Pis'ma v ZhETF, vol.92, iss.9, pp.635-638
© 2010 November 10
Acceleration of particles by nonrotating charged black holes
O.B. Zaslavskii Kharkov V.N. Karazin National University, 61077 Kharkov, Ukraine
Submitted 23 September 2010
Recently, in the series of works a new effect of acceleration of particles by black holes was found. Under certain conditions, the energy in the centre of mass frame can become infinitely large. The essential ingredient of such effect is the rotation of a black hole. In the present Letter, we argue that the similar effect exists for a nonrotating but charged black hole even for the simplest case of radial motion of particles in the Reissner-Nordstrom background. All main features of the effect under discussion due to rotating black holes have their counterpart for the nonrotating charged ones.
I. Introduction. Recently, it was made an interesting observation that black holes can accelerate particles up to unlimited energies Ecm in the centre of mass frame [1]. This stimulated further works in which details of this process were investigated [2 - 6] and, in particular was found that the effect is present not only for extremal black holes but also for nonextremal ones [2]. These results have been obtained for the Kerr metric (they were also extended to the extremal Kerr-Newman one [3] and the stringy black hole [4]). In the work [7] generalization of these observations was performed and it was demonstrated that the effect in question exists in a generic black hole background (so a black hole can be surrounded by matter) provided a black hole is rotating. Thus, rotation seemed to be an essential part of the effect. It is also necessary that one of colliding particles have the angular momentum L\ = Ei/wh [7], where E is the energy, wh is the angular velocity of a generic rotating black hole. If wh 0, L1 —t oo, so for any particles with finite L the effect becomes impossible. Say, in the Schwarzschild space-time, the ratio Ecm/m (to is the mass of particles) is finite and cannot exceed 2y/5 for particles coming from infinity [8].
Meanwhile, sometimes the role played by the angular momentum and rotation, is effectively modeled by the electric charge and potential in the spherically-symmetric space-times. So, one may ask the question: can we achieve the infinite acceleration without rotation, simply due to the presence of the electric charge? Apart from interest on its own., the positive answer would be also important in that spherically-symmetric spacetimes are usually much simpler and admit much more detailed investigation, mimicking relevant features of rotating space-times. As we will see below, the answers is indeed "yes"! Moreover, in [1-7] rotation manifested itself in both properties of the background metric and
in the nonzero value of angular momentums of colliding particles. However, below we show that both manifestations of rotation can be absent but nonetheless the effect under discussion reveals itself. This is demonstrated for the radial motion of particles in the Reissner-Nordstrom black hole, so not only wh = 0 but also L1 = L2 = 0 for both colliding particles. It is surprising that the effect reveals itself even in so simple situation (which is discussed even in textbooks).
Formally, the results for the accleration of charged particle by the Reissner-Nordstrom black hole can be obtained from the corresponding formulas for the Kerr-Newman metric. Although the Kerr-Newman metric was discussed in [3], only motion of uncharged particles with angular momenta was analyzed, the metric being extremal, so there was no crucial difference from the acceleration in the Kerr metric. However, now we are dealing with the situation when a particular case is in a sense more interesting than a general one since it reveals a qualitatively different underlying reason of acceleration to infinite energies. We also discuss both the extremal and nonextremal metrics.
II. Basic formulas. Consider the metric of the Reissner-Nordstrom black hole
da2 = -dt2f + dr2/f + r2dw2
(1)
Here du)2 = sin2 6>#2 + d92, f = 1 - 2M/r + Q2/r2 where M is the black hole mass, Q is its charge. The event horizon lies at r r// M — s/M2 — Q2. Consider a radial motion of the particle having the charge q and rest mass to. Then, its equations of motion read
to«0 = mi =
E■
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TO r
2^2= IE
r
r
m f.
(2)
(3)
/
636
O. B. Zaslavskii
Here, E is the conserved energy, dot denotes differentiation with respect to the proper time r, is the four-velocity. In what follows, we assume that the difference E — qQ/rn > 0, so it is positive for all r > 77/ (motion "forward in time") .
Let two particles (labeled by i = 1,2) fall into the black hole, so fi < 0, r-2 < 0. The relevant quantity which we are interested in is the energy in the centre of mass frame [1 - 7] which is equal to
Ecm = mV 2( It follows from (2)-(4) that
X1X2
UlpU
2
E2
cm
2 TO2
where
Xi — Es
= 1 ■
qiQ
r
ZIZ2
fm2
Zi = sfx2^m2f.
(4)
(5)
(6)
III. Limiting transitions for energy. Now, we are going to examine what happens in different limiting transitions which involve the near-horizon region where
/-► 0.
1) Let / 0. Then, we obtain from the (5), (6) that
rp2 cm(H)
2 to2
= 1+2
12(H) ~ <12 9l(if) ~ 111
Ql(H) ~ 9l 92(H) ~ <122
(7)
where qi(H) = EiTh/Q. It is worth noting that, as rH >Q (for the definiteness, we take Q > 0), the critical charge q^ > E. If a particle falls from infinity, E > m whence q^n) > m, so a particle with the charge q^u) is overcharged in this sense.
2) If, say, q\ = 9i(jj)(l — with S -C 1 and 92 92(H) 5 the energy Ecm^u) ~ 1 /VS can be made as large as one likes. Thus, we have that
lim lim Ecm = 00.
<H^<h(H) r->rH
(8)
3) Let now q\ = q\(u) from the very beginning, 92 q2(H)- Then, X1 = Ei(l — th/t). For the nonex-tremal horizon, in the vicinity of the horizon the expression inside the square root is dominated by the term —to2/ and becomes negative. This means that the horizon is unreachable, so this case is irrelevant for our purposes. Instead, let us consider the extremal horizon, M = Q = rji, / = (1 rji/r)2. After simple manipulations, we find that near the horizon,
E2
cm
2 TO2
= 1-
X2(h)
to2(1^—) r
x LE,
VW
to-
7]+0 ((!-?)). W
Thus, Ecm diverges in the horizon limit: lim lim Ecm = 00.
(10)
4) For completeness, we should also consider the case qi = <7i(ir)> 92 = q2(H) f°r the extremal horizon. Then, Xi = Ei( 1 - rH/r), Zi = (l- rH/r)s/Ef — to2 and we obtain that
E2
cm _ -j^
E1E2 - VEj - TO2 VE
TO
2 to2 ' to2 ' ^ ^
so the energy remains finite and this case is of no interest in the present context.
IV. Limiting transitions for time and conditions of collision. In the above consideration, we showed that the energy Ecm can be made as large as one likes provided q\ Qi(h) &n(l collision occurs near the horizon. Meanwhile, it is also essential to be sure that collision itself can be realized. Preliminarily, it can be understood that this is indeed the case, by analogy with the Kerr case where this was issue traced in detail [1-6].
Consider what happens in more detail. Let at the moment of the coordinate time t = 0 particle 1 starts to move towards the horizon at the point r^, at the later moment t = to > 0 particle 2 does the same (the precedent history of particles is unimportant). Then, the condition of collision at the point r = rf reads
to = h^t2> 0,
ii =
t2 =
I
drX1
r, f^x2^m2f
drX2
(12)
,r, fy/x2 - m2f
To this end, it is sufficient (say, for Q > 0) to take q> < qt, E-> > E|. Then, X2 > X1 for any r, and it is obvious that indeed the time to > 0.
Then rf rji, each of integrals in (12) diverges in accordance with the well known fact that when the horizon is approached, the time measured by clocks of a remote observer is infinite. Let us discuss what happens to to in the limiting situations l)-4) discussed above
If we take the horizon limit 1) we find that to is finite for qt ± qi(H), 92 # 92(h)- Both proper times n, r2 are also finite. If, afterwards, we consider q1 = 9i(jj)(l — 5) with 5 -C 1, the time t is still finite, the allowed region for particle 1 near the horizon shrinks since the positivity the allowed region for particle 1 near the horizon shrinks since the positivity of (3) entails that rH < r < rH + A52 where A = —^M-( —)2. The
proper time n ~ 5, t2 ~ 52.
r
Acceleration of particles by nonrotating charged black holes
637
In case 3) the horizon is external and q = q^. Then, one can obtain the exact explicit expressions:
ti =
E
r—2r// In■
' H
rH
rt
(13)
e2 Q
m2f
qi + = ei + e2.
(16)
(17)
TO
T\ =
TO
Vf + In
rH
Tf - rH
(14)
If rf —t th, t\ ~ (rf — fjj)-1 ~ ¿2- The proper time Ti diverges logarithmically, T2 is finite, so that the situation is very similar to the case of the extremal rotating black holes (cf. [2, 7]). Moreover, calculating the second derivative r from (3), one can see that in the case under discussion both r and r asymptotically vanish as the particle approaches the horizon, so particle 1 halts in the sense that r almost does not change (in terms of the proper distance I, the derivative dl/dr is finite but
1 itself diverges for the extremal horizon). Correspondingly, particle 2 will inevitably will come up with a slow falling particle 1 and will collide with it.
Thus, we checked that in all cases of interest particle
2 can indeed overtakes particle 1, so collision will occur. This happens for a finite (or even almost vanishing) interval of the proper time of particle 2 after the start of motion in point ri.
V. Extraction of energy after collision. Up to now, we discussed the effect of infinitey growing energy in the centre of mass frame. Meanwhile, for observations in laboratory, it is important to know what can be seen by an observer sitting at infinity. This poses a question about the possibility of extraction of the energy after collision. Below, we suggest preliminary analysis f
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