# 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 r_{2} from M is computed in (8), v2 = 2GM/r2. Reversing (8), r_{2} = 2GM/v2. The Schwarzschild radius of
the event horizon of M is r_{s} such that the Schwarzschild factor equals
infinity, see (10).

When v_{2} equals the speed of light c, we apply v_{2}
= c into (9) and we obtain (11) where r_{2} = r_{s}. So, r_{2}
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