The static free-field equations in isotropic form: The Riemann tensor and Kretschmann scalar

We revisit the known free-field spacetime metrics with spherical symmetry in their isotropic form, using both Cartesian (x, y, z) and spherical coordinates (r, θ, ϕ). We analytically obtain the Riemann tensor in Cartesian coordinates and derive the corresponding Kretschmann scalar, K, which is uniquely defined by the three spatial coordinates. Its evaluation along radial directions reveals a non-trivial behavior: K(r) exhibits no singularity and vanishes as r → 0, K(r) ∼ r ⁶ /r ₀ ¹⁰ , in all cases considered, where r ₀ = r _s /4 and rs is the Schwarzschild radius. Furthermore, K(r) reaches its maximum along coordinate axes, but vanishes exactly along the main Cartesian diagonals, K _diag (r) = 0, with x ² = y ² = z ² = r ² /3, results previously unreported in the literature. The concluding remarks discuss possible applications, given the significance of free-field solutions in various physical scenarios.