This question is not what only I am questioning. Searching WEB, you can find some similar opinions. But, because the dominant view seems to take it granted that black holes do exist, So, I try to make the point of discussion clearer.

#### !! CAUTION !! This is “MY QUESTION”. This can be in discord with standard view.

The simplest solution of the Einstein field equations is called Schwarzschild solution, which represents spacetime around a non-spinning gravity source M.

(1)

Where,

(2)

(3)

a: Schwarzschild radius c: speed of light G: gravitational constant M: mass of the gravity source

Deforming eq.(1), relationship between coordinate time dt and proper time dτ can be derived.

(4)

Coordinate time dt is the elapsed time for an Lagrangian observer who stays very far from the gravity source ( ≈ us). Proper time dτ is the elapsed time for an Eulerian observer who is free-falling at the distance r from the gravity source.

According to Wikipedia, a black hole is

a region of spacetime from which gravity prevents anything, including light, from escaping.

Usually, when we talk about it, only escaping is focused on. But here, we focus on falling into it. Eq.(2) shows that at the surface of a black hole ≡ event horizon (r=a) γ becomes 0. With eq.(4), this means that regardless of the elapsed coordinate time dt, proper time dτ does not proceed at all and falling movement freezes observed from us. Thus, I believe, for us, any matter including light have not been able to pass the event horizon to get into the black hole so far and will not be able to do that forever.

In case that there have been black holes from the beginning of the universe and have been gathering huge amount of matter by their gravity, any matter, I think, could have got into the black hole in the strict meaning.

In case that there were not a black hole at the beginning of the universe, I think real black hole will never come into existence, because all terms in eq.(4) are positive because they are squared, elapsed proper time dτ of the falling objects（dr, dθ, dφ ≠ 0） must be smaller than that of stationary objects（dr = dθ = dφ = 0). This is not a strictly mathematical proof, but I suppose that if a celestial body whose density is near that of black hole is formed by gravitational contraction and huge amount of matter keep accreting onto it, time freezes at the brink of forming the event horizon and it will never be formed observed by us.

The same Wikipedia tells that

At the center of a black hole as described by general relativity lies a gravitational singularity, a region where the spacetime curvature becomes infinite.

However, if there was a black hole or a celestial body whose density is near that of a black hole, progress of proper time dτ inside it must be slower than at the surface of it because gravitational potential is lower. Therefore, if a gravitational collapse takes place, it must be observed as freeze-frame for us. I suppose singularity will never be formed no matter how long we wait.

By the way, there are scientists who invented “Kruskal–Szekeres coordinates“. According to this theory, these coordinates

are well-behaved everywhere outside the physical singularity.

- and non-singular at the event horizon.

It is true that object falling to a black hole can reach the center of it within a finite proper time. But at the same time, we must note that progress of the proper time of the object must be observed frozen from us. In my opinion, from the very beginning of the universe till now and forever, nothing has passed the event horizon. So, although this contradicts common view, I think it is only event horizon that is singular and has important physical meaning.

Then, what happens when a huge mass spherical cloud contract by the gravity? For this question, I developed a exact dynamic simulation program of general relativity and carried out numerical simulations. The result is shown below.

According to the simulation result, when the spherically homogeneously distributed non-interacting stationary particle cloud (dust ball) contracts to its last stage in which a black hole is about to be formed, time progress slowed rapidly, and observed from us, falling motion is almost stopped in the air.

To be continued to Simple questions “Is there a singular point at the center of a black hole?”