Inside Black Hole
Core Does Not Slow Down Spinning
In New Explanation of Black Hole Images, Without Holes in Spacetime we came out with a formula of Black Hole’s radius R=4/3×G×M/c² (where G is the gravitational constant, M — the Black Hole’s mass, c — the speed of light). Beyond this radius, significant variations in angular velocities (or in periods of rotation around the Black Hole) of accretion disk gas and dust cause friction among layers of particles, high temperatures and thermal radiation. Now, let’s discuss what is going on inside Black Hole. Graph from the previous article shows:
When we go from R=4/3×G×M/c² in the left direction to R=0, the period of rotation significantly increases up to infinity, meaning the core of the Black Hole stops. Then friction inside the Black Hole will slow down and eventually stop the whole Black Hole. It turns out, that left side of the graph is incorrect, because attracting mass inside the Black Hole is not constant anymore. By Newton’s Shell Theorem, gravitational pull inside a massive hollow sphere (which is in light grey on the drawing below) is 0. For a particle inside the Black Hole at the distance r from the center of Black Hole, outer layer (in light grey) does not attract this particle anymore (by Shell theorem), only dark grey ball part continues attracting the particle. Meaning, this particle is not attracted by the whole mass M=M(R) of the whole Black Hole, but only by mass M(r) of the dark grey part of the Black Hole:
Thus, in the formula for period of rotation
period²(R) = 4π²R³×exp[4G×M/(R×c²)]/(G×M)
instead of constant mass M, we should use variable mass M(r) = 4/3×π×r³×d(r), where d(r) is an average density of the ball of radius r. Density of the ball grows from an average density of the whole Black Hole d(4/3×G×M/c²) to a highest density d(0) at its center. Thus,
period²(r) = 4π²r³×exp[4G×M(r)/(r×c²)]/(G×M(r)) =
4π²r³/(G×4/3×π×r³×d(r)) × exp[4G×4/3×π×r³×d(r)/(r×c²)] =
3π/(G×d(r)) × exp[16/3×π×G×r²×d(r)/c²]
When r — >0, exp[16/3×π×G×r²×d(r)/c²] —> exp(0) = 1, thus, for r — >0
period²(r) ~ 3π/(G×d(r)) ~ 1/d(r)
period(r) ~ 1/sqrt(d(r))
“~” means some numeric constants are dropped for brevity, “sqrt” is square root.
Now we see that period(r) does not grow to infinity when r — >0 (and core does not slow down to a full stop), but on the contrary, because density slowly grows to d(0), we have 1/sqrt(d(r)) going down to 1/sqrt(d(0)):
When r — >0, period(r) goes down to ~1/sqrt(d(0)), and angular velocity ~sqrt(d(r)), which is inverse to the period, goes up to ~sqrt(d(0)).
For slowly changing density, angular velocity changes slowly too, that minimizes friction between concentric layers. Because of friction inside Black Hole, angular velocity inside the core is basically constant (vs. slowly increasing to some finite value at the center).
That solves the puzzle with the left side of the chart about Black Hole core slowing and stopping.
