Pis'ma v ZhETF, vol.92, iss.3, pp. 147-151
© 2010 August 10
On particles collisions in the vicinity of rotating black holes
A. A. Grib, Yu. V. Pavlov* ^
+ Theoretical Physics and Astronomy Department, The Herzen University, 191186 St. Petersburg, Russia *Institute of Problems in Mechanical Engineering, RAS, 199178 St. Petersburg, Russia
Submitted 9 June 2010
Scattering of particles in the gravitational field of rotating black holes is considered. It is shown that scattering energy of particles in the centre of mass system can obtain very large values not only for extremal black holes but also for nonextremal ones. Extraction of energy after the collision is investigated. It is shown that due to the Penrose process the energy of the particle escaping the hole at infinity can be large. Contradictions in the problem of getting high energetic particles escaping the black hole are resolved.
In [1] we put the hypothesis that active galactic nuclei can be the sources of ultrahigh energy particles in cosmic rays observed recently by the AUGER group (see [2]) due to the processes of converting dark matter formed by superheavy neutral particles into visible particles — quarks, leptons (neutrinos), photons. If active galactic nuclei are rotating black holes then in [1] we discussed the idea that "This black hole acts as a cosmic supercollider in which superheavy particles of dark matter are accelerated close to the horizon to the Grand Unification energies and can be scattering in collisions." It was also shown [3] that in Penrose process [4] dark matter particle can decay on two particles, one with the negative energy, the other with the positive one and particles of very high energy of the Grand Unification order can escape the black hole. Then these particles due to interaction with photons close to the black hole will loose energy analogously up to the Greisen-Zatsepin-Kuzmin limit in cosmology [5, 6].
First calculations of the scattering of particles in the ergosphere of the rotating black hole, taking into account the Penrose process, with the result that particles with high energy can escape the black hole, were made in [7, 8]. Recently in [9] it was shown that for the rotating black hole (if it is the critical one) the energy of scattering is unlimited. The result of [9] was criticized in [10, 11] in the sense that it does not occur in nature. The authors of [10, 11] claimed that if the black hole is not a critical rotating black hole so that its dimension-less angular momentum A ^ 1 but A = 0.998 then the energy is limited.
In this paper we show that the energy of scattering in the centre of mass system can be still unlimited in the cases of multiple scattering. In the first part we calculate this energy, reproduce the results of [9-11] and show that in some cases (multiple scattering) the results
of [10, 11] on the limitations of the scattering energy for nonextremal black holes are not valid.
In the second part we obtain the results for the extraction of the energy after collision in the field of the Kerr's metric. It occurs that the Penrose process plays important role for getting larger energies of particles at infinity. Our calculations show that the conclusion of [11] about the impossibility of getting at infinity the energy larger than the initial one in particle collisions close to the black hole is wrong.
The system of units G = c = 1 is used in the paper.
1. The energy of collision in the field of black holes. Let us consider particles falling on the rotating chargeless black hole. The Kerr's metric of the rotating black hole in Boyer-Lindquist coordinates has the form
ds2 = di
(r2 + a2 cos2d)
2 2Mr (dt — a sin20 dip)2
r2 + a2 cos29
dr2
d9'
(r2+ a2) sin20dip2, (1)
where
A = r2 - 2Mr + a2
(2)
M is the mass of the black hole, J = aM is angular momentum. In the case a = 0 the metric (1) describes the static chargeless black hole in Schwarzschild coordinates. The event horizon for the Kerr's black hole corresponds to the value
(3)
r = rH = M + s/M2 - a2.
The Cauchy horizon is
r = rc = M - s/M2 - a2. (4)
The surface called "the static limit" is defined by the expression
e-mail: andrei_grib0mail.ru; yuri.pavlov0mail.ru
r r„ =M+s/M2^a2 cos2 9.
(5)
148
A. A. Gilb, Yu. V.Pavlov
The region of space-time between the horizon and the static limit is ergosphere.
For equatorial (9 = 7r/2) geodesies in Kerr's metric (1) one obtains ([12], §61):
dt_ dï
2 Ma'
dip dï
= £
1 A
2 Ma
2 Ma
L
(6)
(7)
2M , г^2
7F (ae - L)
a2e2
L2
A x
(8)
where ¿i = 1 for timelike geodesies (si = 0 for isotropic geodesies), r is the proper time of the moving particle, e = const is the specific energy: the particle with rest mass to has the energy em in the gravitational field (1); Lrn = const is the angular momentum of the particle relative to the axis orthogonal to the plane of movement.
We denote x = r/M, xh = rg/M, xc = rc/M, A = a/M, ln = Ln/M, Ax = x2 - 2x + A2. For the energy in the centre of mass frame of two colliding particles with angular momenta L1, L2, which are non-relativistic at infinity (ei =£2 = 1) and are moving in Kerr's metric using (1), (6)-(8) one obtains [9]:
1
2 to2
2x2(x^1) + /i/2(2 — a:)
xAx
■ 2a2(x + 1) — 2A(li + /2)
(9)
-y/(2x2 + 2(h-A)2 - l\x) (2x2 + 2(/2-A)2 - l2x)
To find the limit r rn for the black hole with a given angular momentum A one must take in (9) x = xh +a with a 0 and do calculations up to the order a2. Taking into account A2 = xh%c, %h + %c = 2, after resolution of uncertainties in the limit a 0 one obtains
Ec.mjr rH) 2m
f
(h^h)2
2xc(h -lH)(h -ih)
(10)
where Ih = 2s.a ¡A.
For the extremal black hole A = xh = 1, Ih = 2 and the expression (10) is divergent when the dimensionless angular momentum of one of the falling into the black hole particles / 2. The scattering energy in the centre of mass system is increasing without limit [9].
Let's note, that to get the collision with infinite energy one needs the infinite interval of as coordinate as
proper time of the free falling particle. Really, from Eqs. (6), (8) for a particle with dimensionless angular momentum / and specific energy e = 1 falling from some point ro = xqm to the point r/ = xfm > rjj one obtains for the coordinate time (proper time of the observer at rest far from the black hole)
At = M
xo
i
Vi (x3 + A2x + 2A(A - I)) dx
Xf
(x — xh)(x — xc)\/2x2 — l2x
2 (A^l)2 (H)
For the interval of proper time of the free falling to the black hole particle one obtains from (8)
At = M
xo /
B3/2 dx
sj2x2 - l2x + 2(A - I)2
(12)
In extremal case (A = 1, / = 2) the integrals (11), (12) diverges for Xf —t xh = 1 and Ai « M2y/2(xf — l)-1, At « M\ ln(a;/ - 1)|/V2 for xf 1.
From (7), (8) for the angle of the particle falling in equatorial plane of the black hole one obtains
Aip =
xo
i
y/x (xl + 2(A - I)) dx
(x - xh)(x - xc)s/2x2 ^l2x + 2(A^l)2'
(13)
If A ^ 0, then integral (13) is divergent for xf —¥ xh-In extremal case (A = 1, / = 2) Aip « V2(xf — 1)_1 for Xf 1. So before collision with infinitely large energy the particle must commit infinitely large number of rotations around the black hole.
Can one get the unlimited high energy of this scattering energy for the case of nonextremal black hole? Formula (8) leads to limitations on the possible values of the angular momentum of falling particles: the massive particle free falling in the black hole with dimensionless angular momentum A being nonrelativistic at infinity (e = 1) to achieve the horizon of the black hole must have angular momentum from the interval
^2 (l + Vl + A^j =lL < I <Ir = 2(l + Vl - A) •
(14)
Putting the limiting values of angular momenta ii , ir into the formula (10) one obtains the maximal values of the collision energy of particles freely falling from infinity
2 to
Ell(r ^ rH) =
Vl - a2
(15)
■f-
a2 + (1 + vtta + vt^jy 1 + Vl^A2
r
a
r
r
r
r
For A = 1 — e with e —t 0 formula (15) gives:
rH)
(21/4 +
TO
¡174
(16)
So even for values close to the extremal A = 1 of the rotating black hole E™^/m can be not very large as mentioned in [10, 11]. So for j4max = 0.998 considered as the maximal possible dimensionless angular momentum of the astrophysical black holes (see [13]), from (15) one obtains E^/m « 18.97.
Does it mean that in real processes of particle scattering in the vicinity of the rotating nonextremal black holes the scattering energy is limited so that no Grand Unification or even Planckean energies can be obtained? Let us show that the answer is no! If one takes into account the possibility of multiple scattering so that the particle falling from infinity on the black hole with some fixed angular momentum changes its momentum in the result of interaction with particles in the accreting disc and after this is again scattering close to the horizon then the scattering energy can be unlimited.
The limiting value of the angular momentum of the particle close to the horizon of the black hole can be obtained from the condition of positive derivative in (6) dt/dr > 0, i.e. going "forward" in time. So close to the horizon one has the condition I < e2xh/A which for e = 1 gives the limiting value ih-
From (8) one can obtain the permitted interval in r for particles with e = 1 and angular momentum I = = Ih — S. To do this one must put the left hand side of (8) to zero and find the roots:
»1,2 =
I2 ± ^Z4 - 16(A - I)2
(17)
In the second order in 5 close to the horizon one obtains
I = lH-6
X<XH
S2i
ixH^/l — A2
(18)
The effective potential defined by the right hand side of (8) leads to the following behaviour of the particle. If the particle goes from infinity to the black hole it can achieve the horizon if the inequality (14) is valid. However the scattering energy in the centre of mass frame given by (15) is not large. But if the particle is going not from the infinity but from some distance defined by (18) then due to the form of the potential it can have values of i = ih—5
Для дальнейшего прочтения статьи необходимо приобрести полный текст. Статьи высылаются в формате PDF на указанную при оплате почту. Время доставки составляет менее 10 минут. Стоимость одной статьи — 150 рублей.