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International scientific-online conference
38
ENERGETIC PROCESSES AROUND REGULAR BLACK HOLES
Xudoyberdieva M.K.
Juraeva N.B.
National University of Uzbekistan, Tashkent 100174, Uzbekistan
Ulugh Beg Astronomical Institute, Tashkent 100052, Uzbekistan
Xudoyberdiyeva94@Inbox.Ru
https://doi.org/10.5281/zenodo.14060033
INTRODUCTION .
The investigation of high-energetic particles collisions
in the vicinity of rotating black holes was initiated in [1] where the collisional
version of the Penrose process [2] was investigated. The new urge to
considering such processes came from an interesting observation made in Ref.
[3]. It was found there that two particles which move towards the horizon of the
extremal black holes can produce an infinity energy in the centre of mass frame
Ec.m.. This effect (called the BSW one after the names
of its authors) provoked a
large series of works and is under active study currently. The most part of them
was restricted to the investigation of the vicinity of the horizon where collision
occurs.
The spacetime around a RBH can be obtained using GR coupled to nonlinear
electrodynamics (NED) and thecorresponding action for these coupled fields is
written
𝑆 =
1
16𝜋
∫ 𝑑𝑥
4
√−𝑔 (𝑅 − 𝐿(𝐹)) (1)
where
𝐹 = 𝐹
𝜇𝜈
𝐹
𝜇𝜈
is the electromagnetic field invariantand
𝐹
𝜇𝜈
= 𝐴
𝜈,𝜇
− 𝐴
𝜇,𝜈
is
the electromagnetic field tensorand
𝐴
𝜇
is the electromagnetic field four potential.
The spacetime around the RBH has been found bycoupling Einstein’s theory of
gravity to NED where theLagrangian is found as a function of the
electromagnetic
field invariant.
𝐿(𝐹) =
4𝑛
𝛼
(𝛼𝐹)
𝑘+3
4
[1 + (𝛼𝐹)
𝑘
4
]
1+
𝑛
𝑘
(2)
For the case
𝑘 = 1
and
𝑛 ≥ 3
, where
n
is assumed to be an integer [1], the
metric tensor is,
𝑑𝑠
2
= −𝑓𝑑𝑡
2
+ 𝑓
−1
𝑑𝑟
2
+ 𝑟
2
𝑑𝛺
2
(3)
SCIENCE AND INNOVATION IN THE
EDUCATION SYSTEM
International scientific-online conference
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The effective potential for a charged particle a constant plane (
𝜃 = 𝑐𝑜𝑛𝑠𝑡
and
𝜃̇ = 0)
can be found by solving equation
ℰ = 𝑉
𝑒𝑓𝑓
(taking
𝑟̇ = 0)
and we have
and four-velocities of the charged particle
𝑡̇ =
1
𝑓
(ℰ − 𝑞𝐴
𝑡
)
𝑟̇
2
= (ℰ − 𝑞𝐴
𝑡
)
2
− 𝑓[1 + (
𝑙
𝑟𝑠𝑖𝑛𝜃
−
𝑞𝐵
2
𝑟𝑠𝑖𝑛𝜃)
2
]
𝜙̇ =
𝑙
𝑟
2
𝑠𝑖𝑛
2
𝜃
−
𝑞𝐵
2
(4)
In this section, we will study the centre-of-mass energy of two particles in the
case of charged-charged, chargedneutral particles collisions. The expression for
the centreof-mass energy for two particle system with mass
𝑚
1
and
𝑚
2
, in a
given gravitational field is as a sum of two-momenta
{𝐸
𝑐𝑚
, 0,0,0} = 𝑚
1
𝑢
1
𝜇
+ 𝑚
2
𝑢
2
𝜇
(5)
where,
𝑢
1
𝛼
and
𝑢
2
𝛽
are four-velocity of the two colliding particles and the
velocities satisfy the condition
𝑢
𝜇
𝑢
𝜇
= −1
. Keeping the condition one can
square
(5)
and we have.
𝐸
𝑐𝑚
2
= 𝑚
1
2
+ 𝑚
2
2
− 2𝑚
1
𝑚
2
𝑔
𝜇𝜈
𝑢
𝜇
𝑢
𝜈
(6)
Let us consider simple estimation, assuming that the mass of the particles is
different from each other
N
times, i.e.
𝑚
1
= 𝑁𝑚
2
,
N
can not be zero,
obviously that
N >
1 corresponds to
𝑚
1
> 𝑚
2
, and vice versa
N <
1 to
𝑚
1
<
𝑚
2
.Thus, the expression for center-of-mass energy (
6
) takes the following form.
ℰ
𝑐𝑚
2
=
𝐸
𝑐𝑚
2
𝑚
2
= 1 + 𝑁
2
− 2𝑁𝑔
𝜇𝜈
𝑢
1
𝜇
𝑢
2
𝜈
(7)
Using ( 4) and (7 ) conducted an analysis.
Here we will consider the collision of the charged particles with the same mass
𝑚
1
=
𝑚
2
= 𝑚
(charge might be different, for example, electron and positron) and
initial energy
ℰ
1
= ℰ
2
= 1
, then the expression for the centerof-mass energy
takes the following form Now we will study in detail, center-of-mass energy
of two colliding (neutral/charged) particles with different cases, i.e. particles
with the same mass (
𝑚
1
=
𝑚
2
= 𝑚
) and different mass
𝑚
2
≠ 𝑚
1
( assuming
𝑚
1
= 𝑁𝑚
2
, here
N
is some non-zero number) and the angular momentum
(
𝑙
1
= −𝑙
2
=l and
𝑙
1
≠ 𝑙
2
), and initial energies
ℰ
1
= ℰ
2
= 1
in the equatorial
plane using equations of motion charged particles .In figure
1
radial
dependence of center-of-mass is plotted in different values of Q and n.
SCIENCE AND INNOVATION IN THE
EDUCATION SYSTEM
International scientific-online conference
40
FIG. 1: Radial dependence of center-of-mass energy for charges with value
q
=
−
100 and
q
= 2
One can see from left panel of this figure that the center-of mass energy
increases as the charge of RBH increase, but the increase of n cause to decrease
the energy (middle panel), left panel is plotted for extreme charged RBH case
for given values of n obviously that the maximum of the energy is at horizon
when
𝑟 → 𝑟
ℎ
Let us consider that two charged particle having the same charge and the
same angular momentum collision with opposite direction. The question that
what
is
the
minimum values of charge
q
and angular momentum
l
that the center-of-mass
energy
ℰ > 100
can be greater than 100.
References:
1.
B. Toshmatov, Z. Stuchl´ık, and B. Ahmedov, Phys. Rev.D 98, 028501
(2018), arXiv:1807.09502 [gr-qc] .
2.
R. M. Wald, Phys. Rev. D. 10, 1680 (1974).
3.
A. N. Aliev and A. E. G¨umr¨uk¸c¨uoˇglu, Phys. Rev. D 71,104027 (2005),
hep-th/0502223 .
4.
A. A. Abdujabbarov, B. J. Ahmedov, and V. G. Kagramanova, General
Relativity and Gravitation 40, 2515(2008), arXiv:0802.4349 [gr-qc] .
5.
A. Abdujabbarov and B. Ahmedov, Phys. Rev. D 81,044022 (2010),
arXiv:0905.2730 [gr-qc] .
6.
A. A. Grib and Y. V. Pavlov, Gravitation and Cosmology17, 42 (2011),
arXiv:1010.2052 [gr-qc] .
