BH and the other being ejected to the exterior. The second case can correspond to a creation of particles, say e− and e+ from an incoming photon (δ = 0) interacting with another inside 0.1 0.2 0.3 0.4 0.5 0.6 0.7 the ergosphere. We do not present here all the possible cases, Fig. 3. Penrose process. Plots of the ingoing particle (dashed line) com- which are exhaustively studied, especially for AGN, in Williams ρ →∞ = . ing asymptotically, for , from z1 1 58116, to the turning point (1995, 2004). There is also observational evidence for a close ρ = . = . ( 0 23, z 0 59) located inside the ergosphere, whose boundary is correlation between the disappearance of the unstable inner ac- indicated, where it decays into an infalling particle and an outgoing par- ρ = ρ = . cretion disk and some subsequent ejections from microquasars ticle (bold line) following a geodesic asymptotic to 1 1 64341 + when z →∞. The parameters of the black hole are M = 1anda = 0.5. such as GRS 1915 1105 (Mirabel & Rodriguez 1994, 1999), which from our point of view could correspond to the instabil- The parameters of the three geodesics are Ein = 200, Eout = 202,

Efall = −2, Lz,in = Lz,fall = −100, Lz,out = 0, Q = Q = 100 000. ity causing disk material to fall through the ergosphere and to then give rise to a burst of ejecta from Penrose-like processes. Here we are mainly interested in the outgoing particles which where follow geodesics that tend asymptotically towards a parallel to the z axis, as described in the earlier Sect. 3. These events are = − N2 Pz S (51) closely dependent on the possibilities allowed by the conservation equations. In the case when the incoming particle splits into is the relevant factor in the numerator of the right side of two (Rees et al. 1976), the conservation equations of the energy Eq. (21). As remarked earlier we need only consider = 1. For 2 and angular momentum are large ρ these turning points are approximately at z = z1(1 − k/ρ), ρ , z from Eq. (28). The set of these turning points { 4 4}forms Ein = Eout + Efall, (53) a curve of the points of each geodesic with horizontal tangent L , = L , + L , . (54) (i.e. parallel to the ρ−axis). Each geodesic has a positive slope z in z out z fall / ρ> ρ<ρ ρ>ρ dz d 0for 4, a negative one for 4 and a maximum We know from Eq. (29) that for the outgoing particles we study at the turning point. There exists a limit-incoming geodesic with L , = 0. The particles falling irreversibly into the BH have ρ →∞ → z out its turning point at the infinite 4 for z4 z1, i.e. with the energy E and angular momentum L , . The energy and the = ρ →∞ fall z fall asymptote z z1,when . angular momentum of a particle in the ergoregion can be nega- Q A test particle with parameters E , Lz,and coming from tive which is the basis of the Penrose process. The particle falling infinity (in practice from the accretion disk) parallel to z = 0to- into the black hole has negative energy, and hence the outgoing wards the axis of the black hole corresponds to a geodesic which, particle, leaving the ergosphere, has a bigger energy than the in- ρ →∞ = = in the limit , has an asymptote defined by z z1 con- coming particle, stant where z1 is the impact parameter. Therefore it is a limit- = + | |. incoming particle with z < z1,˙z < 0,ρ< ˙ 0, 1 = 2 = 1. In the Eout Ein Efall (55) limit ρ z,dz/dρ = 0, so the tangent has to be parallel to the ρ axis and Eq. (27) produces We have plotted numerically the geodesics for incoming, outgoing and falling particles with the following values for the pa- / Q 1 2 rameters: a/M = 1/2, Ein = 200, Eout = 202, Efall = −2, 1 5 z = · (52) L , = L , = −100 and Q = Q = 10 . These values are cho- 1 2 − z in z fall M E 1 sen in such a way that the geodesics meet inside the ergoregion We have plotted in Fig. 3 (see Sect. 5) an example of a geodesic situated in the interior of the ergosphere ρ of an incoming particle. We see that, unlike 1, z1 does not de- ρ ρ pend upon the black hole parameter a. The incoming particles 2 = − ρ2 − − , z 1 e 1 ρ 1 ρ (56) come from the accretion disk, which means that their energy E e e is rarely very big, which means rarely as big as a or M,which and they produce for the asymptotes of the outgoing particles characterize the black hole energy, but instead of the same order √ ρ1 = 1.64341 and for the incoming z1 = 1.58116. The curves are as 1 or Q .WhenE → 1thenz√1 →∞if Q 0. However, if built with the initial conditions z[1.55] = 20 for outgoing parti- 4 E is very big compared to 1 and Q ,thenz1 → 0. cles and z[0.25] = 0.60 for incoming particles. Their intersection Q For a given , the most energetic incoming particles are is situated at the point ρi = 0.23 and zi = 0.59 inside the ergo- those with a small impact parameter z1, near to zero. Hence only sphere which we can take as the initial condition for the falling a thin slice of the accretion disk can participate with the greatest particle and from where we trace the three curves (see Fig. 3). efficiency in producing Penrose processes leading to the most The exhibition of these numerical solutions with an outgo- intense possible jet. The point√ where the ergosphere surface in- ing geodesic which leaves the ergosphere after the Penrose pro- tersects the z axis is ze = A.Thevalueofze, for the incoming cess and has vertical asymptote with the value ρ1 precisely equal particles, does not play a role like that of ρe for the outgoing to Eq. (40) confirms that a model based on such geodesics is particles (compare Eq. (40)toEq.(52)). possible.

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6. Implications for jet formation pressure exerted by the numerous coherent (b)-particles of the jet (a narrow parallel beam is more incisive). The (a)-particles We have shown that to obtain a jet of particles close to the ro- are probably more rapidly thermalised than the (b)-particles of tation axis, it must be formed from particles with (almost) zero the jet. So, one might expect that many particles ejected at lower angular momentum, Lz = 0. If we consider only particles with = latitudes never attain infinity (neither the height of the jet), and Lz 0, there is among them a subset which give a perfectly col- most of them feed the medium, framing a halo around the BH, limated jet, i.e. a set of geodesics exactly parallel to the axis: for falling inside again, or returning to the accretion disk. each allowed value of Q they form a ring of radius We noted also that geodesics with Q < 0 can be asymptotic Q 1/2 to lines with constant θ. These asymptotes allow us to define ρ = ρ 1 + , (57) another type of jet which is bigger and less collimated than the 1 e 2 2 − a (E 1) previous one. It is interesting to remark that recent observations whwre ρe = a/M. Note that ρ = ρe is the circumference of (Sheth et al. 2003; Sauty et al. 2002) suggest the existence of the ergosphere in the equatorial plane. For large E the set of two different types of jets precisely of these sorts, i.e. narrowly all these geodesics will give a jet of radius of the order of ρe, and broadly collimated. with a density of particles dependent on the distribution in Q There exists an ensemble of geodesics that tend asymptoti- for given E, which remains perfectly collimated all the way to cally to these conical characteristics. The unbounded geodesics infinity. have mainly been discussed, however, by using Boyer-Lindquist θ We note that all other geodesics with Lz = 0 will spread out coordinates r and by the majority of authors. If we rewrite our from the axis along lines z = Kρ. An astrophysical jet will of results, using these coordinates, we may interpret our results and course be of only finite extent and not perfectly collimated, so compare to those of other authors. However, as we show below, it could include such geodesics for suitably large K,aswellas these coordinates are not as well-suited to the issues we have geodesics with a small Lz 0. discussed. Thus forming a collimated jet of particles from a Penrose- Geodesics with Lz 0 may reach low values of ρ/z,ifb2 like process, this jet having a narrow opening angle, for a ro- is large enough, but must be bounded away from | cos θ| = 1 tating black hole without an electromagnetic field, depends on (i.e. θ = 0orθ = π), since those values would imply θ˙2 < 0, the initial distribution of particles leaving the ergosphere, or of from Eq. (3), (cf. Chandrasekhar 1983, p. 348). In practice this some non-gravitational collimating force, even if we consider means that a narrow jet along the axis must be composed of par- only particles with Lz = 0. ticles with very small Lz. Particles with non-zero Lz could only On the other hand, outgoing particles with small energies, lie within a jet with bounded ρ for a limited distance, because namely of the order of their rest energy, E ≈ 1, and Q > 0have large enough z would imply θ˙2 < 0. If Q≥0, the orbits re- θ asymptotes parallel to the z axis with ρ1 ρe. verse the sign of ˙ and reach the equatorial plane, and would This predicted scale of the region of confined highly ener- thus be expected to be absorbed by the accretion disk. For Q < 0 getic particles might provide a test if the accretion disk param- they are confined to a band of values of θ given by the roots of eters provided values for the BH mass and angular momentum, S 2 = 0. These are the “vortical” trajectories of de Felice et al. in a manner such as discussed in McClintock et al. (2006)and (de Felice & Calvani 1972; de Felice & Curir 1992; de Felice papers cited therein, and if the transverse linear scale of the jet & Carlotto 1997). Depending on the maximum opening angle θ, near the BH could be measured (Particles of equally high energy these may still hit, and presumably be absorbed by, a thick ac- may exist in ρ>ρ1 but will spread out away from the axis). cretion disk (de Felice & Curir 1992). Such orbits can be ad- Let us make a brief qualitative remark about the observability equately populated by Penrose-like processes (Williams 1995, of the two species, (a) and (b), of geodesics outgoing from the 2004), and might undergo processes which reduce the opening ergosphere, studied in Sect. 3 (after Eq. (45)). As illustrated by angle (de Felice & Curir 1992; de Felice & Carlotto 1997). A jet the Fig. 2, for each fixed value of ρ1 there is one (b)-geodesic composed of such particles would tend to be hollow and would only, which is the limit of many (one infinity of) (a)-geodesics have a larger radius ρ at large z than is obtained for orbits with when the turning point tends to the infinity (z2 →∞, ρ2 → ρ1). Lz = 0, and hence be observationally distinguishable. The pres- However, the (a)-type geodesics, though much more numerous ence of these escaping trajectories spiralling round the polar axis than the (b)-type geodesics, are, directly or indirectly (i.e. by can be associated with the gravitomagnetic effects due to the ro- radiation, if charged), much more difficult to observe. tation of the hole, one of whose consequences is that even curves Indeed, contrary to the set of (b)-particles framing the jet with Lz = 0 have a non-zero dφ/dt at finite distances. in one direction (collimation along the poles), the (a)-particles Thus although an infinitely extended jet of bounded ρ radius ejected from the ergosphere along unbound geodesics at lower would only contain particles with Lz = 0, which we would ex- latitudes are dispersed into the whole 3D-space (4π steradians). pect to be a set of measure zero among all particles ejected, we The (a)-particles never produce a beam into one privileged di- shall consider this as a good model even for real jets. In prac- rection but instead dilute in the whole space. Observed from the tice, interactions with other forces and objects, which would af- infinity in one line of sight (θ = constant, φ = constant), one fect the jet both by gravitational and other forces, have to be single (a)-particle could directly be detected. While, from the in- taken into account once the jet is well away from the BH, and finity in the line of sight z (θ = 0, ∀φ), the observer will see one these influences might or might not improve the collimation. In infinity (each point of the perimeter of the circle of radius ρ1) (de Felice & Carlotto 1997), the authors discussed possible im- of (b)-particles. The result is reinforced when we extend it to proved collimation for particles of low Lz using forces which all the possible values of ρ1. Encircling the foot of the (b)-jet, have a timescale long compared with the dynamical timescale of the (a)-particles frame a gerb, from the basis of which a possible the geodesics, and which act to move particles to new geodesics indirect effect of isotropic radiation emission (from accelerated with changed parameters. It should be noted that if the object charged particles) could be observed, during the jet eruption. producing the jet is modelled as a rotating black hole, produc- Besides, by their dispersion, the pressure the (a)-particles tion of a collimated jet only arises naturally if the object throws locally exert on the ambient medium is much weaker than the out energetic particles with low Lz, since our discussion shows

Page 7 of 9 A&A 515, A15 (2010) that other particles cannot join such a jet unless there is some concerned, namely the beginning of the jet (parsec scale for mi- other strong collimating influence away from the BH. croquasars,while some hundred parsecs for AGN, depending on However previous authors have not pointed out the existence the BH mass), where it is strongly collimated, our approximation of asymptotes ρ = ρ1, presumably because they are less obvious of test-particles along geodesics is relevant. Indeed, the observed when using coordinates r and θ. In fact, considering z →∞,the jets stemming from active galactic nuclei ejected along the po- θ ≈ − ρ2/ 2 + −3 v = . expressions (12)and(15) produce cos 1 ( 1 2z ) O(z ), lar axis have ultrarelativistic speeds, typically j 0 99995c. θ ≈ ρ / + −4 ≈ + + ρ2/ 2 + −3 The ejected particles, forming the jets, are thermalized with tem- sin ( 1 z) O(z )andr z 1 ( 1 2z ) O(z ). With these expansions it is clear that in the limit θ = 0 one would have peratures of the order 105 K(Filloux 2009) producing a lateral to take the limit of r sin θ to allow ρ1 to be determined. force from the pressure gradient between the thermal energy In the same vein, to find the values of asymptotes z = z1 0 of the particles in the outflow and the low density enveloping near the equatorial plane θ = π/2 for the incoming particles medium (Punsly 1999a,b). The internal particle trajectories to (see Eq. (52)) one has to study r cos θ if one uses the coordi- these jets expand laterally at the speed of sound, being of the v = −1 nates r and θ. In fact, one finds for the asymptotic expansion ρ → order s 30 km s (Filloux 2009), asymptotically forming a −3 ∞ the following expressions, cos θ ≈ (z1/ρ)+O(ρ ), sin θ ≈ 1− conical shape with an opening angle of the order of the inverse − − v /v = −4 (z2/2ρ2) + O(ρ 3)andr ≈ ρ + 1 + [z2 + 1 − (a/M)2]/2ρ+ O(ρ 3). Mach number ( s j) 10 radians. As we can see (Punsly 1 1 1999b, Appendix), the more realistic trajectories corresponding to such corrective terms represent only a small perturbation to 7. Conclusion the geodesics. The model that we present to explain the formation and col- Our main results are the following. limation of jets arises essentially from relativistic strong gravi- There are projections of geodesics all over the meridional tational field phenomena without resort to electromagnetic phe- planes. Among these geodesics there are some, with vertical nomena. From this point of view the model could be interesting asymptotes parallel to z which can form a perfectly collimated also for understanding observational evidence of neutral parti- jet. There are, as well, geodesics with horizontal asymptotes par- cles emitted from the inner jet itself. For example, the recent allel to the radial coordinate ρ, that can represent the paths of observations of Ultra High Energy Cosmic Rays (difficult to incoming particles leaving the accretion disk. explain, implying neutral particles such as neutrinos, or H or These two types of geodesics have intersection points that Fe atoms, etc. Auger 2007a,b; Dermer et al. 2009; and HESS can be situated inside the ergosphere. At these points a Penrose collab. 2009, and references therein) is a new challenge. To ex- process can take place, producing the ejection of particles along plain the Very High Energy of such neutral (massive) particles, the axis with bigger energies than the energies of incoming parti- especially neutrinos (Auger 2007a,b, which are able to travel cles close to the equatorial plane. The energies of outgoing parti- freely over large distances), our model very naturally suggests cles are significantly larger than the ones of the incident particles that they could be directly coming from the collimated inner jet, for the asymptotically vertical geodesics near the scale a/M of which would privilege sources (BH) with rotational z-axis along the ergosphere diameter in the coordinate ρ, so such particles can the line of sight of the observation. Massless particles (photons) show collimation around the surface of a tube of diameter 2a/M would be emitted by charged particles accelerated along the col- centred on the axis of symmetry. Such collimated outgoing par- limated inner jet (Dermer et al. 2009; HESS collab. 2009), which ticles have to have a zero orbital momentum Lz = 0, which im- would privilege sources (BH) with rotational z-axis perpendicu- plies, from the Penrose process, that the incoming particles have lar to the line of sight of the observation. < a negative orbital momentum, Lz 0. Thus the jet has to be fed Our model is sufficiently general to fit various types of ob- from incoming particles with retrograde orbits in the accretion served jets, like GRB, jets ejected from AGN or from micro- disk. There is evidence for the existence of substantial coun- quasars, whenever they are energetic enough to be explained by terrotating parts of accretion disks (Koide et al. 2000; Thakar just a rotating black hole fed by an accretion disk in an axisym- et al. 1997), and such counterrotations could explain the viscos- metric configuration. The main drawback is the need to prefer- ity inducing the instabilities which trigger the falling of matter entially populate the geodesics which can form such collimated towards the ergosphere. It is now known (Mirabel & Rodriguez jets. Work is in progress on this question to determine a possible 1994, 1999; Mirabel 2006) that there is a close connection be- confrontation of the model with observations. Our preliminary tween instabilities in the accretion disk and the genesis of jets studies led us to understand the fundamental role of the function for quasars and microquasars. P(r) of the geodesics equations (See Eqs. (2) and (19)). As an The most energetic incoming particles are those near the example, in the special case where the equation P(r) = 0hasa equatorial plane. Hence the incoming particles which produce real double root, there exist only two narrow ranges of ρ1 val- the most energetic outgoing particles by a√ Penrose process in the ues for large values of E. In this case, we can evaluate from the ergosphere, whose maximum size is ze = A, are those with an- power, for example of radio loud extragalactic jets (Willott et al. < gular momentum Lz 0 and a very small impact parameter z1. 1999), or of microquasars jets (Fender et al. 2004), the particle Also, the limiting diameter of the core of a perfectly colli- density, the mean kinetic energy by particle, the mean veloc- mated jet depends upon the size of the ergosphere. The effective ity and the Lorentz factor of the jets. These results, since they thickness of this part of the jet in this case is of the order of require a long presentation, deserve a separate paper which is 2ρe = 2a/M. under preparation. Our idealised model is based on the well-behaved vacuum The existence of vacuum solutions of the Einstein equa- stationary exact solution of Einstein’s equations with axial sym- tions of Kerr type but with a richer, not connected, topolog- metry, namely the Kerr metrics, which does not take into ac- ical configuration of the ergosphere (see Gariel et al. 2002, count the ambient medium. Though this medium is very dilute, Figs. 7–10), allows us to propose the existence of double jets, it plays a non-negligible role on the more complex global sce- because they are expected to come out from the ergosphere. nario for jets like progressive widening of the beam, advent of These bipolar jets have been observed (see for instance Skinner knots, lobes, etc. However, for the scenario that we are here et al. 1997; Sahai et al. 1998,Fig.1;Fargion 2003,Fig.2;and

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