Mathematical Principles of Theoretical Physics

7.3. BLACK HOLES 439

If theδ-factor satisfies that

(7.3.26)

4

π

g^2 n

̄hc

≤δ<

4

π

g^2 q

̄hc

,

then the neutrons will be crushed to become quarks and gluons. The equality (7.3.25) shows
that the star satisfying (7.3.26) must be a black hole which is composed of quarks and gluons,
and is called quark black hole.
Ifδsatisfies

(7.3.27)

4

π

g^2 q

hc ̄

≤δ<

4

π

g^2 w

̄hc

,

then the quarks are crushed into weaktons, and the body is called weakton black hole.
In summary, we infer from (7.3.24), (7.3.26) and (7.3.27) the following conclusions:

(7.3.28) a body=







a neutron star ifδ<δncandm> 1. 4 M⊙,

a quark black hole ifδnc<δ<δqc,

a weakton black hole ifδqc<δ<δwc,

where

δcj=

4

π

g^2 j

̄hc

forj=n,q,w.

5.Upper limit of the radius.Weaktons are elementary particles, which cannot be crushed.
Therefore, there is no star withδ-factor greater thanδwc. Thus there exists an upper limit for
the radiusrcfor astronomical bodies with massm, determined by

2 MG

c^2 rc

=δwc.

Namely, the upper limit of the radiusrcreads

(7.3.29) rc=
πmG
2 c^2

hc ̄

g^2 w

.

Finally we remark that since black holes cannot be compressed, theδ-factor of any as-
tronomical object cannot be less than one:δ≥1. Therefore, by (7.3.25) and (7.3.26), there
exist no weakton black holes.

7.3.4 Origin of stars and galaxies

The closeness and innateness of black holes provide an excellent explanation for the origin
of planets, stars and galaxies.
In fact, all black holes are inherent. Namely, black holes exist at the very beginning of
the Universe. During the evolution of the Universe, each black hole forms a core and adsorbs
a ball of gases around it. The globes of gases eventually evolve into planets, stars and galaxy
nuclei, according to the radii or masses of the inner cores ofblack holes. Of course, it is
possible that several black holes can bound together to forma core of a bulk of gases.