# Schwarzschild Naked Singularity Creation

In the Schwarzschild metrics (Plank units): $$ds^2=-\left(1-\frac{2m}{r}\right)dt^2+\left(1-\frac{2m}{r}\right)^{-1}dr^2+r^2\left(d\theta^2+\sin^{2}\theta d\phi^{2}\right),$$ we can have a Black Hole (from here SBH) if $$m>0$$ or a Naked Singularity (from here SNS) if $$m<0$$, from definition a Naked Singularity is a curvature singularity without any event horizon.

Denoting with $$A$$ the area of the event horizon, the 2nd Law of Black Hole Mechanics states that any process involving a Black Hole must have $$\delta A>0$$, this is implies that a SBH cannot bifurcate in two SBH.

I did a bit of math and theoretically a SBH can bifurcate in a NSN plus a SBH, it seems a bit strange so I think that there is an error but I can't found where, here is the derivation:

Starting with a SBH with $$A_3=4\pi(2 M_3)^2$$ it can bifurcate in a NSN with mass $$-|M_1|$$ (it has area $$A_1=0$$) plus a SBH with $$A_2=4\pi(2 M_2)^2$$.

Has stated we must have $$\delta A\geq 0$$, i.e: $$A_{3}\leq A_2+A_1=A_2$$ or using the equation for the areas: $$(M_3)^2\leq(M_2)^2$$.

Energy also has to be conserved so $$M_3=-|M_1|+M_2+E_{\text{rad}}$$, and from $$M_2 \geq M_3$$ it becomes: $$M_2\geq-|M_1|+M_2+E_{\text{rad}}$$ i.e. the only real restriction is: $$|M_1| \geq E_{\text{rad}}$$.

So it seems that a SBH should naturally radiates gravitational energy and "emitting" SNS until it eventually expires.