«Radius of a black hole for relativity and Newtonian mechanics»

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PengKuan Em

Aug 16, 2022, 7:54:36 AMAug 16
I have studied black hole and found that Newtonian mechanics can compute Schwarzschild radius fairly easily. Below is a short explanation how Newtonian mechanics and relativity work with black hole.

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  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. The Schwarzschild radius of the event horizon of M is rs such that the Schwarzschild factor equals infinity. When v2 equals the speed of light c, rs= infinity and Newtonian mechanics gives rs = 2GM/c2.

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 in the paper at the end. The formula is the equation (18) of the paper.

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).

The equations and figure are in the paper below
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