Lie Group Analysis of the General Karmarkar Condition

The Karmarkar embedding condition in different spherically symmetrical metrics is studied in general using Lie symmetries. In this study, the Lie symmetries for conformally flat and shear-free metrics are studied which extend recent results. The Lie symmetries for geodesic metrics and general spherical spacetimes are also obtained for the first time. In all cases group invariant exact solutions to the Karmarkar embedding condition are obtained via a Lie group analysis. It is further demonstrated that the Karmarkar condition can be used to produce a model with interesting features: an embeddable relativistic radiating star with a barotropic equation of state via Lie symmetries.