For Newtonian mechanics

The Schwarzschild radius is the radius of the event horizon of a black hole. Amazingly, we can compute it with Newtonian mechanics, which is explained below. Consider a big mass M which creates the gravitational acceleration a for a small mass m at the distance r from M, see figure 1 and equation (1) for gravitational acceleration a=GM/r2. For computing v the radial velocity of m in the gravitational field of M we integrate equation (1).

We compute for the case where m freefalls from infinitely far starting with zero velocity, see (7). With these conditions the radial velocity of m at the distance r₂ from M is computed in (8), v2 = 2GM/r2. Reversing (8), r₂ = 2GM/v2. The Schwarzschild radius of the event horizon of M is r_s such that the Schwarzschild factor equals infinity, see (10).

When v₂ equals the speed of light c, we apply v₂ = c into (9) and we obtain (11) where r₂ = r_s. So, r₂ equals r_s and the Schwarzschild radius is computed with Newtonian mechanics.

For relativity

Although the Schwarzschild radius r_s is a relativistic quantity, in the above it is derived completely with Newtonian mechanics, which is somewhat weird. What will be its value if we apply relativistic principle?

In the following derivation we will use the formula for relativistic transformation of acceleration which is derived in the paper « Relativistic kinematics » linked here: https://www.academia.edu/44582027/Relativistic_kinematics

The formula is the equation (18) of the paper.

Here, this formula is given by (12) in which the gravitational acceleration of m is a and the acceleration in space is a_r and the radial velocity is computed by integration. With the same conditions as (7), the constant of integration k equals 0, see (16). Then, using k = 0 in (15), v is expressed with r in (17).

In the case where the small mass m approaches M, the distance r approaches 0, the radial velocity of m approaches the speed of light c, see (18). So, v the radial velocity of m does not become bigger than the speed of light c for r > 0. The Schwarzschild radius r_s is the radius such that v = c. So, r_s = r = 0, see (19).

Conclusion

When the relativistic principles are applied correctly, the Schwarzschild radius r_s equals zero. The gravitational force on m approaches infinity near M. But the speed of m never reaches the speed of light c, which is true for any force however strong it is and for time of acceleration however long it is. The Schwarzschild radius r_s must obey relativistic principle and is shown to be zero.

That the Schwarzschild radius r_s equals zero means that the geometrical size of a black hole should be zero and thus, a black hole should not have an interior. I have reached this conclusion and explained it in the paper « Gravitational time dilation and black hole » in which I have also shown that point masses could not coalesce to form a black hole whose volume is zero and that observations support this conclusion. This paper is linked here:

https://www.academia.edu/45434676/Gravitational_time_dilation_and_black_hole

The equations and figure are in the paper below

https://www.academia.edu/84798805/Radius_of_a_black_hole_for_relativity_and_Newtonian_mechanics

For more detailed information, I invite you to read the two cited papers:

« Gravitational time dilation and black hole » https://www.academia.edu/45434676/Gravitational_time_dilation_and_black_hole

« Relativistic kinematics » https://www.academia.edu/44582027/Relativistic_kinematics