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

TL;DR

This paper explores the geometric constraints of Karmarkar's condition and conformal flatness in static, spherically symmetric spacetimes, revealing that these conditions lead to known solutions like Schwarzschild interior and de Sitter, but also limit their physical applicability.

Contribution

It demonstrates that Karmarkar's condition combined with conformal flatness uniquely determines certain solutions, questioning its suitability for realistic stellar models without additional factors.

Findings

Schwarzschild interior and de Sitter solutions follow from the constraints

Pressure anisotropy and complexity factor vanish under these conditions

Karmarkar's condition alone may not suffice for realistic models

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.