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.


Rather than being some fundamental statement about particles and horizons the quoted passage simply is a comment about the failure of approximation referred to in a previous paragraph to include general relativistic effects.

From the previous paragraph:

In analogy with the hydrogen atom, one would expect that the Klein-Gordon equation would possess bound states in the presence of a point mass. At least this is so in the Newtonian limit.

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.

Instead static bound states must be found by solving KG-Einstein equations without horizons, and as a result the Newtonian analogy of a hydrogen atom with a point mass at the center and scalar particles around it is no longer applicable.


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