20 CHAPTER 1. GENERAL INTRODUCTION
is called the Schwarzschild radius, at which the metric displays a singularity. There have
been many confusions about black holes throughout the history of general relativity and black
holes.
In this section, we present the black hole theorem proved in Section7.3, and clarify some
of the confusions in the literature. This black hole theoremleads to important insights to many
problems in astrophysics and cosmology, which will be addressed in details in Chapter 7.
Blackhole Theorem (Ma and Wang,2014a)Assume the validity of the Einstein
theory of general relativity, then the following assertions hold true:1) black holes are closed: matters can neither enter nor leave their interiors,
2) black holes are innate: they are neither born to explosion ofcosmic objects,
nor born to gravitational collapsing, and
3) black holes are filled and incompressible, and if the matter field is non-
homogeneouslydistributed in a black hole, then there must be sub-blackholes
in the interior of the black hole.This theorem leads to drastically different views on the structure and formation of our
Universe, as well as the mechanism of supernovae explosion and the active galactic nucleus
(AGN) jets. Here we only make a few remarks, and refer interested readers to Section7.3for
the detailed proof.
1.An intuitive observation.One important part of the theorem is that all black holes are
closed: matters can neither enter nor leave their interiors. Classical view was that nothing
can get out of blackholes, but matters can fall into blackholes. We show that nothing can get
inside the blackhole either.
To understand this result better, let’s consider the implication of the classical theory that
matters can fall inside a blackhole. Take as an example the supermassive black hole at the
center of our galaxy, the Milky Way. By the classical theory,this blackhole would contin-
uously gobble matters nearby, such as the cosmic microwave background (CMB). As the
Schwarzschild radius of the black holer=Rsis proportional to the mass, then the radiusRs
would increase in cubic rate, as the massMis proportional to the volume. Then it would be
easy to see that the black hole will consume the entire Milky Way, and eventually the entire
Universe. However, observational evidence demonstrates otherwise, and supports our result
in the blackhole theorem.
2.Singularity at the Schwarzschild radius is physical.One important ingredient is that
the singularity of the space-time metric at the Schwarzschild radiusRsis essential, and cannot
be removed by any differentiable coordinate transformations. Classical transformations such
as those by Eddington and Kruskal arenon-differentiable, and are not valid for removing
the singularity at the Schwarzschild radius. In other words, the singularity displayed in both
7.1.2 Schwarzschild and Tolman-Oppenheimer-Volkoff (TOV) metrics.
(1.8.3) ds^2 =−euc^2 dt^2 +
(
1 −
r^2
R^2 s)− 1
dr^2 +r^2 dθ^2 +r^2 sin^2 θdφ^2 ,is a true physical singularity, and defines the black hole boundary.