While the standard Schwarzschild metric is overwhelmingly employed in general relativity (GR) as the starting point for various spherical spacetime metric calculations, its isotropic (ISO) form is mentioned in more specialized contexts and its derivation is barely discussed in published GR literature. In this work, we review the isotropic metric, stressing that it stands out as a useful spherically symmetric metric to be employed also in traditional GR problems. We start by deriving the ISO metric through solving the vacuum field equations in Cartesian coordinates, thereby obtaining the Ricci tensor also in spherical coordinates. We then analytically calculate the Riemann tensor in Cartesian coordinates, proving its consistency with the Ricci tensor calculation for pedagogical reasons. Finally, from the Riemann tensor we exactly evaluate the Kretschmann scalar, which lacks metric singularities, a result consistent with the known singular behavior of the standard Schwarzschild metric. We conclude that the isotropic metric naturally emerges as a suitable candidate for modeling static neutron stars and regular black holes, thereby complementing the present attempts to understand these rapidly evolving research fields.
Hector Eduardo Roman (Sun,) studied this question.