Compact anisotropic stars with membrane - a new class of exact solutions to the Einstein field equations

TL;DR

This paper introduces a new class of exact solutions to Einstein's equations describing compact, anisotropic stars with a membrane boundary, expanding the understanding of static, charged, self-gravitating objects in general relativity.

Contribution

It presents a novel set of solutions featuring a membrane boundary and anisotropic interior, unifying and extending classical solutions like Schwarzschild and Reissner-Nordström.

Findings

Solutions include a membrane with surface tension at the boundary.

Interior matter exhibits anisotropic pressure.

One solution links membrane stress-energy to the gravitating mass.

Abstract

A new class of solutions to Einstein's classical field equations of general relativity is presented. The solutions describe a non-rotating, spherically symmetric, compact self gravitating object, residing in a static electro-vacuum space time. The solutions generally have an interior non-zero matter-distribution. A wide class of interior solutions can be constructed for any specific set of exterior parameters (mass, charge). The original Schwarzschild and Reissner-Nordstroem solutions constitute special cases within the variety of interior solutions. An outstanding feature of the new solutions is a non-continuous boundary of the matter-distribution, accompanied by a two dimensional membrane at the boundary. The membrane consists of an infinitesimally thin spherical shell of tangential pressure (surface tension/stress). The interior matter state generally has a locally anisotropic pressure. A general procedure for generating the new solutions is given. A few solutions are derived and discussed briefly. In order to identify the physically most promising solutions, a selection principle is formulated, based on the holographic principle. One solution of particular interest emerges. It is characterized by the property, that the "stress-energy content" of the membrane is equal to the gravitating mass of the object.