“Black holes as supercolliders

of elementary particles”

A. A. Grib a, b, Yu. V. Pavlov a, c

aA.Friedmann Laboratory for Theoretical Physics,bRussian State Pedagogical University (Herzen University),

cInstitute for Problems in Mechanical Engineering RAS,

St. Petersburg, Russia

2

[1] A.A. Grib, Yu.V. Pavlov, Pis’ma v ZhETF 92, 147 (2010).

[2] A.A. Grib, Yu.V. Pavlov, Astropart. Phys. 34, 581 (2011).

[3] A.A. Grib, Yu.V. Pavlov, Gravitation & Cosmology 17, 42 (2011).

[4] A.A. Grib, Yu.V. Pavlov, O.F. Piattella, Int. J. Mod. Phys. A 26, 3856 (2011);Gravitation & Cosmology 18, 70 (2012).

[5] A.A. Grib, Yu.V. Pavlov, Theor. Math. Phys. 176, 881 (2013).

[6] A.A. Grib, Yu.V. Pavlov, EPL (Europhysics Letters) 101, 20004 (2013).

[7] M. Banados, J. Silk and S.M. West, Phys. Rev. Lett. 103, 111102 (2009).Kerr black holes as particle accelerators to arbitrarily high energy.

[8] M.Banados, B.Hassanain, J.Silk, S.M.West, Phys. Rev. D 83, 023004 (2011).

[9] O. B. Zaslavskii, Class. Quantum Grav. 28, 105010 (2011); Phys. Rev. D 82,083004 (2010); 84, 024007 (2011); 85, 024029 (2012).

[10] T. Harada and M. Kimura, Phys. Rev. D 83, 024002 (2011); 83, 084041(2011); 84, 124032 (2011).

3

1. The collision energy close to the black hole

The Kerr’s metric of the rotating black hole in Boyer–Lindquist coordinates has the form:

ds2 = dt2 − 2Mr

ρ2(dt − a sin2θ dϕ)2 − ρ2

⎛⎜⎝dr2

Δ+ dθ2

⎞⎟⎠− (r2 + a2) sin2θ dϕ2, (1)

where

ρ2 = r2 + a2 cos2θ, Δ = r2 − 2Mr + a2, (2)

M is the mass of the black hole, aM its angular momentum. The rotation axis direction corre-

sponds to θ = 0, i.e. a ≥ 0. The event horizon of the Kerr’s black hole corresponds to

r = rH ≡ M +√

M 2 − a2. (3)

The surface of the static limit is defined by

r = r1(θ) ≡ M +√

M 2 − a2 cos2 θ. (4)

In case a ≤ M the region of space-time between the static limit and event horizon is called

ergosphere.

4

Geodesics in Kerr’s metric

ρ2 dt

dλ= −a

(aE sin2θ − J

)+

r2 + a2

ΔP, (5)

ρ2dϕ

dλ= −

⎛⎝aE − J

sin2θ

⎞⎠ +

aP

Δ, (6)

ρ2 dr

dλ= σr

√R, ρ2 dθ

dλ= σθ

√Θ, (7)

P =(r2 + a2

)E − aJ, (8)

R = P 2 − Δ[m2r2 + (J − aE)2 + Q], (9)

Θ = Q − cos2θ

⎡⎢⎣a2(m2 − E2) +

J2

sin2θ

⎤⎥⎦ . (10)

Here E is conserved energy (relative to infinity) of the probe particle, J is conserved angular

momentum projection on the rotation axis of the black hole, m is the rest mass of the probe

particle, for particles with nonzero rest mass λ = τ/m, where τ is the proper time for massive

particle, Q is the Carter’s constant. The constants σr, σθ in formulas (7) are equal ±1 and are

defined by the direction of particle movement in coordinates r, θ. For massless particles one must

take m = 0 in (9), (10).

5

Equatorial geodesics (θ = π/2, m �= 0) in Kerr metric

mdt

dτ=

1

Δ

⎡⎢⎢⎢⎢⎢⎣

⎛⎜⎜⎜⎜⎜⎝r

2 + a2 +2Ma2

r

⎞⎟⎟⎟⎟⎟⎠ E − 2Ma

rJ

⎤⎥⎥⎥⎥⎥⎦ , (11)

mdϕ

dτ=

1

Δ

⎡⎢⎢⎢⎣2Ma

rE +

⎛⎜⎜⎜⎝1 − 2M

r

⎞⎟⎟⎟⎠J

⎤⎥⎥⎥⎦ , (12)

m2⎛⎜⎜⎜⎝dr

dτ

⎞⎟⎟⎟⎠2

= E2 +2M

r3 (aE − J)2 +a2E2 − J2

r2 − Δ

r2 m2 , (13)

whereΔ = r2 − 2Mr + a2 , (14)

6

The energy of collision in the centre of mass frame

(Ec.m., 0 , 0 , 0 ) = p i(1) + p i

(2), (15)

where p i(n) are 4-momenta of particles (n = 1, 2). Due to p i

(n)p(n)i = m2n one has

E 2c.m. = m2

1 + m22 + 2p i

(1)p(2)i. (16)

Note that the energy of collisions of particles in the centre of mass frame is always positive (while

the energy of one particle due to Penrose effect can be negative!) and satisfies the condition

Ec.m. ≥ m1 + m2. (17)

This follows from the fact that the colliding particles move one towards another with some

velocities.

It is important to note that Ec.m. for two colliding particles is not a conserved value differently

from energies of particles (relative to infinity) E1, E2.

7

Expression of the energy through relative velocity vrel

If m1 �= 0,m2 �= 0, in the reference frame of the first particle one has

ui(1) = (1, 0, 0, 0), ui

(2) =

⎛⎜⎜⎝

1√1 − v2

rel

,vrel√

1 − v2rel

⎞⎟⎟⎠. (18)

So

ui(1)u(2)i =

1√1 − v2

rel

, vrel =

√√√√√√√√√1 − 1(ui

(1)u(2)i

)2 . (19)

These expressions evidently don’t depend on the coordinate system.

E 2c.m.

(m1 + m2)2= 1 +

2m1m2

(m1 + m2)2

⎛⎜⎜⎜⎜⎜⎜⎝

1√√√√1 − v2rel

− 1

⎞⎟⎟⎟⎟⎟⎟⎠ (20)

and the nonlimited growth of the collision energy in the centre of mass frameoccurs due to growth of the relative velocity to the velocity of light.

8

For the free falling particles with energies E1, E2 and angular momentum pro-jections J1 = j1M,J2 = j3M one obtains

E 2c.m. = m2

1 + m22 + 2p i

(1)p(2)i. (21)

p i(1)p(2)i =

1

xΔx

{E1E2

(x3 + A2(x + 2)

) − 2A (j1E2 + j2E1) +

+ j1j2(2 − x) −√2E2

1x2 + 2(j1 − E1A)2 − j2

1x + (E21 − m2

1)xΔx ××

√2E2

2x2 + 2(j2 − E2A)2 − j2

2x + (E22 − m2

2)xΔx

}. (22)

in general case(Harada, Kimura, PRD 2011)

E 2c.m. = m2

1 + m22 +

2

ρ2[P1P2 − σ1r

√R1 σ2r

√R2

Δ

− (J1 − aE1 sin2θ)(J2 − aE2 sin2θ)

sin2θ− σ1θ

√Θ1 σ2θ

√Θ2 ]. (23)

9

Banados–Silk–West effect for extremal black hole

In the limit r → rH for E1 = E2 = m = m1 = m2

M. Banados, J. Silk and S.M. West, Phys. Rev. Lett. 103, 111102 (2009)

a = M ⇒ Ec.m.(r → rH) =√

2 m

√√√√√√√√l2 − 2

l1 − 2+

l1 − 2

l2 − 2. (24)

showing the unlimited increasing of the energy of collision when the an-gular momentum of one of falling particles goes to the limitingpossible value equal to J = 2mM.

Limitations on the possible values of the angular momentum of falling particles:(E = m) to achieve the horizon of the black hole

−2(1 +

√1 + A

)= lL ≤ l ≤ lR = 2

(1 +

√1 − A

). (25)

10Non-extremal black hole A = 1 − ε , ε → 0

E maxc.m. = 2

(21/4 + 2−1/4

) √m1m2

ε1/4+ O(ε1/4). (26)

K. S.Thorne, Astrophys.J.191, 507 (1974) Amax = 0.998 ⇒ E maxc.m.√

m1m2≈ 19

[1] E. Berti, V. Cardoso, L. Gualtieri, F. Pretorius and U. Sperhake,

Phys. Rev. Lett. 103, 239001 (2009).

[2] T. Jacobson and T. P. Sotiriou, Phys. Rev. Lett. 104, 021101 (2010).

11

Another form of limiting formula (E1 = E2 = m1 = m2 = m) at the first time

A.A.Grib, Yu.V.Pavlov, JETP Lett. 92 (2010) 125

Ec.m.(r → rH)

2m=

√√√√√√√√√1 +(l1 − l2)2

2xC(l1 − lH)(l2 − lH), (27)

where

lH =2ErH

am=

E

mMΩH, xC = 1 −

√1 − A2 . (28)

ΩH = A/2rH is horizon angular velocity.

lH is the greatest possible specific orbital moment near to horizon

x → xH ,dt

dτ> 0 ⇒ l ≤ lH . (29)

12For l = lH − δ in the second order in δ close to the horizon one obtains:

l = lH − δ ⇒ x < xδ ≈ xH +δ2x2

C

4xH

√1 − A2

. (30)

Figure 1: The effective potential for A = 0.95, ε = 1 and lR ≈ 2.45, l = 2.5, lH ≈ 2.76.

Allowed zones for l = 2.5 are shown by the green color.

1

2

⎛⎝dr

dτ

⎞⎠

2

+ Veff(r, l) = 0 , Veff(x, l) = −1

x+

l2

2x2− (A − l)2

x3, ε = 1 . (31)

13

If the particle falling from the infinity with l ≤ lR arrives to the region defined by (30) and

here it interacts with other particles of the accretion disc or it decays on more light particle so

that it gets the larger angular momentum l1 = lH − δ, then due to (27) the scattering energy in

the centre of mass system is

l1 = lH − δ ⇒ Ec.m. ≈ 1√δ

√√√√√√√2m1m2(lH − l2)

1 −√1 − A2

(32)

and it increases without limit for δ → 0.

Amax = 0.998 , l2 = lL ⇒ Ec.m. ≈ 3.85√

m1m2√δ

. (33)

Note that for rapidly rotating black holes A = 1− ε the difference between lH and lR is not large

lH − lR = 2

√1 − A

A

(√1 − A +

√1 + A − A

)≈ 2(

√2 − 1)

√ε , ε → 0 . (34)

Amax = 0.998 , ⇒ lH − lR ≈ 0.04 . (35)

So the possibility of getting small additional angular momentum in interaction close to the

horizon seems much probable.

14

The time of movement before the collision with unboundedenergy

From equation of the equatorial geodesic one obtains∣∣∣∣∣∣dt

dx

∣∣∣∣∣∣ =M

√x((x3 + A2x + 2A2)ε − 2Al

)

Δx

√2ε2x2 − l2x + 2(Aε − l)2 + (ε2 − 1)Δx

. (36)

So the coordinate time of the particle falling from some point r0 = x0M to the point rf =

xfM > rH in the case of the extremal black hole A = 1, and the limiting value

l = 2, ε = 1 is equal to

Δt =M√

2

⎛⎜⎝2

√x (x2 + 8x − 15)

3(x − 1)+ 5 ln

√x − 1√x + 1

⎞⎟⎠∣∣∣∣∣∣∣x0

xf

(37)

and it diverges as (xf−1)−1 for xf → 1. So for all possible values of l and A to get the collision

with infinitely growing energy in the centre of mass system needs infinitely large

time Δt.

The interval of proper time in this case is

Δτ =M

3√

2

⎛⎜⎝2√

x (3 + x) + 3 ln

√x − 1√x + 1

⎞⎟⎠∣∣∣∣∣∣∣x0

xf

(38)

and it diverges logarithmically when xf → 1. So to get the collision with infinite energy

one needs the infinite interval of as coordinate as proper time of the free falling

particle.

15Some quantitative estimates

In case of the extremal rotating black hole A = 1 and the limiting angular momentum l1 = 2ε1

the time of movement in the vicinity of events horizon up to the point of collision with radial

coordinate xf → xH = 1 is

Δt ∼ 4Mε1

(xf − 1)√3ε2

1 − 1. (39)

The time of movement before collision with a given value of the energy E in the centre of mass

frame

Δt ∼ E2

m1m2

2Mε1

(2ε2 − l2)√3ε2

1 − 1 (2ε1 −√3ε2

1 − 1 ). (40)

For ε1 = ε2 = 1, l2 = 0 one gets

Δt ∼ E2

m1m2

M

2(√

2 − 1)≈ 6 · 10−6 M

M�

E2

m1m2s, (41)

where M� is the mass of the Sun.

So to have the collision of two protons with the energy of the order of the

Grand Unification one must wait for the black hole of the star mass the time

∼ 1024 s, which is large than the age of the Universe ≈ 1018 s. However for

the collision with the energy 103 larger than that of the LHC one must wait

only ≈ 108 s. If r0 = 100rH and the energy of collision is 100m, then the time of falling before

collision is ≈ 0.1 s.

16

Non-extremal case

From (36) one has for l < lH = 2εxH/A, A < 1 and x0 close to xH (for example x0 = 2xH)

Δt ∼ 2MxH

xH − xCln

1

xf − xH, xf → xH. (42)

Remind that for the extremal black hole and the critical value of the angular momentum of the

falling particle this interval is divergent as 1/(r − rH).

For collisions of two particles with l1 = lH − δ close to horizon at the point xδ

Δt ∼ 8MxH

xH − xCln

Ec.m.√m1m2

. (43)

So for

A = 0.998, Δt ∼ 3.2 · 10−4 M

M�ln

Ec.m.√m1m2

s. (44)

Taking the value of the Grand Unification energy Ec.m./√

m1m2 = 1014 and the mass of the

black hole 108M� typical for Active Nuclei of galaxies one obtains Δt ∼ 106 s, i.e. of the order of

12 days.

This time is much smaller than that for the extremal case.

17

A.A. Grib, Yu.V. Pavlov,Theor. Math. Phys. 176, 881–887 (2013);

EPL (Europhysics Letters) 101, 20004 (2013);arXiv:1301.0698 [gr-qc]

On the energy of particle collisions in the ergosphereof the rotating black holes.

It is shown that the energy in the centre of massframe of colliding particles in free fall at any pointof the ergosphere of the rotating black hole can growwithout limit for fixed energy values on infinity.

The effect takes place for large negative values of theangular momentum of one of the particles.

18

Limitations on the values of particle angularmomentum close to the Kerr’s black hole

The permitted region for particle movement is defined by conditions

R =⎛⎝⎛⎝r2 + a2

⎞⎠ E − aJ

⎞⎠2 − Δ[m2r2 + (J − aE)2 + Q] ≥ 0, (45)

Θ = Q − cos2θ

⎡⎢⎢⎢⎢⎢⎣a

2(m2 − E2) +J2

sin2θ

⎤⎥⎥⎥⎥⎥⎦ ≥ 0, (46)

and movement “forward in time” dt/dλ > 0.The condition (46) gives possible values of Carter’s constant Q. From (7) it

follows that for movement with constant θ it is necessary and sufficient that Θ = 0.It is always possible to choose this value, so that for geodesics in the equatorialplane (θ = π/2) Carter’s constants Q = 0.

Let us find limitations for the particle angular momentum from the conditionsR ≥ 0, dt/dλ > 0 at the point (r, θ), taking the fixed values of Θ.

19

Outside the ergosphere r2 − 2rM + a2 cos2θ > 0 one obtains

E ≥ 1

ρ2

√(m2ρ2 + Θ)(r2 − 2rM + a2 cos2θ), (47)

J ∈ [J−, J+] , (48)

J± =sin θ

r2 − 2rM + a2 cos2θ

[−2rMaE sin θ

±√Δ (ρ4E2 − (m2ρ2 + Θ)(r2 − 2rM + a2 cos2θ))

]. (49)

Boundary values of J± correspond to dr/dλ = 0.On the boundary of ergosphere

r = r1(θ) ⇒ E ≥ 0, (50)

J ≤ E

⎡⎢⎢⎣Mr1(θ)

a+ a sin2θ

⎛⎜⎜⎝1 − m2

2E2− Θ

4Mr1(θ)E2

⎞⎟⎟⎠⎤⎥⎥⎦ . (51)

20

Inside ergosphere

rH < r < r1(θ) ⇒ (r2 − 2rM + a2 cos2θ) < 0, (52)

J ≤ J−(r, θ) =sin θ

−(r2−2rM +a2 cos2θ)[2rMaE sin θ −

−√√√√Δ

(ρ4E2 − (m2ρ2 + Θ)(r2 − 2rM + a2 cos2θ)

)]. (53)

So on the boundary and inside ergosphere there exist geodesicson which particle with fixed energy can have arbitrary large inabsolute value negative angular momentum projection.

21

From (53) one can see that for negative energy E of the particle in ergosphereits angular momentum projection on the rotation axis of the black hole must bealso negative. This is a well known Penrose effect. However rotation in Boyer–Lindquist coordinates for any particle in ergosphere has the same direction as therotation of the black hole (the effect of the “dragging” of bodies by the rotatingblack hole). Really for timelike geodesics ds2 > 0 leads to dϕ/dt > 0. Soit is incorrect to say (as it is said in some test books on black holes (seeS.Chandrasekhar ”Mathematical Theory of Black Holes”, p. 368)that “only counter-rotating particles can have negative energy”!Inside the ergosphere the usual intuition which is true far outside it that thechange of the sign of angular momentum projection on Z-axis means the changeof the rotation on counter-rotation following from usual formula J = r × p isincorrect.

From equations of geodesics (6) one can see the peculiar properties of the cor-respondence between the direction of rotation and the angular momentum pro-jection. Let us rewrite it as

22

ρ2 sin2θdϕ

dλ=

E2Mra sin2θ

Δ+ J

r2 − 2rM + a2 cos2θ

Δ. (54)

For large r outside the ergosphere one gets the standard expression for theangular momentum projection in Minkowski space J = mr2 sin2θ dϕ/dτ andJ = mr2 dϕ/dτ for θ = π/2.

But when one is ingoing inside the ergosphere one sees that the coefficient for Jin (54) becomes zero on its surface so that the angular velocity dϕ/dλ and dϕ/dτis defined by the energy and does not depend on J at all.

Inside the ergosphere the coefficient for J in (54) becomes negative, the angularvelocity is still positive and one comes to unusual conclusion: if the energy ofthe particle in ergosphere is fixed particles with negative angularmomentum projection are rotating in the direction of rotation ofthe black hole with greater angular velocity!

So the constant characterizing the geodesic which coincides with the usual an-gular momentum definition taken from Newtonian physics outside the black holedoes not coincide with it in ergosphere.

23

2. Energy of collision with a particle with large angular momen-tum

Let us find the asymptotic of (23) for J2 → −∞ and some fixed value r in ergosphere supposing

the value of Carter’s constant Q2 to be such that (46) is valid and Θ2 � J22 . Then from (23) one

obtains

E 2c.m. ≈

−2J2

ρ2Δ[

J1

sin2θ

⎛⎝r2 − 2rM + a2 cos2θ

⎞⎠

+ 2rMaE1 −σ1rσ2r

√R1

sin θ

√√√√−(r2−2rM +a2 cos2θ)]. (55)

This asymptotic formula is valid for all possible E1, J1 (see (53)) for rH < r < r1(θ) and for

E1 > 0 and J1 satisfying (51) for r = r1(θ). The poles θ = 0, π are not considered here because

the points on surface of static limit are on the event horizon.

24

Note that expression in brackets in (55) is positive in ergosphere. This is evident for r = r1(θ)

and follows from limitations (53) for rH < r < r1(θ), and inside ergosphere (55) can be written

as

E 2c.m. ≈ J2

r2 − 2rM + a2 cos2θ

ρ2Δ sin2θ

(σ1r

√J1+ − J1 − σ2r

√J1− − J1

)2. (56)

So from (55) one comes to the conclusion that when particles fall on therotating black hole collisions with arbitrarily high en-ergy in the centre of mass frame are possible at anypoint of the ergosphere if J2 → −∞ and the energiesE1, E2 are fixed.

The energy of collision in the centre of mass frame is growingproportionally to

√|J2|.Note that for large −J2 the collision energy close to horizon in the centre of mass frame

depending on values E1, J1 can be as large as less then for collisions at the other points of

ergosphere.

25

Outside ergosphere the collision energy is limited for given r, but for r → r1 it can be large if

one of the particles gets in intermediate collisions the angular momentum close to J− (see (49)).

On fig. 2 the dependence of collision energy in the centre of mass frame on the coordinate r is

shown for particles with E1 = E2 = m1 = m2, J1 = 0 and J2 = J− moving in equatorial plane

of black hole with a = 0.8M .

Figure 2: The collision energy in the centre of mass frame for particles with J1 = 0 and J2 = J− out of the ergosphere.

26

Note that large negative values of the angular momentum projection are forbidden for fixed

values of energy of particle out of the ergosphere. So particle which is nonrelativistic on space

infinity (E = m) can arrive to the horizon of the black hole if its angular momentum projection

is located in the interval

−2mM⎡⎢⎣1 +

√√√√√1 +a

M

⎤⎥⎦ ≤ J ≤ 2mM

⎡⎢⎣1 +

√√√√√1 − a

M

⎤⎥⎦ . (57)

The left boundary is a minimal value of the angular momentum of particles with E = m

capable to achieve ergosphere falling from infinity. That is why collisions with J2 → −∞ do

not occur for particles following from infinity. But if the particle came to ergosphere and there

in the result of interactions with other particles is getting large negative values of the angular

momentum projection (no need for getting high energies!) then its subsequent collision with the

particle falling on the black hole leads to high energy in the centre of mass frame.

Getting superhigh energies for collision of usual particles (i.e. protons) in such mechanism occur

however physically nonrealistic. Really from (55) the value of angular momentum necessary for

getting the collision energy Ec.m. has the order

J2 ≈ −aE 2c.m.

2E1. (58)

27

So from (57) absolute value of the angular momentum J2 must acquire the order E 2c.m./(m1m2)

relative to the maximal value of the angular momentum of the particle incoming to ergosphere

from infinity. For example if E1 = E2 = mp (the proton mass) then |J2| must increase with a

factor 1018 for Ec.m. = 109mp. To get this one must have very large number of collisions with

getting additional negative angular momentum in each collision.

However the situation is different for supermassive particles. In our early papers we discussed

the hypothesis that dark matter contains stable superheavy neutral particles with mass of the

Grand Unification scale created by gravitation in the end of the inflation era. These particles are

nonstable for energies of interaction of the order of Grand Unification and decay on particles of

visual matter but are stable at low energies. But in ergosphere of the rotating 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 universe. The member of intermediate collisions for them is not very large (of the order

of 10).

Large absolute values of angular momenta can be obtained due to electromagnetic forces in

ergosphere. One can also speculate about particles with negative energies and momenta radiated

by the black hole to ergosphere (see next part!)

28

A.A.Grib, Yu.V.Pavlov, arXiv:1304.7360 [gr-qc]

Geodesics with negative energy in the ergosphere ofrotating black holes.

It is shown that the geodesics with negative energyfor rotating black holes cannot originate or terminateinside the ergosphere. Their length is always finite andthis leads to conclusion that they must originate andterminate inside the gravitational radius of the ergo-sphere.

29

For particles with negative relative to infinity energy there areno orbits inside ergosphere with constant r or with r changingfor all geodesic inside the interval r1 ≥ r ≥ rH.

Define the effective potential by the formula

Veff = − R

2ρ4(59)

(R see (9)). For effective potential one has

1

2

⎛⎜⎝dr

dλ

⎞⎟⎠2

+ Veff = 0,d2r

dλ2= −dVeff

dr. (60)

Permitted zones of movement of particles are defined by dt/dλ > 0 and

Veff ≤ 0. (61)

To prove our statement it is sufficient to show that for dt/dλ > 0 and

E < 0, r > rH, Veff(r) = 0 ⇒ V ′eff(r) > 0. (62)

30

The example of proof for movement in equatorial plane.For Veff(r) = 0 one obtains

V ′eff =

1

x2

⎡⎢⎢⎣(AE − j)2

x2− E2x + (x − 1)m2

⎤⎥⎥⎦ . (63)

The condition dt/dλ ≥ 0 leads to the inequality

−j + AE ≥ −Ex

2A(x2 + A2) . (64)

E < 0 ⇒ (−j + AE)2 ≥ E2x2

4A2(x2 + A2)2 (65)

so that

V ′eff ≥ 1

x2

⎡⎢⎢⎣E2Δx

2+

E2(x4 − A4)

4A2+ (x − 1)m2

⎤⎥⎥⎦ . (66)

So in case when Veff(x) = 0 the derivative of the effective potential outside thehorizon (x > xH) is positive there are no circular orbits for Penrosetrajectories in Kerr’s black holes. The permitted zone for suchparticles in ergosphere can have only upper boundary.

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The time of movement of particles with negative energy in theergosphere

Δλ = 2rb∫

rH

dr

|dr/dλ| = 2rb∫

rH

dr√−2Veff(r). (67)

In the vicinity of the upper point rb:∣∣∣∣∣∣∣dr

dλ

∣∣∣∣∣∣∣ =√−2Veff ≈

√2(rb − r)V ′

eff(rb). (68)

As it was shown for the boundary point V ′eff(rb) > 0, so

∫ dr

|dr/dλ| ∼∫ dr√

2(rb − r)V ′eff(rb)

(69)

is convergent and the proper time is finite.

Due to the fact that permitted zones for particles with negative energies in ergosphere can have

only upper boundary there are no zeros for dr/dλ in the other points of the trajectory. So the

integral (67) is convergent and the proper time of movement along geodesicin the ergosphere for the particle with the negative energy is finite.

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Finiteness of the proper time for movement of particles with negative energyin ergosphere of the black hole leads to the problem of the origination andtermination of such trajectories. These lines cannot arrive to ergosphere fromthe region outside of the ergosphere. So geodesics with negative energymust originate and terminate inside the gravitational radius. Thismeans that they originate as “white hole” geodesics originating inside the horizon.

Any particle intersecting the event horizon must achieve the Cauchy horizon.After going through the Cauchy horizon the particle can achieve singularity. Thenecessary condition for this is that Carter constant Q ≤ 0 (see (7)–(10)). Forparticles with negative energy in ergosphere this is true only for movement inequatorial plane, i.e. Q = 0.

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From (7)–(46) one can see that all light like particles (photons) withnegative energy falling in equatorial plane from ergosphere achievesingularity. Massive particles also achieve singularity for exampleif E = −m or the angular momentum is such that

J ≤ JH

⎛⎜⎜⎜⎝1 +

a

2M

√√√√√√1 +m2

E2

⎞⎟⎟⎟⎠ . (70)

The proofs of all these results is the same for dr/dλ > 0 and dr/dλ < 0. That is why

the same conditions are valid for “white hole”geodesics originatingin Kerr’s singularity, arriving to ergosphere and then going backinside the gravitational radius.

All particles with nonequatorial movement and E ≤ 0 don’tachieve singularity. Particles moving in equatorial plane also do notachieve singularity if for example

|E|m

� rC

M,

|J |mM

� rC

M. (71)

Then after achieving some minimal values of the radial coordinate the particle can turn it’s

movement in the direction of larger r and come back to ergosphere along the white hole geodesics.

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Conclusion

1. Geodesics with negative energy originate inside the grav-itational radius of the rotating black hole and are ”whitehole”geodesics!

2. One can get information about the interior of the gravitationalradius if some particles move along these geodesics! So there is nocosmic censorship for such rotating black holes. The cosmonautcan get direct information about the interior of the gravitationalradius only inside the ergosphere. However If one considers inter-action of negative energy particles radiated by the ”black-white”hole to ergosphere with ordinary positive energy particles escapingthe ergosphere this information can be obtained by any externalobserver.

3. The mass of the ”black-white”hole can grow due to the radi-ation of negative energy particles to ergosphere where these par-ticles interact with positive energy particles.

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Thank You!