From the mathematical aspects of the Schwarzschild's metric, we present different methods of changing variables which tend to prove that the central singularity hypothesis does not exist. The geometric interpretation of the black hole is not mathematically convincing from the Schwarzschild metric. We recall the different mathematical approaches of several authors, the examples of the Painleve metric and its complex variant, then of the Schwarzschild metric, to deduce a metric with a throat sphere which leads to a mirror spacetime. Subsequently, we deduce the possibility of a bi-metric tangent to the Schwarzschild metric's throat sphere. We will also show that a false interpretation of the variables of the Schwarzschild metric can lead to false physical deductions and in particular to the concept of singularity. We compute the general solution of Einstein's equations in the presence of a non-zero energy tensor, i.e., for a homogeneous fluid ball with energy conditions. Our method of resolution involves a reformulation of the Einstein equation and the integration of the differential system. The metrics found are asymptotic to the Schwarzschild metric outside the fluid ball. We will present assumptions for the pressure inside the fluid ball and derive the corresponding metrics. Then, by solving the continuity equation of the energy-impulse tensor, we deduce an expression for the pressure inside the star which permits the expressing of the interior and exterior metrics.