436 CHAPTER 7. ASTROPHYSICS AND COSMOLOGY
In other words, there is no energy fluxPron the Schwarzschild surface, and we have shown
that no external energy can enter into a black hole.
In conclusion, we have shown that black holes are closed: no energy can penetrate the
Schwarzschild surface.
Step 3. Innateness of black holes.The explosion mechanism introduced in Subsection
7.2.6clearly manifests that any massive object cannot generate anew black hole. In other
words, we conclude that black holes can neither be created nor be annihilated, and the total
number of black holes in the Universe is conserved.
Step 4.Assertion 3) follows by applying conclusion (7.5.55) and the fact that sub black-
holes are incompressible. The theorem is therefore proved.
We remark again that the singularity on the boundary of blackholes is essential and cannot
be removed by any differentiable coordinate transformation with differentiable inverse. The
Eddington and Kruskal coordinate transformations are non-differentiable, and are not valid.
Remark 7.16.The gravitational forceFgenerated by a black hole in its exterior is given by
F=
mc^2
2∇g 00 =−mg^11
∂ ψ
∂r,
whereψis the gravitational potential. By (7.3.2) we have the following gravitational force:
(7.3.13) F=−
(
1 −
2 MG
c^2 r)
mMG
r^2.
Consequently, on the boundary of a black hole, the gravitational force is zero:
F= 0 atr=Rs.7.3.3 Criticalδ-factor
Black holes are a theoretical outcome. Although we cannot see them directly due to their
invisibility, they are, however, strong evidences from many astronomical observations and
theoretic studies.
In the following, we first briefly recall the Chandrasekhar limit of electron degeneracy
pressure and the Oppenheimer limit of neutron degeneracy pressure; then we present new
criterions to classify pure black holes, which do not contain other black holes in their interior,
into two types: the quark and weakton black holes, by using theδ-factor.
1.Electron and neutron degeneracy pressures. Classically we know that there are two
kinds of pressure to resist the gravitational pressure, called the electron degeneracy pressure
and the neutron degeneracy pressure. These pressures prevent stars from gravitational col-
lapsing with the following mass relation:
(7.3.14) m<
{
1. 4 M⊙ for electron pressure,
3 M⊙ for neutron pressure.Hence, by (7.3.14), we usually think that a dead star is a white dwarf if its massm< 1. 4 M⊙,
and is a neutron star if its massm< 3 M⊙. However, if the dead star has massm> 3 M⊙, then