Generating the Tolman V (n = 2) solution from the Minkowski one for static spherically symmetric perfect fluid in general relativity

Abstract

A static perfect fluid sphere is one of the exact solutions to the Einstein’s equation. To solve for perfect fluid solutions, a spherical symmetry has to be added to matters in order to reduce the complexity of the Einstein’s equation and an equation of state, which is one relating the pressure to the density of a perfect fluid sphere, has to be chosen. Selecting new equations of state, a number of perfect fluid solutions have been discovered. When several perfect fluid solutions have been found, it is more difficult to obtain new exact solutions by directly solving the Einstein’s equation than before. In this paper, we make use of the property of the Riccati equation to generate new exact static perfect fluid solutions from known ones without directly solving the Einstein’s equation. The result shows that if we start with the Minkowski solution, the new exact solution is the Tolman V (n = 2) which has physical meaning. This means that a pressure and density of a perfect fluid sphere are always positive. Furthermore, the pressure decreases from a central value to zero at the boundary of the perfect fluid sphere and the density also decreases from a central value to a positive value at the boundary of the perfect fluid sphere.