Pis’ma v ZhETF, vol. 111, iss. 5, pp. 300 - 300

March 10

Schwarzschild black hole as accelerator of accelerated particles

O.B.Zaslavskii¹⁾

Department of Physics and Technology, Kharkov V. N. Karazin National University, 61022 Kharkov, Ukraine

Institute of Mathematics and Mechanics, Kazan Federal University, 420008 Kazan, Russia

Submitted 13 October 2019

Resubmitted 27 January 2020

Accepted 30

January 2020

DOI: 10.31857/S0370274X20050033

We consider collision of two particles near the hori-

Here, it is supposed that both particles move in

zon of a nonextremal static black hole. At least one of

the same direction, P being the radial momentum,

∫_∞

them is accelerated. We show that the energy E_c.m. in

X =E-m

dr^′a(r^′).

the center of mass can become unbounded in spite of

If a critical particle 1 collides with a usual particle 2,

the fact that a black hole is neither rotating nor electri-

const

cally charged. In particular, this happens even for the

γ∼

√

(6)

(r₀ - r₊)

Schwarzschild black hole. The key ingredient that makes

it possible is the presence of positive acceleration (repul-

where a constant depend on the details of trajectories.

sion). Then, if one of particles is fine-tuned properly, the

Then, taking r₀ as close to r₊ as one likes, we obtain the

effect takes place. This acceleration can be caused by an

unbounded growth of γ and E^2c.m. that can be thought

external force in the case of particles or some engine in

of as a counterpart of a similar formula for the Kerr

the case of a macroscopic body (“rocket”). If the force is

metric was considered. Thus there is a close analogy be-

attractive, E_c.m. is bounded but, instead, the analogue

tween our case and the BSW effect near nonextremal

of the Penrose effect is possible.

black holes. In particular, now the same difficulties per-

More explicitly, the black hole metric has the form

sist that forbid arrival of the near-extremal particle from

ds² = -fdt² + f^-1dt² + r²(dθ² + sin² θdφ²),

(1)

infinity because of the potential barrier typical of any

nonextremal black hole. Therefore, either such a particle

where the horizon is located at r = r₊, so f(r₊) = 0.

is supposed to be created already in the vicinity of the

We consider pure radial motion with the four-velocity

horizon from the very beginning or one is led to exploit-

u^µ and four-acceleration a^µ with

ing scenarios of multiple scattering. What is especially

interesting is that the effect under discussion is valid for

a_µa^µ ≡ a²,

(2)

the Schwarzschild black hole.

where by definition a ≥ 0. The presence of acceleration

Usually, the factor connected with additional forces

enables one to have fine-tuned (“critical”) particles, such

(like gravitational radiation) are referred to as obstacles

that the energy

to gaining large E_c.m.. To the extent that such influ-

∫ ∞

ence can be modeled by some force, backreaction does

E = m dr^′a(r^′).

(3)

not spoil the effect. Meanwhile, as we saw now, in our

context the presence of the force not only is compatible

Let particles 1 and 2 move from infinity and collide

with the BSW effect but it can be its origin.

in some point r₀. The energy in the center of mass frame

If a black hole is surrounded by external electromag-

netic fields, we can suppose that the described mecha-

E^2c.m. = -(m₁u^µ1 + m₂u^µ2)(m₁u_1µ + m₂u_2µ) =

nism promotes high energy collisions near black holes.

= m²¹ + m²² + 2m₁m₂γ,

(4)

The Schwarzschild metric and radial motion give us the

simplest exactly solvable example but it is quite proba-

where γ = -u_1µu^2µ is the Lorentz factor of relative

ble that qualitatively the similar results hold in a more

motion. It follows from the above equations that

realistic situation as well.

X₁X₂ - P₁P₂

γ =

(5)

Full text of the paper is published in JETP Letters

m₁m₂f

journal. DOI: 10.1134/S0021364020050033

¹⁾e-mail: zaslav@ukr.net

300

Письма в ЖЭТФ том 111 вып. 5 - 6

2020