The beauty of simplicity: conservation of angular momentum and the birth of a black hole

Let us model a (relatively) massive, wild event by making use of plain rotational mechanics and one of the most fundamental constraints of the behavior of the physical (human-comprehensible) world: the change in angular speed of a stellar core collapsing to form a black hole.

Let \Gamma, L, \omega and I be the magnitudes of: total external torque of the system, the net angular momentum, the angular velocity and the moment of inertia with respect to the centre of mass.

As a matter of fact, the angular momentum of a body is constant -a conserved quantity- if \Gamma=0 (no force is causing rotation about the centre of mass: rotational equilibrium). This, since the net torque (rotational effect) is nothing more than the rate of change of the body’s angular momentum (with respect to time).

Assuming that our collapsing core of mass M and radius R is a (densely) uniform sphere and that neither mass nor angular momentum is lost, we can easily find the factor at which the angular speed of the object increases.

First, we calculate the moment of inertia of a solid, uniform (distribution of mass) spherical object, obtaining I= 2\5 MR^{2}

Now, by the conservation of angular momentum, the initial moment of inertia multiplied by the initial angular speed is equivalent to the product of the final moment of inertia and the final angular speed, i.e.:

I_i \omega_i = I_f \omega_f <=>

2\5 MR^{2}_i \omega_i = 2\5 MR^{2}_f \omega_f <=>

\omega_f = \frac{R^{2}_i \omega_i}{R^{2}_f}

Let’s put some numerical values. The radius of our Sun is of approx. 7.0x10^5 km, a massive star has a radius of about 20 times the radius of our Sun. Thus, let R_i = 1.4x10^7 km and R_f ‎ = 1 km, i.e.: the process of collapse compresses the core radius until it reaches one km.

Hence, \omega_f = (1.4x10^7)^2 \omega_i, which means that the final angular speed is almost 2.0 x10^{14} times the initial angular speed. And, a black hole has just emerged.