Schwarzschild Naked Singularity Creation

Question

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.

Answer 1

If there were mechanisms for the production of negative-mass particles, then we would have instability of all forms of matter, not just instability of a positive-mass Schwarzschild black hole. Inconsistency with the laws of black hole thermodynamics would prove that such a process cannot exist, but consistency with them does not prove that it can.

If a positive-mass Schwarzschild black hole could decay into a positive-mass hole and a negative-mass one, then the two products would repel one another gravitationally and fly apart. Once they had been widely separated, each would constitute an asymptotically flat spacetime. But this would violate the positive energy theorem (assuming the dominant energy condition). Therefore the positive-mass Schwarzschild spacetime is not unstable with respect to this type of decay.

(The positive-mass Schwarzschild spacetime has not been proved to be stable, but is known to be linearly stable. The negative-mass solution is known to be unstable.)