The dynamical stability of the static real scalar field solutions to the Einstein-Klein-Gordon equations revisited

Abstract

We re-examine the dynamical stability of the nakedly singular, static, spherically symmetric solutions of the Einstein-Klein Gordon system. We correct an earlier proof of the instability of these solutions, and demonstrate that there are solutions to the massive Klein-Gordon system that are perturbatively stable.

Qualitative and quantitative features of orbits of massive particles and photons moving in wyman geometry

“…In fact, the above results confirmed early studies of D. Christodoulou who pioneered analytical studies of that model [15]. The Wyman's solution is not usually thought to be of great importance for the issue of gravitational collapse because it is static and the naked singularities derived from it are unstable against spherically symmetric linear perturbations of the system [9,10]. On the other hand, as we saw above, from a particular case of the Wyman's solution one may derive the Roberts' one which is of great importance for the issue of gravitational collapse.…”

“…(9)(10)(11), even the numerical solutions are very complicated. On the other hand, one may restrict his attention to the case of radial timelike geodesics, where the massive particle moves only along the radial and time directions.…”

“…The Wyman geometry describes the four dimensional space-time generated by a minimally coupled, spherically symmetric, static, massless scalar field and has been studied by many authors [6][7][8][9][10][11]. From a particular case of the general Wyman's solution, M. D. Roberts showed how to construct a time dependent solution [8].…”

“…On the other hand, as we saw above, from a particular case of the Wyman's solution one may derive the Roberts' one which is of great importance for the issue of gravitational collapse. Also, it was shown that there are nakedly singular solutions to the static, massive scalar field equations which are stable against spherically symmetric linear perturbations [10]. Therefore, we think it is of great importance to gather as much information as we can about the Wyman's solution for they may be helpful for a better understand of the scalar field collapse.…”

Axially symmetric Proca-Higgs boson stars

“…In turn, configurations supported only by a real scalar field are also dynamically unstable, whether they consist of ordinary [39] or ghost [42] scalar field (see, however, Ref. [43] where the stability analysis for the ordinary scalar field was revisited and the possibility of the existence of stable solutions was demonstrated). In this connection, one may naively expect that the mixed Proca-Higgs systems considered here will also be unstable.…”