Static Spherically Symmetric Exact Solutions of Einstein’s Field Equations with Equal Temporal and Radial Metric Factors

We investigate a class of static, spherically symmetric exact solutions of Einstein's field equations in which the temporal and radial metric coefficients are taken to be identical rather than reciprocal. Specifically, we study line elements of the form $ds^2=-f(r)c^2dt^2+f(r)\,dr^2+r^2d\Omega^2$, with special attention to the two cases $f_1(r)=1-\frac{2m}{r}, \quad f_2(r)=\left(1-\frac{m}{r}\right)^2, \quad m=\frac{GM}{c^2}.$ Both cases are derived directly from Einstein's field equations by determining the stress-energy tensor required to support the geometry. We show that neither metric is a vacuum solution; instead, each is sustained by a nonvanishing anisotropic stress-energy tensor with positive density, positive radial pressure, and negative tangential pressure in the corresponding exterior region. The solutions are compared with the Schwarzschild metric, the extremal Reissner--Nordstr\"om geometry, and the minimal Haug--Spavieri metric. Whereas Schwarzschild and extremal Reissner--Nordstr\"om retain the familiar reciprocal relation between temporal and radial coefficients, the present solutions demonstrate that static spherical spacetimes with equal temporal and radial factors are also admitted by general relativity once anisotropic sources are allowed. The relation of the second case to the Haug--Spavieri physical interpretation is also discussed.