## 1. Newtonian Viewpoint

Consider a massive body with mass ${M}$ inside a ball ${B_R}$ of radius ${R}$. The Schwarzschild radius is defined by ${R_s={2GM}/{c^2}.}$

Based on the Newtonian theory, a particle of mass ${m}$ will be trapped inside the ball ${B_R}$ and cannot escape from the ball, if its kinetic energy, ${mv^2/2}$, is smaller than gravitational energy:

$\displaystyle \frac{mv^2}{2} \le \frac{mc^2}{2} \le \frac{mMG}{r},$

which implies that

$\displaystyle r \le R_s =\frac{2GM}{c^2}.$

In other words, if the radius ${R}$ of the ball is less than or equal to ${R_s}$, then all particles inside the ball are permanently trapped inside the ball ${B_{R_s}}$.

It is clear that the main results of the Newton theory of black holes are as follows:

• the radius ${R}$ of the black hole may be smaller than the Schwarzschild radius ${R_s}$,
• all particles inside the ball are permanently trapped inside the ball ${B_{R_s}}$, and
• particles outside of a black hole ${B_{R_s}}$ can be sucked into the black hole ${B_{R_s}}$.

## 2. Einstein-Schwarzschild Theory

#### Black-Holes are closed

Now consider the case where ${R=R_s}$. Based on the Einstein field equations, in the exterior of the body, the Schwarzschild solution is given by

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

and in the interior the Tolman-Oppenheimer-Volkoff (TOV) metric is

$\displaystyle ds^2= - \frac14 \left[1- \frac{r^2}{R^2_s}\right] c^2dt^2 +\left[1-\frac{r^2}{R^2_s}\right]^{-1}dr^2 +r^2d\theta^2+r^2\sin^2\theta d\varphi^2 \qquad \text{ for } r < R_s. \ \ \ \ \ (2)$

The both metrics have a singularity at ${r=R_s}$, which is called the event horizon:

$\displaystyle d\tau = \left[1-\frac{R_s}{r}\right]^{1/2} dt \rightarrow 0, \quad d\tilde r= \left[1-\frac{R_s}{r}\right]^{-1/2} dr \rightarrow \infty \text{ for } r \rightarrow R_s^+, \ \ \ \ \ (3)$

$\displaystyle d\tau = \frac12 \left[1-\frac{r^2}{R_s^2}\right]^{1/2} dt \rightarrow 0, \quad d\tilde r=\left[1-\frac{R_s}{r}\right]^{-1/2} dr \rightarrow \infty \text{ for } r \rightarrow R_s^-, \ \ \ \ \ (4)$

Both (3) and (4) show that time freezes at ${r=R_s}$, and there is no motion crossing the event horizon:

$\displaystyle \tau_1-\tau_2 =d\tau =0\quad \text{ implies } \quad d \tilde r = \tilde r (\tau_1) -\tilde r(\tau_2) =0.$

Consequently the black hole enclosed by the event horizon ${r=R_s}$ is closed: Nothing gets inside a black hole, and nothing gets out of the black hole either.

#### Black holes are filled

We now demonstrate that black holes are filled. Suppose there is a body of matter field with mass ${M}$ trapped inside a ball of radius ${R < R_s}$. Then on the vacuum region ${R< r < R_s}$, the Schwarzschild solution would be valid, which leads to non-physical imaginary time and nonphysical imaginary distance:

$\displaystyle d\tau = i \left|1-\frac{R_s}{r}\right|^{1/2} dt, \qquad d\tilde r= i \left|1-\frac{R_s}{r}\right|^{-1/2} dr \quad \text{ for } \quad R

Also, when ${R< R_s}$, the TOV metric is given by

$\displaystyle ds^2= -\left[ \frac32 \left(1-\frac{R_s}{R}\right)^{1/2} - \frac12 \left( 1- \frac{r^2 R_s}{R^3}\right)^{1/2} \right]^2 c^2dt^2$

$\displaystyle +\left(1-\frac{r^2R_s}{R^3}\right)^{-1}dr^2 +r^2d\theta^2+r^2\sin^2\theta d\varphi^2 \qquad \text{ for } r < R. \ \ \ \ \ (5)$

Then both time and radial distance would become imaginary near ${r=R}$, and this is clearly non-physical.

This observation clearly demonstrates that the black is filled. In fact, we have proved the following black hole theorem:

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 view on the structure and geometry of black holes than the classical theory of black holes.

## 3. Singularity at ${R_s}$ is physical

A basic mathematical requirement for a partial differential equation system on a Riemannian manifold to generate correct mathematical results is that the local coordinate system that is used to express the system must have no singularity.

The Schwarzschild solution is derived from the Einstein equations under the spherical coordinate system, which has no singularity for ${r>0}$. Consequently, the singularity of the Schwarzschild solution at ${r=R_s}$ must be intrinsic to the Einstein equations, and is not caused by the particular choice of the coordinate system. In other words, the singularity at ${r=R_s}$ is real and physical.

## 4. Mistakes of the classical view

Many writings on modern theory of black holes have taken a wrong viewpoint that the singularity at ${r=R_s}$ is the coordinate singularity, and is non-physical. This mistake can be viewed in the following two aspects:

A. Mathematically forbidden coordinate transformations are used. Classical transformations such as e.g. those by Eddington and Kruskal are singular, and therefore they are not valid for removing the singularity at the Schwarzschild radius. Consider for example, the Kruskal coordinates involving

$\displaystyle u= t-r_\ast, \quad v=t + r_\ast, \qquad r_\ast = r +R_s \ln \left(\frac{r}{R_s}-1\right).$

This coordinate transformation is singular at ${r=R_s}$, since ${r_\ast}$ becomes infinity when ${r=R_s}$.

It is mathematically clear that by using singular coordinate transformations, any singularity can be either removed or created at will.

In fact, many people did not realize that what is hidden in the wrong transformations is that all the deduced new coordinate systems, such as the Kruskal coordinates, are themselves singular at ${r=R_s}$:

all the coordinate systems, such as the Kruskal and Eddington-Finkelstein coordinates, that are derived by singular coordinate transformations, are singular and are mathematically forbidden.

B. Confirmation bias. Another likely reason for the perception that a black hole attracts everything nearby is the fixed thinking (confirmation bias) of Newtonian black hole picture. In their deep minds, people wanted to have the attraction, as produced by the Newtonian theory, and were trying to find the needed “proofs” for what they believe.

In summary, the classical theory of black holes is essentially the Newton theory of black holes. The correct theory, following the Einstein theory of relativity, is given in the black hole theorem above.

### Tian Ma & Shouhong Wang

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### 4 Responses to Singularity at the Black-Hole Horizon is Physical

1. hehuijing says:

Early in this year, American scientist claimed that they have discovered gravitation waves which was generated by the merge of two black holes. According to your new theory, what will happen when two black holes merge? Since nothing can penetrate the horizon of a black hole, the two black hole will move around each other? An interesting problem is to develop black-hole dynamics based on your new theory, which focuses on the interaction among black holes and stars.

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2. Ma-Wang says:

The discovery of gravitational waves provides a definitive support for the Einstein view on gravity being the space-time curvature effect. You are right, according to our theory, when two black holes collide, they will likely form a two-body system, spinning around each other. The assertion of two black hole merger is entirely based on empirical models under the assumption that two black holes can merger. However, the assumption on merger is not verified and cannot be derived from the Einstein equations. In other words, what is observed is only the space-time curvature effect of the gravity, but the merger of two black holes is derived based on empirical models and on the unverified assumption.

Indeed, black holes are essential for the formation and evolution of stars, and for supernovae explosion and AGN jets; see the last chapter of our book: Mathematical Principles of Theoretical Physics. Further investigation is certainly important and is very much needed.

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