content_image/438464.jpg content_image/438465.jpg content_image/438466.jpg content_image/438467.jpg content_image/438468.jpg content_image/438469.jpg content_image/438470.jpg content_image/438471.jpg content_image/438472.jpg content_image/438480.jpg content_image/438478.jpg content_image/438479.jpg content_image/438477.jpg content_image/438476.jpg content_image/438475.jpg content_image/438474.jpg content_image/438473.jpg
that the Penrose process plays important role for getting larger energies of particles at infinity. Our calculations show tha t the conclusion of [11] about the impossibility of getting at infi nity 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. 

## 2. 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 

<img src='content_image/438464.jpg'>

where 

<img src='content_image/438465.jpg'>

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 

<img src='content_image/438466.jpg'>

The Cauchy horizon is 

<img src='content_image/438467.jpg'>

The surface called “the static limit” is defined by the expres sion 

<img src='content_image/438468.jpg'>

The region of space-time between the horizon and the static limit is ergosphere. 

For equatorial ( θ = π/ 2) geodesics in Kerr’s metric (1) one obtains ([13], § 61) 2 

<img src='content_image/438469.jpg'>

<img src='content_image/438470.jpg'>

<img src='content_image/438471.jpg'>

where δ$_{1}$ = 1 for timelike geodesics ( δ$_{1}$ = 0 for isotropic geodesics), τ is the proper time of the moving particle, ε = const is the specific energy: the particle with rest mass m has the energy ε m in the gravitational field (1); Lm = const is the angular momentum of the particle relative to the axis orthogo- nal to the plane of movement. 

Let us find the energy E$_{c}$$_{.}$$_{m}$$_{.}$ in the centre of mass system of two colliding particles with the same rest m in arbitrary gravi- tational field. It can be obtained from 

<img src='content_image/438472.jpg'>

where u i = dx$^{i}$/ds . Taking the squared (9) and due to u$^{i}$u$_{i}$ = 1 one obtains 

<img src='content_image/438480.jpg'>

The scalar product does not depend on the choice of the coor- dinate frame so (10) is valid in an arbitrary coordinate system and for arbitrary gravitational field. 

We denote x = r / M , x$_{H}$ = r$_{H}$ / M , x$_{C}$ = r$_{C}$ / M , A = a / M , l$_{n}$ = L$_{n}$ / M . Apply the formula (10) for calculation of the en- ergy in the centre of mass frame of two colliding particles with angular momenta L$_{1}$ , L$_{2}$ , which are nonrelativistic at infinity ( ε$_{1}$ = ε$_{2}$ = 1) and are moving in Kerr’s metric. Using (1), (6)–(8) one obtains [9] 

<img src='content_image/438478.jpg'>

where 

<img src='content_image/438479.jpg'>

To find the limit r → r$_{H}$ for the black hole with a given angu- lar momentum A one must take in (11) x = x$_{H}$ + α with α → 0 and do calculations up to the order α $^{2}$. Taking into account A 2 = x$_{H}$x$_{C}$ , x$_{H}$ + x$_{C}$ = 2, after resolution of uncertainties in the limit α → 0 one obtains 

<img src='content_image/438477.jpg'>

where 

<img src='content_image/438476.jpg'>

In the limit → the expression 

<img src='content_image/438475.jpg'>

given first in [9], showing the unlimited increasing of the en - ergy of collision when the specific angular momentum of one of falling particles goes to the limiting possible value equal to 2 M necessary for achieving the horizon of the extremal 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 mo- mentum A being nonrelativistic at infinity ( ε = 1) to achieve the horizon of the black hole must have angular momentum from the interval 

<img src='content_image/438474.jpg'>

Note that for the mentioned limiting values the right hand side of the formula (8) is zero for 

<img src='content_image/438473.jpg'>

