Radius of a black hole for relativity and Newtonian mechanics

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 rs 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 = rs. So, r2 equals rs and the Schwarzschild radius is computed with Newtonian mechanics.

For relativity

Although the Schwarzschild radius rs 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 ar 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 rs is the radius such that v = c. So, rs = r = 0, see (19).


When the relativistic principles are applied correctly, the Schwarzschild radius rs 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 rs must obey relativistic principle and is shown to be zero.

That the Schwarzschild radius rs 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:


The equations and figure are in the paper below


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

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