научная статья по теме PARTICLE COLLISIONS NEAR THE COSMOLOGICAL HORIZON OF A REISSNER-NORDSTRöM DE SITTER BLACK HOLE Физика

Текст научной статьи на тему «PARTICLE COLLISIONS NEAR THE COSMOLOGICAL HORIZON OF A REISSNER-NORDSTRöM DE SITTER BLACK HOLE»

Pis'ma v ZhETF, vol.94, iss.8, pp.631-634

© 2011 October 25

Particle collisions near the cosmological horizon of a Reissner-Nordstrom de Sitter black hole

C. ZhongS. Gao1) Department of Physics, Beijing Normal University, 100875 P.R., Beijing, China

Submitted 23 August 2011 Resubmitted 8 September 2011

It has recently been shown that black holes can act as particle accelerators and two particles can collide with arbitrarily high center-of-mass (CM) energy under certain critical conditions. In this paper, we investigate particle collisions outside a Reissner-Nordstrom-de Sitter (RN-dS) black hole. We find that infinite CM-energy can be produced near the cosmological horizon for generic spacetime configurations. Remarkably, such infinite CM-energy does not require the black hole to be extremal, in contrast to spacetimes in the absence of cosmological constants. However, since the charge of an astrophysical body is negligible, the required charge to mass ratio of the particle is extremely higher than that of any elementary particle.

1. Introduction. Banados, Silk and West (BSW) [1] showed that Kerr black holes can serve as particle accelerators and infinite center-of-mass energies can in principle arise. The BSW-mechanism was soon extended to different black hole solutions [2-12]. These works suggest that the following features are required for the divergence of the CM-energy: 1) the collision must occur arbitrarily close to the horizon; 2) one of the particles possesses a critical value of angular momentum or charge; 3) the black hole must be extremal. It has been pointed out [2, 3] that condition 3 can not be fulfilled due to the theoretical upper limit on the spin parameter of black hole. So it would be more meaningful to look for infinite CM-energies around a non-extremal black hole. Recently, we have proven [13] that infinite CM-energies cannot be created outside a non-extremal Kerr black hole. However, Wei et al. [14] pointed out that for a Kerr de Sitter black hole, two particles can collide with arbitrarily high CM-energy without imposing the extremal condition. This result indicates that if cosmological constant is taken into account, condition 3 may be released. Since the well-known ACDM-model fits remarkably well with the current cosmological observations, it is worthwhile to further study the accelerating effect for spacetimes with a positive cosmological constant.

In this paper, we investigate particle collisions near a charged black hole in an asymptotically de Sitter spacetime, i.e., Reissner-Nordstrom-de Sitter black hole. Previously, Zaslavskii [4] has studied the radial motion of charged particles in a Reissner-Nordstrom (RN) background and found the result is similar to that in a Kerr background. In particular, infinite CM-energies

cczhong0mail.bnu.edn.cn, sijie0bnu.edn.cn

can only be attained when the black hole is extremal, i.e., the black hole possesses the maximum charge Q = M. However, the situation will be different if a positive cosmological constant A is introduced. In an asymptotically de Sitter black hole, there exists a cosmological horizon which is located at the radius of order A-1/2. By calculating the radial motion of charged particles, we find that infinite CM-energies can be obtained at the cosmological horizon of a generic RN-dS black hole. In this case, the black hole need not be extremal. However, a critical charge is required for one of the particles. By numerical estimation, we find that the charge to mass ratio of the particle is much higher than that of an electron. Thus, the infinite CM-energy is not realizable in the real world.

This paper is organized as follows. In section 2, we discuss the horizons of RN-dS space-times. In section 3, we calculate the CM-energy of two particles near the cosmological horizon of RN-dS black hole. In section 4, we perform some numerical calculations based on the data of astrophysical observations. Finally, conclusions are made in section 5.

2. RN-dS Black Hole Horizons. The RN-dS black hole is described by the following metric

dr2

ds2 = -adt2 + — + r2d92 + r2 sin2 Odtp2, (1) a

where a = 1 — ^ + ^— p- and I = The roots of a = 0 give rise to the location of horizons. We denote the three roots by r 1 and r2 correspond to the

inner and outer horizons of a RN black hole, while r3 is called the cosmological horizon. Due to the smallness of A, r3 can be approximated by I and is much larger than fj and r2■ Fig. 1 depicts the function a(r) in the RN and RN-dS cases. For I M, the radius of cosmo-

IlHCbMa b ?K3T<J> tom 94 Bbin.7-8 2011

631

632

C. Zhong, S. Gao

Fig. 1. The function a(r) for the RN and RN-dS black holes

logical horizon r3 is much larger than the radius of the two black hole horizons r1 and r2.

A RN-dS black hole is called extremal if two or three horizons coincide. The three types of extremal horizons are illustrated in Fig. 2. Using the method suggested by [7], it is easy to check that when

M =

-I,

6

is satisfied, the three horizons coincide at

I

»"1,2,3 = —?=■

(2)

(3)

3. Collision energy in the center-of-mass frame. In this section, we shall study the collision of two particles in the RN-dS spacetime. Suppose that the two particles have the same mass m. and different charges q\ and q2. In the rest of the paper, we shall focus on collisions near the cosmological horizon which is located at r = r3. Collisions near the black hole horizon r = r2 is essentially the same as the case in the absence of the cosmological constant which has been studied by Zaslavskii [4]. Suppose the two particles collide in the region r > r3. The center-of-mass energy is given by [1]

E,m = Tn

where

dx"

1 - 9abUfUb2

d y

dx" )

(4)

(5)

is the four-velocity of particle at the collision point. The motion of a charged particle is determined by the following Lagrangian [15]

r — ^H. • ft • v L. — ^ Qßv^ ^

qAßxß.

(6)

Here = dx11 /dr and A^ are the components of the electromagnetic 4-potential. Substituting the RN-dS metric Eq. (1) into Eq. (6), we find [16],

£ =

TO

»2 ¿2

r2 sin2 I

a

qQ:

t.

(7)

The Euler-Lagrange equation then leads to the constant of the motion

E = m.at+—.

r

(8)

For our purposes, we shall confine our discussion to radial motions, i.e., 0 = 4> = 0. Thus, the normalization condition g^u^u" = yields

97?. y \ r )

m.a.

(9)

Here we have chosen the minus sign because — (d/dr)a is a future-directed timelike vector outside the cosmological horizon.

By direct substitution of Eqs. (8) and (9) in Eq. (4), we find

Ei

2m* 1+

(10)

Our purpose is to find conditions for a possible infinite Ec,m. Eq. (10) suggests that an infinite Ec,m can occur only at a horizon where a = 0. We shall focus on the cosmic horizon located at r = r3. To see if Ec,m diverges at the horizon, we need to calculate the limiting value of the numerator of Eq. (10). The lowest order of the numerator can be obtained by taking a = 0 and r = th, which gives

Ei

Th )

E2

92

Th )

tn)

To simplify this formula, we need to know the sign of E{ - qiQ/rn, where i = 1,2. According to Eq. (8), Ei — qiQ/rH is proportional to t. Note that (d/dt)a is spacelike in the region r > rjf. Thus the sign of t corresponds to an ingoing mode or outgoing mode. If the two particles could take different sign at the horizon, it would mean that one particle falls toward the horizon and the other one escapes from the horizon. This is not possible because even a photon cannot escape from the horizon2^. Thus, E^ — qiQ/rH must have the same sign and then Eq. (11) vanishes. The vanishing of Eq. (11) is important for our following analysis. Otherwise, Ec,m would be generically divergent without

2,The same argument has been used in our paper [13] for the inner horizon of a Kerr black hole.

IlHCbMa b ?K3T<1> tom 94 bmu.7-8 2011

m2a

Ua =

r

Particle collisions near the cosmological horizon of a Reissner-Nordström

633

a

(c)

'1,2,3

Fig. 2. Three types of extremal RN-dS black holes

requiring the particle to possess a critical charge. By expanding the numerator of Eq. (10) around a = 0, we obtain the following limit

E2 1

c. m _ 1 i _

2m2 2

«2

Q

qi

EirH

Q

. «1

EirH

Q

«2

Q

(12)

We see that Ec,m blows up at the horizon if one of the particle takes the critical charge

Qc =

ErH

Q

(13)

and the other particle takes any different value of charge.

Since the divergence of energy occurs at the horizon r = th = r3, we need to check if the particle can actually reach the horizon from infinity. This requires that the square root of Eq. (9) must be positive in a vicinity of the horizon. Note that a < 0 in the region r > r3, it follows immediately that r2 > 0 outside the cosmological horizon. Thus, with the critical charge, the particle can fall all the way from infinity to the horizon. The above argument does not require any fine tuning on the parameters Q, M and A. Thus, an infinite Ec,m is attainable at the cosmological horizon of a generic RN-dS black hole.

4. Numerical estimation. We see from Eq. (13) that the critical charge is proportional to r3, which is a very large quantity. Consequently, it may lead to a very large charge to mass ratio. To estimate the ratio, we choose E in Eq. (13) to be the proper mass m. of the particle. Thus, the required charge to mass ratio in SI units reads

q/m =

4ireor3C2

Q

To estimate r3, write down a in SI units

2 GM GQ2 GAr2

a = 1

47re0c4r2 3c2

(14)

(15)

We take M = 2 • 1032 kg, which is about 100 times the solar mass. In geometrized units, the maximum charge

of a black hole is Q = M. So in SI units, the maximum charge is given by

Q

/

97re0G

M ~ 1.8 • 1022 C.

(16)

Based on Supernovae observations, the value of the cosmological constant is about

A ~ 8TrPvac • 10-120, (17)

where pvac ~ 1O04 kg • m-3 is the vacuum energy density. Then r3 can be solved as r3 ~ 1026 m and the right-hand side of Eq. (14) is found to be

q/m. ~ 5 • 10 C/kg.

(18)

Note that the charge to mass ratio for an elec

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