# Event horizon and the existence of point particles

In this paper by David Kuap that first introduced the concept of Boson stars, he states that when the Einstein-Klein-Gordon system of equations is solved, the solutions obtained do not account for an event horizon.

Furthemore, he states that

[...] if the event horizon does not occur, then we can no longer interpret the system as containing a point particle.

I am not able to find any relation between the existence of an event horizon and the existence of point particles in the system. I would be glad if anyone could indicate any source that would make this issue more clear.

In the nonrelativistic limit the bound state of the system of scalar particles interacting only through gravity with a point mass at the center could be described by solving Schrödinger equation with the potential $$V=G mM / r$$. If we want a similar description in general relativity the first guess would be to replace Schrödinger equation with the KG equation and Newtonian potential with the Schwarzschild metric. But this approximation would fail, because bound solutions of KG equation in the presence of the horizon would have complex energies, indicating that they would be decaying with time as particles have nonzero probability to “fall into” the black hole.