## Remarks on a New Blackhole Theorem

This post makes a few remarks on the following blackhole theorem proved in (Tian Ma & Shouhong Wang, Astrophysical Dynamics and Cosmology, J. Math. Study, 47:4 (2014), 305-378):

Blackhole Theorem (Ma-Wang, 2014) 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 of cosmic objects, nor born to gravitational collapsing, and
3. black holes are filled and incompressible, and if the matter field is non-homogeneously distributed 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. We refer interested readers to the original paper for the detailed proof.

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 black holes, but matters can fall into black holes. We show that nothing can get inside the black hole either.

To understand this result better, let’s consider the implication of the classical theory that matters can fall inside a black hole. Take for example the supermassive black hole at the center of our galaxy, the Milky Way. By the classical theory, this black hole would continuously gobble matters nearby, such as the cosmic microwave background (CMB). As the Schwarzschild radius of the black hole

$\displaystyle R_s = \frac{2 M G}{c^2} \ \ \ \ \ (1)$

is proportional to the mass, then the radius ${R_s}$ would increase in cubic rate, as the mass ${M}$ is proportional to the volume. Then it would not hard to see 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 black hole theorem.

Singularity at the Schwarzschild radius is physical. One important ingredient is that the singularity of the space-time metric at the Schwarzschild radius ${R_s}$ is essential, and cannot be removed by any differentiable coordinate transformations. Classical transformations such as e.g. those by Eddington and Kruskal are non-differentiable, and are not valid for removing the singularity at the Schwarzschild radius. In other words, the singularity displayed in both the Schwarzschild metric

$\displaystyle ds^2= -\left(1-\frac{R_s}{r}\right)c^2dt^2+\left(1-\frac{R_s}{r}\right)^{-1}dr^2 +r^2d\theta^2+r^2\sin^2\theta d\varphi^2, \ \ \ \ \ (2)$

and the Tolman-Oppenheimer-Volkoff (TOV) metric

$\displaystyle ds^2= -e^u c^2dt^2+\left(1-\frac{r^2}{R_s^2}\right)^{-1}dr^2 +r^2d\theta^2+r^2\sin^2\theta d\varphi^2, \ \ \ \ \ (3)$

is a true singularity, and defines the black hole boundary.

Geometric Realization of a black hole. As described in Section 4.1 in the paper, the geometrical realization of a black hole, dictated by the Schwarzschild and TOV metrics, clearly manifests that the real world in the black hole is a hemisphere with radius ${R_s}$ embedded in ${R^4}$, and at the singularity ${r=R_s}$, the tangent space of the black hole is perpendicular to the coordinate space ${R^3}$.

This geometric realization clearly demonstrates that the disk in the realization space ${R^3}$ is equivalent to the real world in the black hole. If one observes from outside that nothing gets inside the black hole, nothing will get in the real world (the black hole).

Tian Ma & Shouhong Wang