Gravitational field of a pit and maximal mass defects

Abstract

A general relativistic solution, composed of a Zel'dovich-Letelier interior made of radial strings matched through a spherical thin shell at radius $r_0$ to an exterior Schwarzschild solution with mass $m$, is presented. It is the Zel'dovich-Letelier-Schwarzschild star. When the radius $r_0$ of the star is shrunk to its gravitational radius $2m$, the solutions have interesting properties. There are solutions with $m=0$ and $r_0=0$ that obey $\frac{2m}{r_0}=1$. The solutions have a horizon, but are not black holes, they are quasiblack holes, though atypical. The proper mass $m_p$ of the interior is nonzero and made of one string. Hence, a Minkowski exterior space hides an interior with matter in a pit. These are the pit solutions and show a maximal mass defect. There are two classes of pit solutions, one a finite string and the other a semi-infinite one. These pits are really string pits, that can be seen as Wheeler bags of gold, albeit squashed bags. There is another class, which is a compact stringy star at the $\frac{2m}{r_0}=1$ limit with $m$ nonzero. It is a typical quasiblack hole and it has maximal mass defect. Generically pit solutions with $\frac{2m}{r_0}=1$ and $m=0$ can exist with maximal mass defects. The Zel'dovich-Letelier-Schwarzschild star at the $r_0=2m$ limit is an instance of it. These three classes of static solutions yield the same spectrum of solutions that appear in critical gravitational collapse, there are solutions that yield naked null singularities, which here are the two string pit classes, there are solutions that yield black holes, which here are represented by the compact stringy stars at the quasiblack hole limit, and the solutions that disperse away in critical collapse here are the static Zel'dovich-Letelier-Schwarzschild stars. Thermodynamics of the string pit and stringy star quasiblack hole solutions is provided, and other connections are mentioned.