Klein-Gordon Geon

TLDR

Klein‑Gordon geons, spherically symmetric solutions of the Klein‑Gordon Einstein equations, possess unique properties distinct from other gravitating systems, with equilibrium states analogous yet thermodynamically unstable because adiabatic perturbations are forbidden. The instability analysis shows that the equations of state are differential, and when stability is examined through infinitesimal field perturbations, Klein‑Gordon geons resist spherically symmetric gravitational collapse. These results demonstrate that the conventional equation‑of‑state concept fails for Klein‑Gordon geons and that they serve as counterexamples to the conjecture that gravitational collapse is inevitable.

Abstract

A study of the spherically symmetric eigenstates of the Klein-Gordon Einstein equations (Klein-Gordon geons) reveals that these geons have properties that are uniquely different from other gravitating systems that have been studied. The equilibrium states of these geons seem analogous to other gravitating systems; but when the question of stability is considered from a thermodynamical viewpoint, it is shown that, in contrast with other systems, adiabatic perturbations are forbidden. The reason is that the equations of state for the thermodynamical variables are not algebraic equations, but instead are differential equations. Consequently, the usual concept of an equation of state breaks down when Klein-Gordon geons are considered. When the question of stability is reconsidered in terms of infinitesimal perturbations of the basic fields, it is then found that Klein-Gordon geons will not undergo spherically symmetric gravitational collapse. Thus, Klein-Gordon geons are counterexamples to the conjecture that gravitational collapse is inevitable.