What happens to the matter already at the very center of the star when it turns to a black hole? I was wondering whether when an object collapses into a black hole, the matter in the position $r=0$, instantly becomes part of the singularity, or does it take time to fall into the singularity, and is it as if the object is falling from any other location?
For example, if you had a star that was 30 times the mass of the sun, and you had some matter at exactly $r=0$, would this instantly become part of the singularity or would it take time for the singularity to form?
 A: According Roger Penrose's single paper that netted him a Nobel Prize, the singularity forms after the event horizon:

Any light cone that has no part pointing away from the singularity is inside the horizon.
A: This is just a long comment.
In all theories of physics, classical mechanics, classical electrodynamics, Newtonian gravity, all potentials lead to singularities, i.e. infinities in the value , since they are $1/r$ potentials and when $r=0$ the value becomes infinite. This has been solved by the quantum mechanical theories , where the positions are no longer calculable, only the probability of the particle's being at $r=0$ is calculable. As quantum mechanics has been validated , and as it can be shown mathematically that the quantum mechanical solution join smoothly with the classical solutions it is an accepted fact that wherever there are singularities in space, quantum mechanics has to be invoked.
This is true in the general relativity model of Big Bang, where effective quantization of gravity is utilized to change the original GR singularity into a spacetime region .

This awaits the theory of definite quantization of gravity within GR, but at the moment  the Big Bang is the mainstream cosmological model.
I expect , in the same way, the r=0 of the singularity defining a black hole will have to be smeared in a similar way, ending in a probabilistic location, so there is no sense in talking about a singularity at exactly $r=0$ and how it behaves.
A: In General Relativity the so-called singularity is not a part of space-time manifold but a lack of it. Consequently, nothing can fall there. It is a consensus among physicists that within framework of classical GR it is not possible to resolve this problem. But let us ask the question what is the meaning of $r=0$? The $r$ is so-called areal radius - a square root of sphere area $A$ divided by $4\pi$. Thus, $r=0$ means a sphere in the limit of zero surface area. It is known that $S^2$ sphere embedded in $R^3$ space degenerates to a point for $r=0$. But a sphere not embedded in any ambient space is not contractile to a point. It remains sphere (retains its topology) despite of zero surface area. Therefore, new insights can arise from definition of spherically symmetric spacetime as a sphere bundle over Lorentz two-manifold (r-t coordinates), as for example proposed by H. P. Künzle in the paper Construction of singularity-free spherically symmetric space-time manifolds, or as discussed in my question about Definition of static spherically symmetric spacetime as fiber bundle. In such spacetime there is no point at $r=0$. The black hole formation starts with the event horizon of zero surface area. Taking into account QM the initial event horizon would probably start with a sphere of Planck surface area.
A: NO, THERE IS NO MATTER INSIDE THE BLACK HOLE
Here is a proof based on existing discoveries of cosmology
To know and understand this black hole, scientists have been making many discoveries for almost a century. But till now no exact answer has been found. In contrast, many mysteries are getting deepened in the context of the universe and black holes. It is a fact of common sense that if an event occurs in the right sequence of A, B and C, then its reverse sequence will be C, B and A, apart from this there can be no other reverse sequence. But in the absence of correct information in the context of black holes, different types of events sequence are told by scientists for right and reverse sequence.
First let us understand the right event sequence. Scientists, through their discoveries, have precisely defined the right event sequence of the formation of the universe from a black hole, which is told in three stages (Chronology of the universe). During the first phase of Planck Epoch (0 to 10^-43 s), the Big Bang explosion occurs in the black hole and a unified Energy comes out of the black hole. Then during the second phase of Grand Unification Epoch (10^-43 s to 10^-36 s), three different types of Energies split from one unified Energy. In this way a total of four types of forces or energies are formed. Then in the third phase of Particle Epoch (10^-36 s to 370,000 years), these four types of forces form the atom particles and the matter of the universe. This right event sequence is perfect and correct. But scientists are not yet able to define the unified Energy consisting of all four forces of Planck Epoch, so this right event sequence is not being considered at all. In fact, it is not possible to define a black hole by science, because there is no matter inside a black hole. The main basis of science is atoms or matter, which are formed in the third phase of the Big Bang explosion, the Particle Epoch.
Due to this anomaly, the reverse event sequence of the formation of a black hole from the universe is told in a very strange way by scientific facts based on matter. While scientifically explaining the formation of a black hole from a galaxy in reverse event sequence, it is often said that space-time curves after the event horizon in a black hole. Due to infinite gravity, the density of the matter going into the black hole becomes infinite and after the event horizon, even light does not come out of it (Black hole). Firstly, the hypothesis of having a Space-Time Curve Fabric is just a theoretical concept, which is hypothesized to explain scientific theories. As we understood in the previous slide (video available separately), light moves in a Spiral Curve Line in the universe, whereas in the scientific concept we have assumed that light moves in a straight line. Hence there is an illusion of space and time being a curve. In reality nothing like the curve of space and time is possible. In the scientific explanation of black holes words like event or time, light, gravity, density, matter, etc., whose reference is possible only if matter exists. There is also no imaginary matter with infinite density inside the black hole, because the particles, atoms and matter are formed only during the third phase of particle epoch after inflation in the black hole.
Most scientists consider the black hole visible as an accretion disk to be a real black hole, which is not correct. What is visible through a telescope as an accretion disk in space is a virtual black hole. In fact, during the process of formation of a black hole, all the atoms of matter are fragmented by antimatter. Due to the fragmentation of atoms, the matter gradually vanishes and the pure Energy of the atoms gets collected at the center of the black hole. As long as the fission process of atoms continues, a virtual black hole appears as an accretion disk due to the light and Energy generated by the friction of remaining atoms and molecules . As soon as all the atoms of the matter are fragmented, the pure Energy of the entire galaxy is collected at the center of the black hole and becomes almost zero. When the matter is exhausted, the light and Energy of the accretion disk also ends and the black hole is not visible. When the entire pure Energy of the galaxy's life and matter converges at near-zero point in space, a real black hole is formed, which is impossible to see through any scientific instrument. A few years ago, scientists experienced something similar during the study of Galaxy Abell-2261 in 2018, when the supermassive blackhole Abell-2261, located 2.7 billion light years away, suddenly disappeared from its place (On the Hunt for a Missing Giant Black Hole). This may have been due to the formation of a real black hole, when all the matter and antimatter of the Galaxy is exhausted in the form of pure Energy to become a zero-sized real black hole. Scientists have proved through their discoveries that the black hole in the center of our galaxy, 13.8 billion years ago, in which the Big Bang explosion took place and our galaxy was formed, the actual size of that black hole was 10 to power -35 m. Simply think of it as the tip of a needle can hold trillions of real black holes.
Black holes are the biggest question mark for cosmologists at present. It is a matter of wonder and amazement as to what happens in this nearly zero-sized black hole singularity, from which such a massive galaxy is formed and then life begins on Earth. To understand black hole singularity, we have to keep in Mind this important scientific discovery that subatomic particles, atoms and matter are formed in the third phase of the Big Bang explosion, that is, the particle epoch. It is clear from this that there is no matter like atoms or matter inside the black hole or in the first phase of Planck Epoch or in second phase of Grand Unification Epoch. This proves that there are non-matter components inside a black hole. What are those non-matter components inside a black hole? Let us know here (Black Hole Singularity ~ What you are looking for is right under your nose).
Scope of the question forces me to limit the answer here. But my findings do not end here. My findings have revealed the secrets of the black hole, that what actually are there inside the black hole. I am the author of this research, if asked I may reveal the same.
