On the Limitations of Karmarkar’s Condition in Static, Conformally Flat Space-Times

Abstract For a static and spherically symmetric space-time, we investigate a class of exact solutions that arise when two fundamental geometric constraints are imposed simultaneously: Karmarkar’s condition and vanishing of the Weyl tensor. These conditions restrict the curvature in such a way that the space-time becomes conformally flat and belongs to the family of embedding class-I solutions. Even though the subsequent solutions, namely, the Schwarzschild interior solution and the de Sitter solution are well known, the novelty of our presentation is that these solutions are shown to be a direct consequence of the imposed geometric constraints. The physical matter composition becomes highly constrained by the associated geometry under such conditions. The Schwarzschild interior solution describes space-time of an incompressible fluid sphere, while the de Sitter solution corresponds to a vacuum energy dominated configuration. Interestingly, pressure anisotropy as well as the ‘‘complexity factor’’ vanish identically once Karmarkar’s condition and the conformal flatness conditions are applied simultaneously. As these two geometric constraints alone are sufficient to determine the background space-time uniquely, application of Karmarkar’s condition might not be a suitable method for the development of realistic stellar models in conformally flat space-time unless one invokes other factors into consideration, such as time-dependent metric potentials.