Документ взят из кэша поисковой машины. Адрес оригинального документа : http://nuclphys.sinp.msu.ru/conf/lpp14/220809/Fil'chenkov.pdf Дата изменения: Wed Sep 16 16:31:16 2009 Дата индексирования: Tue Oct 2 00:45:57 2012 Кодировка:

Moscow 2009

14th Lomonosov Conference

Nonhydrogen-like Graviatom Radiation
Michael L. Fil'chenkov Yuri P. Laptev2
1

1

Institute of Gravitation and Cosmology, Peoples' Friendship University of Russia, Moscow Physics Department, Moscow Bauman Technical University
2

1


Moscow 2009

14th Lomonosov Conference

Bound quantum systems via

electromagnetic interaction · atoms · molecules gravitational interaction

strong interaction · hadrons · nuclei

?
2


Moscow 2009

14th Lomonosov Conference

Quantum systems bound by gravity
Graviatoms are bound quantum systems maintaining particles in orbit around mini-holes (primordial black holes). Particles: mesons, leptons.

3


Moscow 2009

14th Lomonosov Conference

Theoretical solution to the graviatom problem
SchrЖdinger's equation for the graviatom

mc 2 rg mc 2 rg rq 1 d 2 dR pl l (l +1) 2m Rpl = 0 r dr - 2 Rpl + 2 E + 2r - 2 2 4r r dr r

(1)

describes a radial motion of a particle with the mass m and the charge q in the mini-hole potential, taking account of DeWitt's self-interaction, 2 2GM where rg = 2 and rq = q 2 are the mini-hole gravitational radius and mc c the classical radius of a charged particle respectively.
4


Moscow 2009

14th Lomonosov Conference

Graviatom existence conditions
· the geometrical condition L > rg + R , where L is the characteristic size of the graviatom and R is the characteristic size of a particle; · the stability condition given by (a) gr < H , where gr is the graviatom lifetime and H is the minihole lifetime, and (b) gr < p , where p is the particle lifetime (for unstable particles); · the indestructibility condition (due to tidal forces and the Hawking effect) Ed < Eb , where Ed is the destructive energy and Eb is the binding energy. The charged particles satisfying these conditions are the electron, muon, taon, wino, pion and kaon.
5


Moscow 2009

14th Lomonosov Conference

Dependence of mini-hole masses on charged-particle ones

Figure 1. The black curves indicate the range of values related to the geometrical condition (the upper curve) and to Hawking's effect "ionization" (the lower curve). The red curve is related to the particle stability condition ( p = 10 -22 s).
6


Moscow 2009

14th Lomonosov Conference

DeWitt's self-action corrections to the hydrogen-like graviatom
The energy spectrum of a hydrogen-like atom has the form
(0 En )

=-

mc 2 2n
2

2 g

,

(2)

where g = GMm is the fine-structure constant gravity equivalent. The perturbation being due to DeWitt's self-interaction
c

mc2rg rq Vq = . 2 4r

(3)

7


Moscow 2009

14th Lomonosov Conference

Corrections to the hydrogen-like spectrum
(1) Enl
2 where eg = e g .

2 mc2eg g = , n3 l + 1 2

(4)

c

As will be shown below Mm = g m2 , where g = 0.5 ± 0.6. pl The intensity of the electric dipole radiation of a particle with mass m and charge e in the gravitational field of a mini-hole for the transition 2 p 1s is
8


Moscow 2009

I10,21

I1(0,)21 1 0

- 46 9

eg

4

14th Lomonosov Conference

(1- 6
3

eg

)

2

,

(5)

where
(0 I10,)2 1 (0 2 e 12 ) (0) f = 10, 21 3 mc 2

(6)

(0 (0 is the intensity for a hydrogen-like graviatom, 12 ) = ( E1(0) - E2 ) ) / is

the frequency and

f

(0) 10,21

213 = 9 is the oscillator strength. 3

Thus, DeWitt's self- interaction diminishes both frequencies and intensities of the hydrogen-like graviatom.

9


Moscow 2009

14th Lomonosov Conference

Pauli's corrections to the hydrogen-like graviatom
Taking account both DeWitt's self-action and Pauli's spin corrections, the graviatom energy spectrum takes the form
2 mc2 g mc2 + E =- 2n2 2n3 3 g


l+

2e
1 2

2

c j+

-



g + 1
2



4 3mc2 g , 8n4

(7)

where j = l + s , s = 0 for mesons and s = ± 1 for leptons.
2

2 p 1s splits into two transitions: 2 p1/ 2 1s1/ 2 with j = 0 and 2 p3/ 2 1s1/ 2 with j =1. The intensity for the transition 2 p3/ 2 1s1/ 2 is I10,
21

The first nl j levels: 1s1/ 2 , 2s1/2, 2 p1/ 2 , 2 p3/2. The dipole transition

= 2 I1(0,)21 1+ 0 3

15 48



2 g

4



10


Moscow 2009

14th Lomonosov Conference

Graviatoms with slowly rotating miniholes
Lense-Thirring's metric for a slowly rotation black hole has the form 2 2 = 1- rg c2 dt 2 - dr - r 2 d 2 + sin 2 d 2 + 2rg a sin 2 d dt , (8) ds r r r 1- g
r

where a is the specific angular momentum. The perturbation being due to rotation
2 rg a l l +1 rg a l l +1 Va = - . 3 4 r r

(9)
2

The intensity for the transition 2 p 1s

I21 = I

(0) 1+ 21

0.0349 ma



4

1-1.1064

ma



11


Moscow 2009

14th Lomonosov Conference

The correction to the hydrogen-like spectrum of the graviatom taking account of DeWitt's self-force, particle spin and a slow rotation of the minihole reads
2 (1) mc Enl = 2n3 3 g



2 e- l+1 c j 2
2 g

2



g + +1 2

2 4 4 3mc2 g 2 g m2c2a 3n - (l + 1 - + 2 8n4 3 l+1 n ( 2) n2 ( l + 1 2





)

1 2 2

)

2



2 g

(10)

Providing am

1

, we obtain
(1) Enl = (0) En
2 g n

1 - 4am 1 j+1 1 l + 2 2



2



.

(11)

12


Moscow 2009

14th Lomonosov Conference

(1 Enl) 1, if (0 En )

2 g

n , 4am



l+1 , 2

2

j+1 2

1.

As a result, the perturbation theory is valid for a slow rotation of the minihole and higher levels of the particle being constituents of the graviatom.

13


Moscow 2009

14th Lomonosov Conference

Conclusion
The graviatoms can contain only leptons and mesons. The hydrogenlike graviatom is perturbed by DeWitt's self-action and minihole rotation diminishing, whereas particle spin enhancing the dipole radiation intensity.

14