Influence of mass on the space-time’s geometry

In General Relativity, in systems with a dominant mass, the curvature of spacetime is described by the Schwarzschild metric. However, obtaining its metric coefficients of the gravitational field due to this mass generally requires the use of advanced mathematical tools, such as differential geometry and tensor calculus. In this article, to obtain such metric coefficients, we use Gauss’s Egregium Theorem on spatial curvature and the hypothesis that these coefficients coincide, in the limit of zero mass, with those of the Minkowski metric valid for Special Relativity. With this formulation, the article contains, in a didactic way, a first quantitative presentation of the Theory of General Relativity, based on knowledge of Special Relativity, using basic tools of calculus, without the use of tensors. Furthermore, we provide video lectures based on the material of this article.

Keywords:
Theorema Egregium; General Relativity; Schwarzschild metric