Документ взят из кэша поисковой машины. Адрес оригинального документа : http://www.ipa.nw.ru/SPAS/download/PavlovIPA10.pdf Дата изменения: Thu Feb 17 17:04:59 2011 Дата индексирования: Tue Oct 2 00:37:27 2012 Кодировка: Поисковые слова: п п п п п п п п п п п п п п п п п п п п

" "
. .
a, b

,

. .

a, c

a

,
b

. . . ,

c

, -


2

.
1) , . M 1014 1080 .
( , , . . , . . , . . . 1: "" . . .: . , 1988.)


3

Standard Cosmological Model


4

2) , . 3) 10-14 ( ). 4) . A. A. Grib, Yu. V. Pavlov, Int. J. Mod. Phys. D11 (2002), 433­436; Int. J. Mod. Phys. A17 (2002), 4435­4439; Gravit. & Cosmology 8 (2002), Suppl., 148­153. Gravit. & Cosmology 12 (2006) 159­162. Gravit. & Cosmology, 14 (2008), 1­7. Mod. Phys. Lett. A23 (2008), 1151­1159. Gravit. & Cosmology, 15 (2009), 44­48.


5

Pierre Auger Observatory

. 2:

The Pierre Auger Observatory is a hybrid detector. On the hill is one of the 4 Fluorescence Detector buildings and communications tower. In the bottom foreground is one of the 1,600 Surface Detectors ­ water tanks (12,000 liter), each tank is separated from each of its neighbors by 1.5 kilometers.


6

Pierre Auger Col laboration, Science 2007, Vol.318, 938­943. " Correlation of the HighestEnergy Cosmic Rays with Nearby Extragalactic Ob jects"

. 3: Sky mapshowing cosmic rays detected by the Pierre Auger Observatory. Low-energy cosmic rays appear to originate from evenly distributed sources (blue dots), but the origins of the highest-energy events (crosses) correlate with the distribution of lo cal matter as represented by nearby active galactic nuclei (red stars). Thus, active galactic nuclei are a likely source of these rare 18 high-energy cosmic rays. 1 EeV = 10 eV


7

[1] A. A. Grib and Yu. V. Pavlov, Mod. Phys. Lett. A 23, 1151 (2008); arXiv:0712.2667v1. Do active galactic nuclei convert dark matter into visible particles? [2] .., ... . .18. .: , 2008. .161­181. . [3] A. A. Grib and Yu. V. Pavlov. Grav. Cosmol. 15, 44 (2009); arXiv:0810.1724v1. Active galactic nuclei and transformation of dark matter into visible matter. [4] A.A.Grib and Yu.V.Pavlov. Int. J. Mod. Phys. A 24, N8&9, (2009) 1610­ 1619. Superheavy particles as dark matter and their role in creation of visible matter in active galactic nuclei and the early Universe.


8

[1] , , , .
100 % M = 1014 2 · 10-10 , , , M · j 1028 . = 10-4 , , , ma = Mj / , , , .
, . , , = 10
-20

/3. ,

v 108 / (.. ).


9



c = 4

c v

2

2 rg .

(1)

. , . . (1) , , .

rg -- . MBH = 108M (M -- ), ma = cv 3 · 1028 /, j 1038 /.

(r) =

0 (r/r0) (1 + r/r0 )

3-

(2)

= 1 --, = 1.5 , r0 = 45 , 0 = 10-24 /3, ma 2 · 1028 - 1030 /. , .


10



-- A. A. Grib, Yu. V. Pavlov, Mod. Phys. Lett. A23 (2008), 1151; Gravit. & Cosmology, 15 (2009), 44 -- , , , .


11

.
Wald : , v> c 2

. , . , . , , . ( , .)


12


-

2Mr (dt - a sin2d)2 - ds = dt - (r + a )sin d - r2 + a2 cos2 dr2 2 2 2 - (r + a cos ) 2 + d 2 , r - 2Mr + a2
2 2 2 2 2 2

(3)

M -- , aM -- .

r = rH M + r = r0 M +

M 2 - a2 .

(4)

, " ",

M 2 - a2 cos2.

(5)

- .


13

( = /2) (3) dt 1 = d 2Ma r +a + r
2 2 2

2Ma - L, r

(6)

1 2Ma 2M d = + 1- L, d r r dr d
2

(7) (8)

2M a 2 2 - L 2 2 = + 3 (a - L) + - 2 1 , 2 r r r
2

(9) = r2 - 2Mr + a2 , 1 = 1 (1 = 0 ), -- , = const -- : m (3) m; Lm = const -- , .


14

Ec.m. m (Ec.m. , 0 , 0 , 0) = mu
i (1)

+ mu

i (2)

,

(10)

ui = dxi/ds. (10) , uiui = 1

E

c.m.

=m 2

1+ ui u (1)

(2)i

.

(11)

(11) L1, L2, (1 = 2 = 1) . x = r/M , A = a/M , ln = Ln/M , x = x2 - 2x + A2 (3), (6)­(8)
2 1 Ec.m. = 2 2m x

x

2x2(x - 1) + l1l2(2 - x)+2A2(x +1) - 2A(l1 + l2) -
2 2x2 +2(l2 - A)2 - l2 x ,

-

2 2x2 +2(l1 - A)2 - l1 x

(12)

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


15

r rH

a=M

E

c.m.

(r rH ) =



2m

l2 - 2 l1 - 2 + , l1 - 2 l2 - 2

(13)

, 2M , . (8) ( = 1), : -2 1+ 1+ A = lL l lR = 2 1+ 1-A . (14)

(8) xR = 2 1+ 1 - A - A, xL = 2 1+ 1+ A + A (15)

l = lR l = lL .


16

Ec.m. (r rH ) = 2m

( l1 - l2 ) 2 2+ . 2 - 1)(4 - l l ) 8 - 2A(l1 + l2)+ ( 1 - A 12

(16)

lL,lR (16), Ecma.x .m 2m (r rH ) = 4 1 - A2 1 - A2 + 1+ 1+ A + 1+ 1 - A2 1-A
2

.

(17)

. 4: .


17

A = 1 -



0 (17)

A=1-



E

max c.m.

(r rH ) 2 21/4 +2-1


/4

m
1/4



m · 4, 06
1/4

.

(18)

Amax = 0.998

Ecma.x/m 19 . .m

A=0 Ecma.x /m = 2 5 .m A. N. Baushev, Int. J. Mod. Phys. D 18, 1195 (2009). [1] K. S. Thorne, Astrophys. J. 191, 507 (1974). Disk-accretion onto a black hole. I I. Evolution of the hole. [2] E. Berti, V. Cardoso, L. Gualtieri, F. Pretorius and U. Sperhake, Phys. Rev. Lett. 103, 239001 (2009). Comment on "Kerr black holes as particle accelerators to arbitrarily high energy". [3] T. Jacobson and T. P. Sotiriou, Phys. Rev. Lett. 104, 021101 (2010). Spinning black holes as particle accelerators.


18

A.A.Grib, Yu.V.Pavlov, arXiv:1004.0913v1 [gr-qc]:

E

c.m.

(r rH ) = 2m

(l1 - l2)2 , 1+ 2xC (l1 - lH )(l2 - lH )

(19)



2xH 2(1 + 1 - A2 ) = , lH = A A dt >0 d

xC = 1 -

1 - A2 .

(20)

lH --

x xH ,

2xH l< = lH . A

(21)

r = 1 l (8)

x

1,2

=

l2 ±

l4 - 16(A - l)2 . 4

(22)


19

l = lH - , :

l = lH -

2 x2 C x < x xH + . 2 4xH 1 - A

(23)

. 5:

A = 0.95 lR 2.45, l = 2.5, lH 2.76. l = 2.5 . 1 2 dr d
2

+ Veff (r, l) = 0 ,

l2 1 (A - l )2 Veff (x, l) = - + 2 - . x 2x x3

(24)


20

l lR , , (23) , l1 = lH - , , (16) -



l1 = lH -



E

c.m.

m

2(lH - l2) 1 - 1 - A2

(25)

0. 3.85m Ec.m. . (26) , A = 1 - lH lR 1-A lH - lR = 2 1 - A + 1+ A - A 2( 2 - 1) , 0. (27) A Amax = 0.998 , l2 = l
L



lH - lR 0.04 . (28) Amax = 0.998 , .


21

A. A. Grib and Yu. V. Pavlov, Mod. Phys. Lett. A 23, 1151 (2008); arXiv:0712.2667v1. Do active galactic nuclei convert dark matter into visible particles?

This black hole acts as a cosmic supercollider in which superheavy particles of dark matter are accelerated close to the horizon to the Grand Unification energies and can be scattering in collisions.


22

The time of movement b efore the collision with unb ounded energy
From equation of the equatorial geodesic (6), (8) for a particle with dimensionless angular momentum l and specific energy = 1 (i.e. the particle is non relativistic at infinity) falling on the black hole with dimensionless angular momentum A one obtains dr (x - xH )(x - xC ) 2x2 - l2x +2(A - l)2 =- . dt x3 + A2x +2A(A - l) x (29)

So the coordinate time (proper time of the observer at rest far from the black hole) of the particle falling from some point r0 = x0M to the point rf = xf M > rH is equal to x0 x x3 + A2x +2A(A - l) dx t = M . (30) 2 - l 2 x +2(A - l )2 (x - xH )(x - xC ) 2x
xf

In case of the extremal black hole (A = 1, xR = xH = 1) and the limiting value l = 2 x0 x-1 M 2 x (x2 +8x - 15) +5 ln t = 3(x - 1) x +1 xf 2

(31)

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 t.


23

For the interval of proper time of the free falling to the black hole particle one obtains x0 x3 dx = M . (32) 2 - l 2 x +2(A - l )2 2x
xf

If the angular momentum of the particle falling inside then the proper time is finite for xf xH . For A = 1, M 2 x (3 + x)+3 l = 32

the black hole is such that lL < l < lR l = 2 the integral (32)) is equal to x0 x-1 n (33) x +1 xf

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. From (7), (8) for the angle of the particle falling in equatorial plane of the black hole one obtains x0 x (xl +2(A - l)) dx = . (34) 2 - l 2 x +2(A - l )2 (x - xH )(x - xC ) 2x
xf

If A = 0, then (34) is divergent for xf xH . So before collision with infinitely large energy the particle must commit infinitely large number of rotations around the black hole.


24

The Extraction of Energy after the Collision in Kerr's Metrics
Conservation laws in inelastic particle collisions for the energy and momentum lead to

m(u

i (1)

+ ui ) = (v (2)

i (1)

i + v(2)) .

(35)

From (35) for t- and -component due to (6) one obtains m( 1 + 2 ) = ( 1 + 2 ) , m(L 1 + L 2 ) = (L 1 + L 2 ) , (36)

i.e. the sum of energies and angular momenta of colliding particles is conserved in the field of Kerr's black hole. The initial particles in our case were supposed to be nonrelativistic on the infinity: 1 = 2 = 1, so Eq. (35) for r-component becomes
-m =
2 1µ

2M 2M L2 1 2+ - 2+ (a - L1 ) 3 r r r a2 2µ 1 -L r2
2 1µ

2M 2M L2 2 2+ -2 (a - L2 ) 3 r r r 2M (a r3

= -L r2
2 2µ

(37) - . r2

+

2M (a r3

1µ

- L1 µ )2 +

-

- r2



2 2µ

+

2µ

- L2µ )2 +

a2 2µ 2


25

For the case when the collision takes place on the horizon of the black hole (r rH ) the system (36), (37) can be solved exactly 1 = AL1 , 2rH 2 = 2m AL1 , - 2rH L 2 = m (L 1 + L 2 ) - L 1 . (38)

In general case the system of three Eqs. (36)­(37) for four variables 1, 2, L1, L2 can be solved numerically for fixed value of one variable (and fixed parameters m/, L1/M , L2/M , a/M , r/M ). The example of numerical solution is /m = 0.3,l1 = 2.2, l
1

l2 = 2.198,
1

A = 0.99,
2

x = 1.21 , = -0.548 .

= 16.35,

l

2

= -1.69,

= 7.215,

Note that the energy of the second final particle is negative and the energy of the first final particle is larger than the energy of initial particles as it must be in the case of a Penrose process. The limit 2m for the extracted energy for any (including Penrose process) scattering process in the vicinity of the black hole was obtained in T. Jacobson and T. P. Sotiriou, Phys. Rev. Lett. 104, 021101 (2010). Let us show why this conclusion is incorrect.


26

The main assumption collinearity of vectors of hole. They say that these In the limiting case (A 4-velocity of the infalling

made by T.Jacobson and T. P.Sotiriou is the supposition of the 4-momenta of the particles falling inside and outside of the black vectors are "asymptotically tangent to the horizon generator". = 1, l1 = 2) the expressions dt/d , d/d of the components of the particle (6), (7) go to infinity when r rH , but dr/d goes to zero.

grr ur u (1)

r (1)

-2,

r rH .

(39)

For the particle outgoing from the black hole due to exact solution on the horizon (38) one puts 1 = l1/2+ , where is some function of r and l1, such that 0 when r rH . Putting this 1 into (6)­(8) one gets for x = r/M 1
t (1) ut (1)

v

=

(1) u (1)

v

l1 = + , x-1 2

r (1) ur (1)

v

=-

2 2 2 l1 3 2 1 + + l1 - . (x - 1)2 x - 1 8 2

(40)

Due to the condition dt/d > 0 (movement forward in time) the necessary condition for collinearity is that both (40) must be zero, which is not true.

u

(1)

= ut , ur , u , 0 , (1) (1) (1)

v

(1)

t r = v(1), v(1), v(1), 0 .

(41)



28

.
1) , (: 2, 2005) " , , , , . , , , , : . , , , , , -- , !" 2) , . : XXI (.-. ) (: 2, 2007) . 219


29


30


31

3) , . . .. (: , 2008) . 124. ... ( -- ., ..) , . 4) (.: , 1985) . 34 [Regge T Cronache Del l'Universo (Boringhieri, Torino, 1981)] " ( -- ., ..) , « »". 5) (-: . . ., 2002) [Rees M Our cosmic habitat (Princeton: Princeton Univ. Press, 2000)] , , , : ... .


32


33

1.

rg 2 2 dr2 -r ds = 1 - c dt - rg r 1- r
2 2

d2 +sin2 d2 .

(42)

rg = 2Gm/c2 -- , c -- .

dr cd

2

=

rg + 2 - 1, r

dt = , d 1 - rg /r

(43)

= const. r r0 > rg , = 1 - rg 0

rg dr =- 1- cdt r

1 - rg /r 1- 1 - rg /r0

1/2

.

(44)

(44) t - t0 r0 t0 -


34

r < r0: t - t0 = rg c x0 - 1 (2 + x0) arctg (x0 - 1) x + x0 1- x x0 x0 - x + x x (x 0 - x ) + (45)

+ 2 ln

- ln |x - 1| ,

x0 = r0/rg , x = r/rg . - 0 r0 r r0 - 0 = c r0 rg arctg r0 - 1+ r r r - r0 r
2 2 0

.

(46)

, r0 r !


35

. ds = 0 dr rg =± 1- . cdt r r0 - rg r0 - r rg + ln . c c r - rg (47)

t - ts (ts -- ) r0 r t - ts = (48)

x(t) x( ) ( ).
. 6:


36

(48) (45), : ts r0 , r < r0 "", r0 t0 < ts?

rg (2 + x0) x0 - 1 arctg ts - t0 = c + x0 - x (x0 - 1) x -

x0 - x + x x0 - x +2 ln x + x0 x0 - x (x0 - 1) x
0

(49) ,

x = r/rg , x0 = r0 /rg . (49) x 1, .. r rg , , , :

rg ts - t0 = (2 + x0) x0 - 1 arctg c

x0 - 1+ 2 ln 2 - ln x

0

.

(50)


37

, , , . ts - t0

rg x ts - t0 = c

3/2 0

arctg

x0 - x + x

x 0 x (x 0 - x ) - (x 0 - x ) .

(51)

x0/x = r0/r

1 (49) (51)

ts - t0

r0 2c

r0 . rg

(52)


38

. 7:

-.

, , ( ). , , . ( ). , .