Innermost stable circular orbit near dirty black holes in magnetic field and ultra-high energy particle collisions

We consider the behavior of the innermost stable circular orbit (ISCO) in the magnetic field near"dirty"(surrounded by matter) axially-symmetric black holes. The cases of near-extremal, extremal and nonextremal black holes are analyzed. For nonrotating black holes, in the strong magnetic field ISCO approaches the horizon (when backreaction of the field on geometry is neglected). Rotation destroys this phenomenon. The angular momentum and radius of ISCO look model-independent in the main approximation. We also study the collisions between two particles that results in the ultra-high energy $E_{c.m.}$ in the centre of mass frame. Two scenarios are considered - when one particle moves on the near-horizon ISCO or when collision occurs on the horizon, one particle having the energy and angular momentum typical of ISCO. If the magnetic field is strong enough and a black hole is slow rotating, $E_{c.m.}$ can become arbitrarily large. Kinematics of high-energy collision is discussed. As an example, we consider the magnetized Schwarzschild black hole for an arbitrary strength of the field (the Ernst solution). It is shown that backreaction of the magnetic field on the geometry can bound the growth of $E_{c.m.}$.


I. INTRODUCTION
Several years ago, it was shown by Bañados, Silk and West that if two particles collide near the black hole horizon of the extremal Kerr metric, their energy E c.m. in the centre of mass (CM) frame can grow unbound [1]. This is called the BSW effect, after the names of its authors. These findings stimulated further study of high-energy collisions near black holes.
The validity of the BSW effect was extended to extremal and nonextremal more general black holes. It was also found that there exists the version of this effect near nonrotating electrically charged black holes [2]. Another version of ultra-high energy collisions reveals itself in the magnetic field, even if a black hole is neutral, vacuum and nonrotating, so it is described by the Schwarzschild metric [3]. Generalization to the case when the background is described by the Kerr metric was done in [4].
In the BSW effect, one of colliding particle should be so-called critical. It means that the energy and the angular momentum (or electric charge) of this particle should be fine-tuned.
In particular, the corresponding critical condition is realized with good accuracy if a particle moves on a circular orbit close to the horizon. Therefore, an innermost stable circular orbit (ISCO) can play a special role in ultra-high energy collisions in astrophysical conditions. Without the magnetic field, this was considered in [5] for the Kerr black hole and in [6] for more general rotating black holes. Kinematically, the effect is achieved due to collision of a rapid typical so-called usual particle (without fine-tuning) and the slow fine-tuned particle on the ISCO [7] (see also below).
In [3] and [4] collisions were studied just near the ISCO in the magnetic field. In both cases, a black hole was taken to be a vacuum one. Meanwhile, in astrophysical conditions, black holes are surrounded by matter, so in this sense they are "dirty". The aim of our work is two-fold. First, we discuss properties of ISCO near dirty black holes when a magnetic field is present. It is worth noting that ISCO is important in astrophysics by itself [8] (say, in studies of different models of accretion disc [9], [10]), so the description of its characteristics in such circumstances can be of interest not only for applications to the BSW effect and related topics. Second, we consider scenarios of high-energy particle collisions near such orbits.
In both works [3] and [4], it was assumed that the magnetic field is weak in the sense that backreaction of the magnetic field on the metric is negligible but, at the same time, it is strong in the sense that it affects motion of test particles. Such combination is self-consistent since the dimensionless parameter b that controls the magnetic field strength contains a large factor q/m that controls motion of particles. Our approach is model-independent and is not restricted by some explicit background metric. Therefore, the most part of formulas applies also to the metrics which are affected by the magnetic field. From the other hand, if the magnetic field is too strong, its backreaction on the metric can change the properties of E c.m.
itself, as it will be seen below.
It is worth noting that high-energy collisions in the magnetic field were studied also in another context in [11] - [12].
The paper is organized as follows. In Sec. II, the metric and equations of motion are presented. In Section III, we give basic equations that determine ISCO. In Sec. IV, we consider ISCO in the magnetic field for near-extremal black holes and analyze the cases of small and large fields. In Sec. V the case of nonrotating (but dirty) and slowly rotating black hole is discussed. As we have two small parameters (slow rotation and inverse field strength), we consider different relations between them separately. In Sec. VI we show that if a black hole rotates, even in the limit of strong magnetic field ISCO does not tend to the horizon radius. In Sec. VII, it is shown that for extremal nonrotating black holes, for large b, ISCO approaches the horizon radius. In Sec. VIII, it is shown that this property is destroyed by rotation. In Sec. IX, general formulas for E c.m. for particle collisions in the magnetic field are given. In Sec. X, we find the velocity of a particle on ISCO and argue that kinematic explanation of high-energy collision is similar to that for the BSW effect [7]. In Sec. XI, we apply general formulas of collision to different black hole configurations and different scenarios. In Sec. XII, the exact solution of the Einstein-Maxwell equations (static Ernst black hole) is chosen as background for collisions. This enables us to evaluate the role of backreaction of the magnetic field on E c.m. . In Sec. XIII, the main results are summarized.
Throughout the paper we use units in which fundamental constants are G = c = 1.

II. METRIC AND EQUATIONS OF MOTION
Let us consider the metric of the form where the metric coefficients do not depend on t and φ. The horizon corresponds to N = 0.
We also assume that there is an electromagnetic field with the four-vector A µ where the only nonvanishing component equals (2) In vacuum, this is an exact solution with B = const [14]. We consider configuration with matter (in this sense a black hole is "dirty"), so in general B may depend on r and θ.
Let us consider motion of test particles in this background. The kinematic momentum p µ = mu µ , where m is the particle's mass, four-velocity u µ = dx µ dτ , where τ is the proper time, x µ are coordinates. Then, the generalized momentum is equal to q is the particle's electric charge. Due to the symmetry of the metric, P 0 = −E and P φ = L are conserved, where E is the energy, L is the angular momentum.
We consider motion constrained within the equatorial plane, so θ = π 2 . Redefining the radial coordinate r → ρ, we can always achieve that within this plane. Then, equations of motion give uṡ Dot denotes differentiation with respect to the proper time τ . As usual, we assume the forward in time conditionṫ > 0, so X ≥ 0. Hereafter, we use notations Subscripts "+", "0" denote quantities calculated on the horizon and ISCO, respectively.
In what follows, we will use the Taylor expansion of quantity ω near the horizon. We denote x = ρ − ρ + , where ρ + is the horizon radius. Then,
In general, it is impossible to find exact solutions of eqs. (17), (18). Therefore, in next sections we analyze separately different particular situations, with main emphasis made on the near-horizon region. In doing so, we develop different versions of the perturbation theory that generalize the ones of [4]. The radius of ISCO, its energy and angular momentum are represented as some series with respect to the corresponding small parameter, truncated at the leading or subleading terms similarly to [4].

IV. NEAR-EXTREMAL BLACK HOLES
Let a black hole be nonextremal. In what follows, we are interested in the immediate vicinity of the horizon and use the Taylor series for corresponding quantities. Then, near the horizon we have the expansion where κ has the meaning of the surface gravity.
By definition, we call a black hole near-extremal if where x 0 = ρ 0 − ρ + . Then, for the lapse function we have expansion near ISCO Taking into account (18), after straightforward (but somewhat cumbersome) calculations, one can find that where P and dP dx are to be taken at x = x 0 or, with the same accuracy, at x = 0 (i.e., on the horizon). Then, From (16), (21) we have Using (23), (24) and (9) we derive equation for the value of the angular momentum L 0 on ISCO: To have a well-defined limit b = 0, we demand d − D > 0. We are interested in the positive root according to (23). Then, and, in a given approximation, where we neglected the difference between dβ dx + and dβ dx 0 . Eqs. (33) -(37) give the expression for H after substitution into (28). To avoid cumbersome expressions, we leave it in the implicit form. Now, two different limiting cases can be considered.

A. Small magnetic field
that agrees with eq. (44) of Ref. [6]. It follows from (16) and (30) that Let us consider small but nonzero b. We can find from (33) that B. Large magnetic field In doing so, we find from (33), (9), (30) Thus according to (27), in general the radius of ISCO depends on the value of the magnetic field via the coefficient H. However, there is an exception. Let Then, The dependence on the magnetic field is due to the term 1 P dP dx in the denominator.
One can find that However, in general they can differ significantly.
As a result, the ISCO radius (27) also may vary over wide range.

V. SLOWLY ROTATING BLACK HOLE
Now, we assume that κ is not small, so the first term in (19) dominates. Here, we will consider different cases separately.

A. Non-rotating black hole
Here, we generalize the results known for the Schwarzschild black hole [16], [17], to a more general metric of a dirty static black hole. In eqs. (14), (15) we should put a 1 = 0 = a 2 .
For a finite value of the magnetic field parameter b, ISCO lies at some finite distance from the horizon. However, now we will show that in the limit b → ∞, the radius of ISCO tends to that of the horizon with x 0 ∼ b −1 .
We will show that this indeed happens, provided the term with L in (9) is large and compensates the second one with b. Correspondingly, we write where For what follows, we introduce the quantity Then, near the horizon, where x is small, we can use the Taylor expansion where Now, β 0 ≫ 1 but, by assumption, β is finite.
In terms of the variable it can be rewritten as It is clear from the above formulas that the expansion with respect to the coordinate x is equivalent to the expansion with respect to inverse powers of the magnetic field b −1 , so for b ≫ 1 this procedure is reasonable.
Then, after substitution of (62), we can represent (14) and (15) in the form of expansion with respect to β −1 0 : Here, the coefficients at leading powers are equal to Then, in the main approximation we have equations C 0 = 0 and S 1 = 0 which give us To find the corrections, we solve eqs. (64) and (65) perturbatively. In doing so, it is sufficient to substitute these values into further coefficients C 1 and S 0 . After some calculations, one The results with the leading term and subleading corrections read In the particular case of the Schwarschild metric, Eqs.(77), (79) agree with [3] and [4].
It is interesting that in terms of variables u, L 0 R + and b the answer (68), (74) in the main approximation looks model-independent. This can be thought of as manifestation of the universality of black hole physics near the horizon. Dependence on a model reveals itself in higher-order corrections.

B. Extremely slow rotation
Now, we consider rotation as perturbation. Here, the angular velocity of rotation is the most small parameter. Correspondingly, in the expressions (18), (15) we neglect the term L 2 since it contains ω ′2 . More precisely, we assume so from (73), (74) we have In the particular case of the slow rotating Kerr metric, where a = J/M, J is the angular momentum of a black hole, a * = a M . Then, (80) reads There are two kinds of corrections -due to the magnetic field and due to rotation.
In the main approximation, we consider them as additive contributions, so where ε 1 and δ 1 are given by (71), (72). One can check that the presence of rotation leads to the appearance in the series (64), (65) of half-integer inverse powers of β 0 , in addition to integer ones.
Then, solving these equations perturbatively, one finds ε 2 and δ 2 . Omitting details, we list the results: It follows from (8), (16) that For the slow rotating Kerr metric, R + ≈ 2M, In the main approximation the difference between the Boyer-Lindquist coordinate r and quasiglobal one ρ has the same order a 2 and can be neglected. Then, They agree with the results of Sec. 3 B 2 of [4]. It is seen from (92) -(94) that the fractional corrections have the order a * b 3/2 and are small in accordance with (82). In a more general case, the small parameter of expansion corresponds to (81), so it is the quantity √
Now, one can check that, in contrast to the previous case, a finite β is inconsistent with eqs.
(64), (65). Instead, β ∼ b for large b. By trial and error approach, one can find that the suitable ansatz reads where we introduced in this ansatz the dimensionless quantity and the coefficient 4 9 to facilitate comparison to the case of the Kerr metric (otherwise, this coefficient can be absorbed by y).

Kerr metric
In the case of the slow rotating Kerr black hole, eq. (91) entails where we used (118).
One should compare this result to that in [4]. Now, R + = 2M, the horizon radius of the Kerr metric r + ≈ 2M(1 − a * 2 4 ). Eq. (53) of [4] gives us whence x 0 = r 0 − r + ≈ R + 16 a * 2 that coincides with (121). It is seen from (113), (120) that the angular momentum takes the value that coincides with eq. (55) of [4]. Also, one finds that In eq. (112) one should take into account that ω + depends on r + that itself can be expressed in terms of a * and M. Collecting all terms, one obtains from (118) that agrees with eq. (54) of [4].

VI. ISCO FOR ROTATING NONEXTREMAL BLACK HOLES IN THE STRONG MAGNETIC FIELD
In the previous section we saw that in the limit b → ∞ the ISCO radius does not coincide with that of the horizon that generalizes the corresponding observation made in Sec. III B 3 of [4]. Now, we will see that this is a general result which is valid for an arbitrary degree of rotation and finite κ (so, for generic nonextremal black holes). It is worth noting that for b = 0 it was noticed that the near-horizon ISCO are absent [18], [6]. However, for b ≫ 1 the corresponding reasonings do not apply, so we must consider this issue anew.
Neglecting higher order corrections, we can rewrite them in the form 1) Let us suppose that β is finite or, at least, β ≪ b. Then, it follows from (57), (9), (55) impossible to compensate the term with L 2 in (127) having the order b 2 .
2) Let β ∼ L ∼ b. Then, in (126) the first term has the order b 2 and cannot be compensated.
Thus we see that, indeed, in the limit b → ∞ the assumption about x → 0 leads to contradictions, so ISCO radius does not approach the horizon.

VII. EXTREMAL NONROTATING BLACK HOLE
Up to now, we considered the case of a nonextremal black hole, so the surface gravity κ was arbitrary or small quantity but it was nonzero anyway. Let us discuss now the case of the extremal black hole, so κ = 0 exactly. We pose the question: is it possible to get the ISCO such that for b → ∞ the ISCO radius tends to that of the horizon? Now, we will see that this is indeed possible for a nonrotating black hole (ω = 0). We assume that the electric charge that can affect the metric is negligible. The extremal horizon appears due to properties of matter that surrounds the horizon that is possible even in the absence of the electric charge, provided equation of state obeys some special conditions [15].
For ISCO close to the horizon we can use the expansion in which we drop the terms of the order x 3 and higher. Now we show that the case under discussion does exist with a finite quantity β. We can use now (62) in which only the first term is retained, so where u is given by eq. (61). Then, (10) reads where Eqs. (14), (15) They have the solution Correspondingly, eqs. (129), eq. (13) give us We can also find the angular momentum on ISCO Thus for big b there is ISCO outside the horizon that tends to it in the limit b → ∞, so that the quantity x 0 → 0.

VIII. EXTREMAL ROTATING BLACK HOLE
Is it possible to have ISCO in the near-horizon region (as closely as we like) for the extremal BH, when κ = 0? Mathematically, it would mean that Then, (17), (21) with κ = 0 give us for small x that Eq. (18) with terms of the order x 2 and higher neglected, gives rise to Then, the main terms in (141), (142) entail For b ≫ 1, assuming for definiteness that d > D (d is defined according to (32)), one finds from (9) and (143) that The terms x 2 in (141) and x in (142) give us, with (143) taken into account The system is overdetermined, eq. (149) cannot be satisfied in general. This does not exclude some exceptional metrics but we will not discuss this issue further. Generically, the answer to our question is negative, so the ISCO radius does not approach the horizon in the limit b → ∞.

IX. PARTICLE COLLISIONS: GENERAL FORMULAS
Let two particles collide. We label their characteristics by indices 1 and 2. Then, in the point of collision, one can define the energy in the centre of mass (CM) frame as Here, is the total momentum, is the Lorentz factor of their relative motion.
For motion in the equatorial plane in the external magnetic field (2), one finds from the equations of motion (6), (7) that Here, ε = +1, if the particle moves away from the horizon and ε = −1, if it moves towards it.
Now, there are two scenarios relevant in our context. We call them O-scenario and Hscenario according to the terminology of [6]. Correspondingly, we will use superscripts "O" and "H".

A. O -scenario
Particle 1 moves on ISCO. As V 1 (ρ 0 ) = 0 on ISCO, the formula simplifies to As we are interested in the possibility to get γ as large as one likes, we will consider the case when the ISCO is close to the horizon, so N is small. In doing so, we will assume that (X 2 ) = 0, so particle 2 is usual according to the terminology of [6]. We also must take into account eq. (16), whence For ISCO close to the horizon, the first term dominates and we have B. H -scenario Now, particle 1 leaves ISCO (say, due to additional collision) with the corresponding energy E = E(x 0 ) and angular momentum L = L(x 0 ) that corresponds just to ISCO. This particle moves towards the horizon where it collides with particle 2.
Mathematically, it means that we should take the horizon limit N → 0 first in formula (153). We assume that both particles move towards the horizon, so ε 1 ε 2 = +1. Then, where all quantities are to be calculated on the horizon.
For small X 1 , when we see from (157) that Now, where X 0 = E 0 − ω 0 L corresponds to ISCO. With (12), (16) taken into account, in the main approximation Now, we apply these formulas to different cases considered above.

X. KINEMATICS OF MOTION ON ISCO
It is instructive to remind that the general explanation of high E c.m. consists in the simple fact that a rapid usual particle having a velocity close to speed of light, hits the slow particle that has parameters approximately equal the critical values. This was explained in detail in [7] for the standard BSW effect (without considering collision near ISCO). Does this explanation retain its validity in the present case? One particle that participates in collision is usual, so it would cross the horizon with the velocity approaching the speed of light in an appropriate stationary frame (see below). We consider the near-horizon ISCO, so the velocity of a usual particle is close to the speed of light. Now, we must check what happens to the velocity of a particle on ISCO.
To describe kinematic properties, it is convenient to introduce the tetrads that in the local tangent space enable us to use formulas similar to those of special relativity. A natural and simple choice is the tetrad of so-called zero-angular observer (ZAMO) [8]. It reads Here, x 0 = t, x 1 = r, x 2 = θ, x 3 = φ. It is also convenient to define the local three-velocity a = 1, 2, 3.
From equations of motion (13) -(22) and formulas for tetrad components, we obtain the component v (2) = 0 for equatorial motion.
Then, introducing also the absolute value of the velocity v according to one can find that Eq. (172) was derived in [7] for the case when the magnetic field is absent. We see that its general form does not depend on the presence of such a field.
For a circle orbit, eq.(16) should hold. Comparing it with (172), we find that that has the same form as for the static case [3]. Now we can consider different cases depending on the value of the magnetic field and a kind of a black hole.
For large b, accoding to (59), the quantity β is proportional to b,and grows, v → 1.
However, this case is not very interesting since the individual energy (48) itself diverges.
B. Non-rotating or slowly rotating nonextremal black holes According to eq. (68), β ≈ 1 √ 3 . Slow rotation adds only small corrections to this value. Thus, rather unexpectedly, we again obtain that on ISCO v ≈ 1 2 .
This value coincides for the near-extremal Kerr without a magnetic field and a nonrotating or slow rotating dirty black hole in the strong magnetic field.

C. Modestly rotating nonextremal black hole
It follows from (115), (98) that |β| ≫ 1. However, as the energy of a particle on ISCO (118) tends to inifnity, this case is also not so interesting.
To summarize, in all cases of interest (when an individual energy is finite), β remains finite even in the strong magnetic field. Correspondingly, v < 1 on ISCO and the previous explanation of the high E c.m. [7] applies. For less interesting cases, when an individual energy diverges, we have collision between two rapid particles but their velocities are not parallel and this also gives rise to high γ 0 (see eq. 20 in [7]).

XI. CENTRE-OF MASS ENERGY OF COLLISION
A. Near-extremal black hole

O -scenario
Using (156), (21) and (29) one obtains In the strong magnetic field, with b ≫ 1, using the expression (45) for β, we obtain In the near-extremal Kerr case, that coincides with eq. (61) of [4] in which the limit b → ∞ should be taken.

H -scenario
Now, due to (143), eq. (161) gives us X 1 = 0. It means that in the expansion (21) we must retain the first correction in the expression for N, when it is substituted into (161).
As a result, we have There are also terms of the order x 2 0 ∼ κ 4/3 but they are negligible as compared to κ. Correspondingly, (157) gives us In the strong magnetic field, with b ≫ 1, using (45) again we obtain Thus in both versions, for b ≫ 1 the effect is enhanced due to the factor b. For b = 0 we return to [6].
In the Kerr case, that corresponds to eq. (59) of [4] in which b ≫ 1.
B. Extremely slow rotating or nonrotating black hole

H -scenario
Using (88) and neglecting in (161) the second term (rotational part), we get C. Modestly rotating black holes in strong magnetic field

H -scenario
In a similar manner, one can obtain from (159), (161) and (117) that (E H c.m. ) 2 ∼ b with a somewhat cumbersome coefficient that we omit here.
Both these scenarios are less interesting since according to (118), the individual energy r + = 2M is the horizon radius, B is a constant parameter. It follows from (11) (with B replaced withB) and (189) that Many important details of particle's motion in this background can be found in Ref. [20].
Calculating the corresponding coefficients according to (58) -(60) and substituting them into (71) -(76), we obtain It follows from (75) that For the energy of collision we have from (182), (184) where When ξ ≪ 1, there is agreement with the results for the Schwarzschild metric [3], [4] since eq. (192) turns into (77) Otherwise, both energies (197), (198) contain the factor b ξ ∼ q mBM that bounds E c.m. which begins to decrease when B increases. One should also bear in mind that it is impossible to take the limit ξ → ∞ literally since the geometry becomes singular. In particular, the component of the curvature tensor R θφ θφ grows like ξ 2 . The maximum possible E c.m. is achieved when ξ ∼ 1, then z ∼ q m . The example with the Ernst metric shows that strong backreaction of the magnetic field on the geometry may restrict the growth of E c.m. to such extent that even in spite of large b, the effect disappears because of the factor ξ that enters the metric. It is of interest to consider the exact rotating magnetized black hole [21] that generalizes the Kerr metric but this problem certainly needs separate treatment.

XIII. SUMMARY AND CONCLUSION
We obtained characteristics of ISCO and the energy in the CM frame in two different situations. For the near-extremal case, the BSW effect reveals itself with or without the magnetic field, so we gave generalization of previous results to the presence of this field.
Thus this case is elaborated now for dirty magnetized black holes. In doing so, there is qualitative difference between dirty rotating black holes and the Kerr one. Namely, the radius of ISCO depends on the magnetic field strength b already in the main approximation with respect to small surface gravity κ in contrast to the case of the vacuum metric [4], where this dependence reveals itself in the small corrections only.
For extremal black holes, we showed that, due to the strong magnetic field, there exists the near-horizon ISCO that does not have a counterpart in the absence of this field.
Correspondingly, we described the effect of high-energy collisions near these ISCO.
We demonstrated that rotation destroys near-horizon ISCO both for the nonextremal and extremal horizons. so lim b→∞ r 0 (b) = r + . However, if the parameter responsible for rotation is small, E c.m. is large in this limit.
For slowly rotating black holes we analyzed two different regimes of rotation thus having generalized previous results on the Kerr metric [4]. Both the parameters of expansion and the results agree with the Kerr case. In particuar, for modestly slow rotation the individal energy of the particle on ISCO is unbound.
In the main approximation, the expressions for the ISCO radius and angular momentum in dimensionless variables are model-independent, so here one can see universality of black hole physics.
We also found the three-velocity of a particle on ISCO in the ZAMO frame. It turned out that for slowly rotating dirty black holes in the magnetic field it coincides with the value typical of the Kerr metric without a magnetic field, v ≈ 1 2 . Correspondingly, previous explanation of the high E c.m. as the result of collision of very fast and slow particles [7] retains its validity in the scenarios under discussion as well.
In previous studies of the BSW effect in the magnetic field [3], [4], [13], some fixed background was chosen. In this sense, the magnetic field was supposed to be weak in that it did not affect the metric significantly (although it influenced strongly motion of charged particles). Meanwhile, the most part of the formulas obtained in the present work applies to generic background and only asymptotic behavior of the metric near the horizon was used. Therefore, they apply to the backgrounds in which the magnetic field enters the metric itself, with reservation that the surface gravity κ = κ(b), etc. In particular, we considered the static magnetized Ernst black hole and showed that strong backreaction of the magnetic field on the geometry bounds the growth of E c.m. . Throughout the paper, it was assumed that the effect of the electric charge on the metric is negligible. It is of interest to extend the approach of the present work to the case of charged black holes.