On the Limitations of Karmarkar's Condition in Static, Conformally Flat Spacetimes

TL;DR

The paper shows that imposing both the Karmarkar embedding-class I condition and conformal flatness ($C_{abcd}=0$) on static, spherically symmetric spacetimes tightly restricts the spacetime geometry, reducing the metric potentials to determine the solution uniquely. This geometric pairing yields exactly two nontrivial conformally flat solutions: the Schwarzschild interior solution for a homogeneous fluid and the de Sitter spacetime, both of which are isotropic with vanishing anisotropy and complexity. The results highlight that these constraints alone do not support richer stellar models without introducing additional degrees of freedom (e.g., time-dependent potentials or relaxing conformal flatness). The work clarifies the distinct roles of isotropy and conformal flatness and points to relaxing the geometric constraints or extending analyses (e.g., via Lie-group methods) to pursue more general, physically viable configurations.

Abstract

For a static and spherically symmetric spacetime, we investigate the class of exact solutions that arise when two fundamental geometric constraints are imposed simultaneously: the Karmarkar's condition and the vanishing of the Weyl tensor. These conditions restrict the curvature in such a way that the spacetime 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 the spacetime of an incompressible fluid sphere while the de Sitter solution corresponds to a vacuum energy dominated configuration. Interestingly, pressure anisotropy as well as `complexity factor' vanish identically once the Karmarkar's condition and the conformal flatness conditions are applied simultaneously. As these two geometric constraints alone are sufficient to determine the background spacetime uniquely, Karmarkar's condition might not be a suitable method for the development of realistic stellar models in a conformally flat spacetime unless one invokes other factors into consideration such as time-dependent metric potentials.