Non-extremal Kerr black holes as particle accelerators

Sijie Gao* and Changchun Zhong†

Department of Physics, Beijing Normal University, Beijing 100875, China(Received 3 May 2011; published 2 August 2011)

It has been shown that extremal Kerr black holes can be used as particle accelerators and arbitrarily

high energy may be obtained near the event horizon. We study particle collisions near the event horizon

(outer horizon) and Cauchy horizon (inner horizon) of a non-extremal Kerr black hole. First, we provide a

general proof showing that particles cannot collide with arbitrarily high energies at the outer horizon.

Second, we show that ultraenergetic collisions can occur near the inner horizon of a Kerr black hole with

any spin parameter a.

DOI: 10.1103/PhysRevD.84.044006 PACS numbers: 04.70.Bw, 97.60.Lf

I. INTRODUCTION

Whether Kerr black holes can serve as particle accelera-tors with infinite collision energy has recently been dis-cussed. Banados, Silk and West (BSW) [1] showed thatparticles falling from rest outside an extremal Kerr blackhole (a ¼ M) can collide with arbitrarily high center-of-mass energies when the collision occurs arbitrarily close tothe horizon. The BSW mechanism has been further dis-cussed and generalized to different spacetimes (See e.g.[2–7]). Jacobson and Sotiriou [8] pointed out that infiniteenergies for the colliding particles can only be attained atinfinite time. Authors also argued [9] that such a highenergy may not be realizable due to the theoretical upperbound a=M ¼ 0:998 [10]. The BSW mechanism would bemore realistic if it worked for non-extremal Kerr blackholes (a <M). Although numerical analysis indicates thatthe collision energies are finite near the horizon of a non-extremal Kerr black hole, no rigorous proof has been given.In this paper, we prove analytically that infinite center-of-mass energies can never be attained outside a non-extremalKerr black hole. Compared with previous literature, ourproof is general in the following senses. First, the collisiontakes place anywhere outside the black hole, not confinedto the equatorial plane � ¼ �

2 . Second, we allow the 4-

velocities of the two particles to be arbitrary. In particular,the 4-velocities have nonvanishing _� components. Finally,we release the restriction that particles fall from infinity,allowing them to fall from anywhere outside the blackhole. Our analysis shows that infinite energies can onlybe attained at the horizon and one of the particles musthave a critical angular momentum. However, with such anangular momentum, there always exists a potential barrieroutside the black hole preventing the particle from ap-proaching the event horizon.

Since the event horizon of a non-extremal Kerr blackhole cannot serve as a particle accelerator creating infinitecollision energies, it is natural to ask whether the inner

horizon can make a difference. This issue has recently beenexplored by Lake [11]. The author claimed in the originalversion that the center-of-mass energy for two collidingparticles is generically divergent at the inner horizon andno fine tuning is required. Then in the Erratum this claimwas withdrawn because physical constraints forbid suchcollisions. We reexamine this issue in details and arrive atthe following conclusions. We first confirm, using differentarguments, that a generic divergence at the inner horizon isnot possible. We further show that for a critical angularmomentum, the center-of-mass energy diverges at the in-ner horizon. Such a divergence is similar to that proposedby BSW where a critical angular momentum also is re-quired. The difference is that there is no restriction on thespin parameter a. So in principle, infinite collision energiescan be obtained near the inner horizon of any non-extremalKerr black hole.

II. COLLISIONS NEAR THE EVENTHORIZON r ¼ rþ

In this section, we consider two particles collidingoutside a Kerr black hole and show that in any case, theycannot collide with arbitrarily high center-of-mass ener-gies. The Kerr metric is given by [12]

ds2 ¼ ���� a2sin2�

�

�dt2 � 2asin2�ðr2 þ a2 ��Þ

�dtd�

þ�ðr2 þ a2Þ2 � �a2sin2�

�

�sin2�d�2

þ�

�dr2 þ �d�2; (1)

where

� ¼ r2 þ a2cos2�; � ¼ r2 þ a2 � 2Mr: (2)

Without loss of generality, we shall choose

M ¼ 1 (3)*sijie@bnu.edu.cn†cczhong@mail.bnu.edu.cn

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in the rest of this paper. We shall deal with the non-extremal case, i.e.,

0< a< 1: (4)

Suppose a particle of mass m moves in the spacetimewith 4-velocity ua ¼ ð @@�Þa, where � is the proper time.

The 4-velocity of one of the particles at the point ofcollision takes the general form

ua ¼ _t

�@

@t

�a þ _r

�@

@r

�a þ _�

�@

@�

�a þ _�

�@

@�

�a: (5)

The geodesic motion is determined by the following con-served quantities [12]

E ¼ �gabua

�@

@t

�b ¼

�1� 2r

�

�_tþ 2arsin2�

�_�; (6)

L ¼ gabua

�@

@�

�b

¼ � 2arsin2�

�_tþ ðr2 þ a2Þ2 � �a2sin2�

�sin2� _�; (7)

� 1 ¼ gabuaub; (8)

where E is the conserved energy per unit mass and L is theangular momentum per unit mass. Solving Eqs. (6) and (7)yields

_t ¼ a4E� 4aLrþ 2Er4 þ a2Erð2þ 3rÞ þ a2E�cos2�

2��;

(9)

_� ¼ að2Er� aLÞ þ L�csc2�

��: (10)

Since ua is a future-directed timelike vector, it follows that_t > 0 near the horizon r ¼ rþ [12]. By expanding thenumerator of Eq. (9) around r ¼ rþ and requiring theleading term to be non-negative, we find

L � 2E

að1þ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffi1� a2

pÞ: (11)

Note that Eq. (11) was also derived in [13] from the nullenergy condition.

Then Eq. (8) yields

_r ¼ � 1

2ffiffiffi2

p�

ffiffiffiS

p; (12)

where

S ¼ 8��ð�1� _�2�Þ � csc2�½�a4E2 þ 4a2L2

� 6a2E2rþ 16aELr� 16L2r� 5a2E2rþ 8L2r2

� 4E2r4 þ a2E2� cos4�þ 4ð�4aELrþ E2r4

þ a2L2 þ a2E2rð2þ rÞÞ cos2��: (13)

Note that we have chosen the minus sign for _r, referring toingoing geodesics. We shall discuss the plus sign at the endof this section.Obviously, a physically allowed trajectory satisfies

S � 0: (14)

Suppose two particles with the same mass m collideeach other. The center-of-mass energy is given by [1]

Ec:m: ¼ mffiffiffi2

p ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1� gabu

aub2

q: (15)

For simplicity, we define the effective center-of-massenergy

Eeff ¼ �gabuaub2 : (16)

We find

Eeff ¼ �� _� _�2 � E0

8��; (17)

where

E0 ¼ ð�a4EE2 þ 4a2LL2 � 6a2EE2rþ 8aE2Lr

þ 8aEL2r� 16LL2r� 5a2EE2r2 þ 8LL2r

2

� 4EE2r4 þ 4ð�2aðE2Lþ EL2Þrþ EE2r4

þ a2ðLL2 þ EE2rð2þ rÞÞÞ cos2�þ a2EE2� cos4�Þcsc2�þ ffiffiffi

Sp ffiffiffiffiffi

S2p

: (18)

Here S2 is obtained by replacing L, E, _� with L2, E2, _�2 inEq. (13).Our purpose is to examine whether an infinite Eeff

defined in Eq. (17) exists under the constraints (4), (11),and (14). As physical requirements, all the constants E, L,etc. should be finite. Otherwise, infinite collision energiescan be produced even in Minkowski spacetime. It is alsoreasonable to assume that _� and _�2 are finite at the horizon.This is because one can define

L� � gabua

�@

@�

�b ¼ � _� (19)

as the ‘‘angular momentum with respect to �.’’ Note thatunlike L, L� is not constant along geodesics. However, it isplausible to require that L� be finite everywhere, particu-larly at the horizon. It then follows from Eq. (19) that _� isfinite at the horizon.

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Hence, it is easy to see from Eq. (17) that an infinite Eeff

cannot be obtained unless � ¼ 0, i.e., the collision mustoccur at the outer horizon r ¼ rþ or the inner horizonr ¼ r�. To check if Eeff could be infinite, we expand E0at r ¼ rþ and find

E0 ¼ �0 þ �1ðr� rþÞ þ . . . ; (20)

where

�0 ¼ 8½�a2LL2 þ Eð�8þ 4a2 � 8ffiffiffiffiffiffiffiffiffiffiffiffiffiffi1� a2

pÞE2

þ 2aðEL2 þ E2LÞð1þffiffiffiffiffiffiffiffiffiffiffiffiffiffi1� a2

pÞ

þ a2

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi�L� 2Eð1þ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffi1� a2

pÞ

a

�2

vuut

�ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi�L2 � 2E2ð1þ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffi1� a2

pÞ

a

�2

vuut �: (21)

Using Eq. (11), the square root terms can be simplified andone finds

�0 ¼ 0: (22)

The vanishing of �0 is important because it rules out thedivergence of Ec:m: for generic angular momentums.

Now it is obvious that Eeff cannot be infinite unless �1 is

infinite. SinceffiffiffiS

p jr¼rþ appears in the denominator of �1,

an infinite �1 requires

Sjr¼rþ ¼ 2a2ðL� LcÞ2 ¼ 0: (23)

Thus, if we choose

L ¼ Lc ¼ 2Eð1þffiffiffiffiffiffiffiffiffiffiffiffiffiffi1� a2

pÞ

a; (24)

Ec:m: will become arbitrarily large at the horizon r ¼ rþ.However, to make sure that the particle with this criticalangular momentum can actually reach the horizon,Eq. (14) must hold outside the horizon. By Taylor expan-sion, we find

S ¼ a0 þ a1ðr� rþÞ þ . . . : (25)

For L ¼ Lc, a0 ¼ 0 and a1 is given by.

a1 ¼ b1 þ�

2E2

a2sin2�þ 2 _�2

�b2; (26)

where

b1 ¼ �32þ 32a2 � 32ffiffiffiffiffiffiffiffiffiffiffiffiffiffi1� a2

pþ 16a2

ffiffiffiffiffiffiffiffiffiffiffiffiffiffi1� a2

p

� 16a2ffiffiffiffiffiffiffiffiffiffiffiffiffiffi1� a2

pcos2�; (27)

b2 ¼ �64þ 80a2 � 16a4

þffiffiffiffiffiffiffiffiffiffiffiffiffiffi1� a2

pð�64þ 48a2 � 3a4 � a4 cos4�Þ

þ 4a2ð�4þ 4a2 � 4ffiffiffiffiffiffiffiffiffiffiffiffiffiffi1� a2

pþ 4a2

ffiffiffiffiffiffiffiffiffiffiffiffiffiffi1� a2

pÞ cos2�:

(28)

Simple analysis shows that both b1 and b2 are negative for0< a< 1. Therefore,

a1 < 0; (29)

which means S is negative near the horizon and conse-quently the particle with L ¼ Lc cannot approach thehorizon. The barrier outside the horizon is illustrated inFig. 1.In the above argument, we have chosen the minus sign in

Eq. (12) for both particles. The same choice was made byBSW. What if the two particles choose different signs?

In that case, the plus sign in front offfiffiffiS

pin Eq. (18) will

become a minus sign and consequently �0 in Eq. (20) willnot vanish. Therefore, such two particles will collidewith infinite energy even without a fine turning on angularmomentum. But to make this happen, one of the particlesmust be outgoing ( _r > 0) on the horizon. For a non-extremal black hole, even holding a particle still at thehorizon requires an infinite local force [12]. Therefore, theconfiguration with _r > 0 should be ruled out.

III. COLLISIONS NEAR THE INNERHORIZON r ¼ r�

Now we discuss the motions in the region r� < r < rþ.Because of the different natures of the inner and outerhorizons, some of the arguments in section II will breakdown in this section. We shall highlight the differences.As illustrated in Fig. 2, we consider two particles cross-

ing the event horizon of the right-hand universe and hittingeach other at the Cauchy horizon of the left-hand universe.The general equations (5)–(10) remain unchanged.

FIG. 1. The plot of SðrÞ in the region r > rþ. The parametersare chosen as: a ¼ 0:999, � ¼ �

2 , L ¼ Lc.

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However, Eq. (11) is derived from the fact that rat is apast-directed timelike vector, which is no longer true out-side the inner horizon. Instead, rar becomes past-directedin this region and any particle must fall in the direction of_r < 0, i.e.,

_r ¼ � 1

2ffiffiffi2

p�

ffiffiffiS

p< 0: (30)

It should be noticed that Eq. (12) holds because we havechosen the ingoing mode for both particles in the regionr > rþ, while Eq. (30) holds for any particle in the regionr� < r < rþ. Note that Eqs. (13)–(18) remain unchanged.Then we expand E0 at r ¼ r�. Corresponding to Eq. (20),we have

E0 ¼ �0 þ �1ðr� r�Þ þ . . . ; (31)

where

�0 ¼ 8

��a2LL2 þ Eð�8þ 4a2 þ 8

ffiffiffiffiffiffiffiffiffiffiffiffiffiffi1� a2

pÞE2

þ 2aðEL2 þ E2LÞð1�ffiffiffiffiffiffiffiffiffiffiffiffiffiffi1� a2

pÞ

þ a2

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi�L� 2Eð1�

ffiffiffiffiffiffiffiffiffiffiffiffiffiffi1� a2

pÞ

a

�2

vuut

�ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi�L2 � 2E2ð1�

ffiffiffiffiffiffiffiffiffiffiffiffiffiffi1� a2

pÞ

a

�2

vuut �: (32)

This equation can be simplified by choosing the ingoingmode for both particles, as depicted in Fig. 2. Here in theregion r� < r < rþ, the ingoing mode means _t < 0 (By thesame argument as given at the end of section II, _t > 0 atthe left-hand Cauchy horizon is not physically realizable).Then using the same argument which led to Eq. (11),we find

�L� 2Eð1�

ffiffiffiffiffiffiffiffiffiffiffiffiffiffi1� a2

pÞ

a

��L2 � 2E2ð1�

ffiffiffiffiffiffiffiffiffiffiffiffiffiffi1� a2

pÞ

a

�� 0;

(33)

which leads to

�0 ¼ 0: (34)

Again, the necessary condition for Eeff blowing up at thehorizon is that �1 blows up, which requires

Sjr¼r� ¼ 8a2ðL� L0cÞ2 ¼ 0; (35)

where

L0c ¼ 2Eð1�

ffiffiffiffiffiffiffiffiffiffiffiffiffiffi1� a2

pÞ

a: (36)

Therefore, an infinite collision energy near the inner hori-zon requires that one of the particles has the momentum

L ¼ L0c (37)

and the other particle has any different angular momentum.For the collision near the event horizon, we have shownthat a potential barrier always exists. But the geodesicmotions near the inner horizon are very different. We shallshow that there is no potential barrier in the vicinity of theinner horizon. For our purposes, it is sufficient to considerthe motions in the equatorial plane and let E ¼ 1, in whichcase, S given in Eq. (13) reduces to

S ¼ 16r½a4 � 4ð1�ffiffiffiffiffiffiffiffiffiffiffiffiffiffi1� a2

pÞðr� 2Þ þ a2ðr2 þ 2rþ 4

ffiffiffiffiffiffiffiffiffiffiffiffiffiffi1� a2

p� 8Þ�

a2: (38)

By Taylor expansion, we find

S ¼ 32

a2

�5� a2 � 3

ffiffiffiffiffiffiffiffiffiffiffiffiffiffi1� a2

p� 4ð1�

ffiffiffiffiffiffiffiffiffiffiffiffiffiffi1� a2

pÞ

a2

�ðr� r�Þ þ . . . : (39)

FIG. 2. Penrose diagram for the Kerr spacetime in the regionr� � r � rþ. Two particles cross the event horizon and collideat the inner horizon. The dashed lines refer to surfaces ofconstant r.

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By plotting the coefficient of r� r� as a function of a, wesee immediately that S > 0 for any 0< a< 1 in the vicin-ity of r ¼ r�.

Now we show, by a concrete example, that such acollision can be realized at the Cauchy horizon for twoparticles falling from rest at infinity.

The orbits of the two particles are confined to theequatorial plane � ¼ �

2 . The parameters are chosen as

E ¼ E2 ¼ 1; (40)

L ¼ L0c; (41)

L2 ¼ L0c

2: (42)

As depicted in Fig. 3, the center-of-mass energy blowsup at r ¼ r�. Figure 4 shows S > 0 and S2 > 0 for allr > r�, i.e., the two particles can fall from infinity all theway to the Cauchy horizon. Figure 5 plots _tðrÞ in the regionr > r�. We see that _t and _t2 are positive in the regionr > rþ, as expected. For r� < r < rþ, _t and _t2 are nega-tive. This means both particles must hit the left-handCauchy horizon, as depicted in Fig. 2. Otherwise, therewill be a turning point _t ¼ 0 in this region. It should also benoticed that _t2 ! �1, meaning that this particle will crossthe Cauchy horizon. But for the first particle which has acritical angular momentum, _t is finite as r ! r�. Thus,FIG. 3. Plot of the center-of-mass energy.

FIG. 5. Plots of _tðrÞ and _t2ðrÞ.

FIG. 4. Plots of the functions SðrÞ and S2ðrÞ. Both functions are positive in the range r > r�.

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instead of crossing the Cauchy horizon, this particle spiralsasymptotically onto the horizon. This is the necessarymechanism for an infinite center-of-mass energy as pointedout in [8]. One can also check that _r vanishes on theCauchy horizon while _r2 does not, which is consistentwith the behaviors of _t and _t2.

IV. CONCLUSIONS

We have provided a rigorous proof showing that thecenter-of-mass energy cannot be divergent at the eventhorizon of a non-extremal black hole. This proof is generaland exhausts all of the possibilities. The motion of particlesis not confined to the equatorial plane and the particles maybe released from any point outside the black hole. We findthat a critical angular momentum is required for the diver-gence of Ec:m: at the horizon. However, with this angularmomentum, the particle can never reach the horizon. In

relation to Lake’s work, we first prove that no divergence ofenergy occurs at the inner horizon for particles with genericangular momentums. A critical angular momentum is re-quired for the divergence. We show with an explicit ex-ample that two particles can fall from infinity all the way tothe inner horizon and collide with an arbitrarily high center-of-mass energy. We have shown that such arbitrarily highenergies can be obtained in any non-extremal Kerr space-time, unlike the case in [1] that requires the black hole to beexactly extremal. Since extremal black holes do not exist innature, the ultraenergetic collisions near Cauchy horizonsmay be more practical and realizable.

ACKNOWLEDGMENTS

This research was supported by NSFC GrantNos. 10605006, and 10975016 and by ‘‘the FundamentalResearch Funds for the Central Universities.’’

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