All fundamental electrically charged thin shells in general relativity: From star shells to tension shell black holes, regular black holes, and beyond

We classify all fundamental electrically charged thin shells in general relativity, i.e., static spherically symmetric perfect fluid thin shells with a Minkowski spacetime interior and a Reissner-Nordström spacetime exterior, characterized by the spacetime mass $M$, which we assume positive, and the electric charge $Q$, which without loss of generality in our analysis can always be assumed as being the modulus of the electric charge, be it positive or negative. The fundamental shell can exist in three states, namely, nonextremal when $\frac{Q}{M}<1$, which includes the Schwarzschild $\frac{Q}{M}=0$ state, extremal when $\frac{Q}{M}=1$, and overcharged when $\frac{Q}{M}>1$. The nonextremal state, $\frac{Q}{M}<1$, allows the shell to be located in such a way that the radius $R$ of the shell can be outside its own gravitational radius $r_{+}$, i.e., $R>r_{+}$, where $r_{+}$ is given in terms of $M$ and $Q$ by $r_{+}=M+\sqrt{M^2-Q^2}$, or can be inside its own Cauchy radius $r_{-}$, i.e., $R<r_{-}$, where $r_{-}$ is given in terms of $M$ and $Q$ by $r_{-}=M-\sqrt{M^2-Q^2}$. The extremal state, $\frac{Q}{M}=1$, allows the shell to be located in such a way that the radius $R$ of the shell can be outside its own gravitational radius $r_{+}$, i.e., $R>r_{+}$, where now $r_{+}=r_{-}$, or can be inside its own gravitational radius, i.e., $R<r_{+}$, or can be at its own gravitational radius $r_{+}$, i.e., $R=r_{+}$. The overcharged state, $\frac{Q}{M}>1$, allows the shell to be located anywhere $R\ge 0$. There is yet a further division; indeed, one has still to specify the orientation of the shell, i.e., whether the normal out of the shell points toward increasing radii or toward decreasing radii. For the shell’s orientation, the analysis in the nonextremal state is readily performed using Kruskal-Szekeres coordinates, whereas in the extremal and overcharged states the analysis can be performed in the usual spherical coordinates. There is still a subdivision in the extremal state $r_{+}=r_{-}$ when the shell is at $r_{+}$, $R=r_{+}$, in that the shell can approach $r_{+}$ from above or approach $r_{+}$ from below. The shell is assumed to be composed of an electrically charged perfect fluid characterized by the energy density, pressure, and electric charge density, for which an analysis of the energy conditions, null, weak, dominant, and strong, is performed. In addition, the shell spacetime has a corresponding Carter-Penrose diagram that can be built out of the diagrams for Minkowski and Reissner-Nordström spacetimes. Combining these two characterizations, specifically, the physical properties and the Carter-Penrose diagrams, one finds that there are fourteen cases that comprise a bewildering variety of shell spacetimes, namely, nonextremal star shells, nonextremal tension shell black holes, nonextremal tension shell regular and nonregular black holes, nonextremal compact shell naked singularities, Majumdar-Papapetrou star shells, extremal tension shell singularities, extremal tension shell regular and nonregular black holes, Majumdar-Papapetrou compact shell naked singularities, Majumdar-Papapetrou shell quasiblack holes, extremal null shell quasinonblack holes, extremal null shell singularities, Majumdar-Papapetrou null shell singularities, overcharged star shells, and overcharged compact shell naked singularities.