Solution generating algorithms in general relativity.
We conduct a comprehensive investigative review of solution generating algorithms
for the Einstein field equations governing the gravitational behaviour of an isolated
neutral static spherical distribution of perfect fluid matter. Traditionally, the master
field equation generated from the condition of pressure isotropy has been interpreted
as a second order ordinary differential equation. However, since the pioneering work
of Wyman (1949) it was observed that more success can be enjoyed by regarding
the equation as a first order linear differential equation. There was a resurgence
of the ideas of Wyman in 2000 and various researchers have been able to generate
complete solutions to the field equations up to certain integrations. These have
been accomplished by working in Schwarzschild (curvature) coordinates, isotropic
coordinates, area coordinates and a coordinate system written in terms of the redshift
parameter. We have utilised Durgapal–Banerjee (1983) coordinates and produced a
new algorithm. The algorithm is used to generate new classes of perfect fluid solutions
as well as to regain familiar particular solutions reported in the literature. We find
that our solution is well behaved according to elementary physical requirements.
The pressure vanishes for a certain radius and this establishes the boundary of the
distribution. Additionally the pressure and energy density are both positive inside
the radius. The energy conditions are shown to be satisfied and it is particularly
pleasing to have the causality criterion satisfied to ensure that the speed of light is
not exceeded by the speed of sound. We also report some new solutions using the
algorithms proposed by Lake (2006).