On the energy of particle collisions in the ergosphere of the rotating black holes

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by  A. A. Grib

On the energy of particle collisions in the ergosphere of the ro-
tating black holes
A. A. Grib(a)1,2,3 and Yu. V. Pavlov(b)1,2,4
1 A. Friedmann Laboratory for Theoretical Physics - Saint Petersburg, Russia
2 Copernicus Center for Interdisciplinary Studies - Krak´ow, Poland
3 Theoretical Physics and Astronomy Department, The Herzen University - Moika 48, Saint Petersburg 191186,
Russia
4 Institute of Problems in Mechanical Engineering, Russian Academy of Sciences - Bol’shoy pr. 61, V.O., Saint
3
Petersburg 199178, Russia
1
0
2

n PACS 04.70.-s–Physics of black holes
Ja PACS 04.70.Bw–Classical black holes
PACS 97.60.Lf–Black holes
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Abstract–Itisshownthattheenergyinthecentreofmassframeofcollidingparticlesinfreefall
]
c at any point of the ergosphere of the rotating black hole can grow without limit for fixed energy
q values on infinity. The effect takes place for large negative values of the angular momentum of
- oneof theparticles.
r
g
[

1
v
8
Introduction. – Recently many papers were pub- black hole there is no limit for negative values of the mo-
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lished [1]– [10] in which some specific properties of colli- mentum and arbitrary large energy in the centre of mass
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0sionsofparticlesclosetothe horizonofthe rotatingblack frame is possible for large angular momentum of the two
.holes were discussed. In [1] the effect of unlimited grow- colliding particles. In a sense the effect is similar to the
1
ing energy of colliding two particles in the centre of mass BSW-effect. Oneofthe particlewithlargenegativeangu-
0
3frame for critical Kerr rotating black holes (call it BSW lar momentum can be called “critical”, the other particle
1effect) was discovered. In our papers [2]– [4] the same ef- “ordinary”.
:fect was found for noncritical black holes when multiple The system of units G=c=1 is used.
v
icollisions took place.
X General formulas for the collision energy close
All this shows that close to the horizonthere is natural
to the black hole. – The Kerr’s metric of the rotating
r
asupercollider of particles with the energies up to Planck’s blackhole[11]inBoyer–Lindquistcoordinates[12]hasthe
scale. However its location close to the horizon makes
form:
difficult due to the large red shift get some observed ef-
2Mr
fects outside of the ergospherewhen particles go from the ds2 = dt2 (dt asin2θdϕ)2
“black hole supercollider” outside. Here we shall discuss − ρ2 −
someothereffectvalidatanypointoftheergosphere. The dr2
ρ2 +dθ2 (r2+a2)sin2θdϕ2, (1)
energy in the centre of mass frame can take large values − (cid:18) ∆ (cid:19)−
for large negative values of the angular momentum pro-
where
jection on the rotation axis of the black hole. The prob-
lem is how to get this large (in absolute value) negative ρ2 =r2+a2cos2θ, ∆=r2 2Mr+a2, (2)
values. That is why first we shall obtain limitations on −
thevaluesoftheprojectionofangularmomentumoutside M is the mass of the black hole, aM its angular momen-
ergosphere for particles falling into the black hole and in- tum. The rotation axis direction corresponds to θ = 0,
sidetheergosphere. Itoccursthatinsidetheergosphereof i.e. a 0. The event horizon of the Kerr’s black hole
≥
corresponds to
(a)E-mail: andrei [email protected]
(b)E-mail: [email protected] r =rH M + M2 a2. (3)
≡ −
p
p-1
A. A. Grib, Yu. V. Pavlov
The surface of the static limit is defined by For the free falling particles with energies E , E and
1 2
angularmomentumprojectionsJ ,J from(5)–(9)oneob-
1 2
r =r (θ) M + M2 a2cos2θ. (4) tains [5]:
1
≡ −
p
Ilinmciatsaenad≤evMentthheorriezgoinonisocfaslpleadcee-rtgimosephbeertew.eenthe static Ec2.m. = m21+m22+ ρ22(cid:20)P1P2−σ1r∆√R1σ2r√R2
For geodesics in Kerr’s metric (1) one obtains (see [13], (J aE sin2θ)(J aE sin2θ)
1 1 2 2
− −
Sec. 62 or [14], Sec. 3.4.1) − sin2θ
ρ2dt = a aEsin2θ J + r2+a2 P, (5) −σ1θ Θ1σ2θ Θ2(cid:21). (14)
dλ − − ∆ p p
(cid:0) (cid:1)
Limitationson the valuesofparticle angularmo-
ρ2dϕ = aE J + aP, (6) mentum close to the Kerr’s black hole. – The per-
dλ −(cid:18) − sin2θ(cid:19) ∆ mitted region for particle movement is defined by condi-
tions
dr dθ
ρ2 =σr√R, ρ2 =σθ√Θ, (7) J2
dλ dλ Θ=Q cos2θ a2(m2 E2)+ 0, (15)
− (cid:20) − sin2θ(cid:21)≥
P = r2+a2 E aJ, (8)
−
(cid:0) (cid:1) R 0,andmovement“forwardintime” leadsto dt/dλ>
≥
R=P2 ∆[m2r2+(J aE)2+Q], (9) 0[16]. Thecondition(15)givespossiblevaluesofCarter’s
− −
constant Q. From (7) it follows that for movement with
J2 constant θ it is necessary and sufficient that Θ = 0. It is
Θ=Q cos2θ a2(m2 E2)+ . (10)
− (cid:20) − sin2θ(cid:21) always possible to choose this value, so that for geodesics
in the equatorial plane (θ = π/2) Carter’s constants Q =
Here E is conserved energy (relative to infinity) of the
0.
probe particle, J is conservedangular momentum projec-
Let us find limitations for the particle angular momen-
tion on the rotation axis of the black hole, m is the rest
tum from the conditions R 0, dt/dλ > 0 at the point
mass of the probe particle, for particles with nonzero rest ≥
(r,θ),takingthefixedvaluesofΘ. Outsidetheergosphere
mass λ = τ/m, where τ is the proper time for massive r2 2rM +a2cos2θ >0 one obtains
particle, Q is the Carter’s constant. The constants σ ,σ −
r θ
informulas (7)are equal 1 andaredefined by the direc- 1
± E (m2ρ2+Θ)(r2 2rM +a2cos2θ), (16)
tionofparticlemovementincoordinatesr,θ. Formassless ≥ ρ2 −
p
particles one must take m=0 in (9), (10).
One can find the energy in the centre of mass frame J ∈[J−, J+], (17)
of two colliding particles E with rest masses m , m
c.m. 1 2
taking the square of sinθ
J± = 2rMaEsinθ
r2 2rM +a2cos2θ(cid:20)−
(E ,0,0,0)=pi +pi , (11) −
c.m. (1) (2)
∆(ρ4E2 (m2ρ2+Θ)(r2 2rM +a2cos2θ)) . (18)
± − − (cid:21)
where pi are 4-momenta of particles (n = 1,2). Due to p
(n)
pi p =m2 one has Boundary values of J± correspond to dr/dλ=0.
(n) (n)i n
On the boundary of ergosphere
E2 =m2+m2+2pi p . (12)
c.m. 1 2 (1) (2)i r =r (θ) E 0, (19)
1
⇒ ≥
Notethattheenergyofcollisionsofparticlesinthecentre Mr (θ) m2 Θ
of mass frame is always positive (while the energy of one J E 1 +asin2θ 1 .
≤ (cid:20) a (cid:18) − 2E2 − 4Mr (θ)E2(cid:19)(cid:21)
1
particle due to Penrose effect [15] can be negative!) and
(20)
satisfies the condition
Inside ergosphere
Ec.m. ≥m1+m2. (13) rH <r<r1(θ) ⇒ (r2−2rM +a2cos2θ)<0, (21)
Thisfollowsfromthefactthatthecollidingparticlesmove
one towards another with some velocities. sinθ
J J−(r,θ)= 2rMaEsinθ
It is important to note that E for two colliding par- ≤ (r2 2rM+a2cos2θ)(cid:20)
c.m. − −
ticles is not a conserved value differently from energies of
∆(ρ4E2 (m2ρ2+Θ)(r2 2rM +a2cos2θ)) . (22)
particles (relative to infinity) E1, E2. − − − (cid:21)
p
p-2
On the energy of particle collisions in the ergosphere
So on the boundary and inside ergosphere there exist This asymptotic formula is valid for all possible E , J
1 1
geodesics on which particle with fixed energy can have ar- (see (22)) for r < r < r (θ) and for E > 0 and J
H 1 1 1
bitrarylargeinabsolutevaluenegativeangularmomentum satisfying (20) for r = r (θ). The poles θ = 0,π are not
1
projection. considered here because the points on surface of static
From(22)onecanseethatfornegativeenergyE ofthe limit are on the event horizon.
particle in ergosphere its angular momentum projection Note that expression in brackets in (24) is positive in
on the rotation axis of the black hole must be also neg- ergosphere. This is evident for r = r1(θ) and follows
ative. This is a well known Penrose effect [15]. However from limitations (22) for rH <r <r1(θ), and inside ergo-
rotation in Boyer–Lindquist coordinates for any particle sphere (24) can be written as
in ergosphere has the same direction as the rotation of r2 2rM +a2cos2θ
the black hole (the effect of the “dragging” of bodies by Ec2.m. ≈J2 − ρ2∆sin2θ
the rotatingblackhole[14]). Reallyfor timelike geodesics
2
ds2 > 0 leads to dϕ/dt > 0. So it is incorrect to say (as σ1r J1+ J1 σ2r J1− J1 . (25)
it is said in some test books on black holes [13], p. 368) ×(cid:16) p − − p − (cid:17)
that“onlycounter-rotatingparticlescanhavenegativeen- So from (24) one comes to the conclusion that when
ergy”! Inside the ergosphere the usual intuition which is particles fall on the rotating black hole collisions with ar-
true far outside it that the change of the sign of angular bitrarily high energy in the centre of mass frame are pos-
momentum projection on Z-axis means the change of the sible at any point of the ergosphere if J2 and the
→ −∞
rotationon counter-rotationfollowing from usual formula energies E1,E2 are fixed. The energy of collision in the
J=r p is incorrect. centre of mass frame is growing proportionally to J2 .
From×equationsofgeodesics(6)onecanseethepeculiar Note that for large J2 the collision energy cplos|e t|o
−
properties of the correspondence between the direction of horizon in the centre of mass frame depending on values
rotation and the angular momentum projection. Let us E1,J1canbeaslargeaslessthenforcollisionsattheother
rewrite it as points of ergosphere.
Outside ergosphere the collision energy is limited for
dϕ E2Mrasin2θ r2 2rM +a2cos2θ given r, but for r r it can be large if one of the parti-
ρ2sin2θ = +J − . → 1
dλ ∆ ∆ clesgetsinintermediatecollisionstheangularmomentum
(23) close to J− (see (18)). On fig. 1 the dependence of colli-
For large r outside the ergosphere one gets the stan- sion energy in the centre of mass frame on the coordinate
dard expression for the angular momentum projection r is shown for particles with E =E =m =m , J =0
1 2 1 2 1
in Minkowski space J = mr2sin2θdϕ/dτ and J = andJ2 =J− movinginequatorialplaneofblackholewith
mr2dϕ/dτ for θ =π/2. a=0.8M.
But when one is ingoing inside the ergosphere one sees
E (cid:143)!m!!!!!! m!!!!!!!
thatthecoefficientforJ in(23)becomeszeroonitssurface c.m.(cid:144) 1 2
so that the angular velocity dϕ/dλ and dϕ/dτ is defined
30
by the energy and does not depend on J at all.
Inside the ergosphere the coefficient for J in (23) be- 25
comes negative, the angular velocity is still positive and 20
one comes to unusual conclusion: if the energy of the par- 15
ticle in ergosphere is fixed particles with negative angular
10
momentum projection are rotating in the direction of ro-
5
tation of the black hole with greater angular velocity! r
€€€€€€€€
So the constant characterizingthe geodesic which coin-
2.2 2.4 2.6 2.8 M
cides with the usual angular momentum definition taken
from Newtonian physics outside the black hole does not
coincide with it in ergosphere. Fig. 1: The collision energy in the centre of mass frame for
particles with J1 =0 and J2 =J− out of theergosphere.
Energy of collision with a particle with large an-
gularmomentum.– Letusfindtheasymptoticof(14) Note that large negative values of the angular momen-
forJ andsomefixedvaluerinergospheresuppos- tum projection are forbidden for fixed values of energy of
2
→−∞
ing the value of Carter’sconstantQ to be such that (15) particle out of the ergosphere. So particle which is non-
2
is valid and Θ J2. Then from (14) one obtains relativistic on space infinity (E = m) can arrive to the
2 ≪ 2
horizonofthe blackhole if its angularmomentum projec-
2J J tion is located in the interval
E2 − 2 1 r2 2rM +a2cos2θ
c.m. ≈ ρ2∆ (cid:20)sin2θ − a a
(cid:0) (cid:1) 2mM 1+ 1+ J 2mM 1+ 1 .
+2rMaE σ1rσ2r√R1 (r2 2rM+a2cos2θ) . (24) − (cid:20) r M (cid:21)≤ ≤ (cid:20) r − M (cid:21)
1− sinθ − − (cid:21) (26)
p
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A. A. Grib, Yu. V. Pavlov
The left boundary is a minimal value of the angular mo- [6] Zaslavskii O.B., Class. Quantum Grav., 28 (2011)
mentum of particles with E = m capable to achieve er- 105010.
gosphere falling from infinity. That is why collisions with [7] Zaslavskii O.B., Phys. Rev. D, 84 (2011) 024007.
J do not occur for particles following from in- [8] Grib A.A., Pavlov Yu.V.andPiattella O.F.,Int. J.
2 → −∞ Mod. Phys. A, 26 (2011) 3856.
finity. But if the particle came to ergosphere and there
[9] Grib A.A., Pavlov Yu.V. and Piattella O.F., Grav.
in the result of interactions with other particles is getting
Cosmol., 18 (2012) 70.
largenegativevaluesoftheangularmomentumprojection
[10] Zaslavskii O.B., Phys. Lett. B, 712 (2012) 161.
(no need for getting high energies!) then its subsequent [11] Kerr R.P., Phys. Rev. Lett., 11 (1963) 237.
collision with the particle falling on the black hole leads [12] Boyer R.H. and Lindquist R.W., J. Math. Phys., 8
to high energy in the centre of mass frame. (1967) 265.
Getting superhigh energies for collision of usual parti- [13] Chandrasekhar S., The Mathematical Theory of Black
cles(i.e. protons)insuchmechanismoccurhoweverphys- Holes (Clarendon Press, Oxford University Press, New
ically nonrealistic. Really from (24) the value of angular York,Oxford) 1983.
momentumnecessaryforgettingthecollisionenergyE [14] Frolov V.P. and Novikov I.D., Black Hole Physics:
c.m.
has the order Basic Concepts and New Developments (Kluwer Acad.
aE2 Publ., Dordrecht) 1998.
J2 c.m.. (27) [15] Penrose R., Rivista Nuovo Cimento, Num. Spec., I
≈− 2E
1
(1969) 252.
So from (26) absolute value of the angular momentum [16] Wald R.M., General Relativity (The University of Chi-
J2 must acquire the order Ec2.m./(m1m2) relative to the cago Press, Chicago and London) 1984.
maximal value of the angular momentum of the parti- [17] GribA.A.andPavlov Yu.V.,Int.J. Mod. Phys. D,11
cle incoming to ergosphere from infinity. For example if (2002) 433.
E =E =m (the proton mass) then J must increase [18] Grib A.A. and Pavlov Yu.V., Mod. Phys. Lett. A, 23
1 2 p 2
with a factor 1018 for E = 109m .| T|o get this one (2008) 1151.
c.m. p [19] GribA.A.andPavlov Yu.V.,Int. J. Mod.Phys. A,24
musthaveverylargenumberofcollisionswithgettingad-
(2009) 1610.
ditional negative angular momentum in each collision.
Howeverthe situation is different for supermassive par-
ticles. In [17]– [19] we discussed the hypothesis that dark
matter contains stable superheavy neutral particles with
mass of the Grand Unification scale created by gravita-
tion in the end of the inflation era. These particles are
nonstableforenergiesofinteractionofthe orderofGrand
Unification and decay on particles of visual matter but
are stable at low energies. But in ergosphere of the rotat-
ing black holes such particles due to getting large relative
velocities can increase their energy from 2m to values of
3m and larger so that the mechanism considered in our
paper can lead to their decays as it was in the early uni-
verse. The member of intermediate collisions for them is
not very large (of the order of 10).
∗∗∗
The research is supported by the grant from The John
Templeton Foundation.
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