General-Relativistic Interior Metric for a Stable Static Charged Matter Fluid with Largeem

Abstract

The general-relativistic field equations are solved for a spherically symmetric static charged matter fluid in isotropic coordinates, and matched at the boundary $R$ to the external Reissner-Nordstr\"om metric. No pressure, i.e., stress of undefined origin, is involved. There turns out to be one more unknown than equation, but consistency and continuity conditions severely restrict the solutions. It is found that the charged fluid can be in equilibrium even with a large $\frac{e}{m}$, such as for an electron considered as a fluid, but only if (1) there is a singularity in ${g}_{44}$ at $r=0$, (2) the matter density ${\ensuremath{\rho}}_{m}$ in the energy-momentum tensor is negative, and (3) the ratio $\frac{{\ensuremath{\rho}}_{e}}{{\ensuremath{\rho}}_{m}}$ of charge to matter density is a variable function. The energy density (as opposed to the matter density) can be made everywhere positive in the coordinates used. A simple picture of how the forces balance can be made.