The Relation between General Relativity’s Metrics and Special Relativity’s Gravitational Scalar Generalized Potentials and Case Studies on the Schwarzschild Metric, Teleparallel Gravity, and Newtonian Potential

This paper shows that gravitational results of general relativity (GR) can be reached by using special relativity (SR) via a SR Lagrangian that derives from the corresponding GR time dilation and vice versa. It also presents a new SR gravitational central scalar generalized potential <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>V</mi><mo>=</mo><mi>V</mi><mo stretchy="false">(</mo><mi>r</mi><mo>,</mo><mover><mi>r</mi><mo>.</mo></mover><mo>,</mo><mover><mi>ϕ</mi><mo>.</mo></mover><mo stretchy="false">)</mo></mrow></semantics></math></inline-formula>, where <i>r</i> is the distance from the center of gravity and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mover><mi>r</mi><mo>.</mo></mover><msub><mo>,</mo><mrow></mrow></msub><mover><mi>ϕ</mi><mo>.</mo></mover></mrow></semantics></math></inline-formula> are the radial and angular velocity, respectively. This is associated with the Schwarzschild GR time dilation from where a SR scalar generalized potential is obtained, which is exactly equivalent to the Schwarzschild metric. Thus, the Precession of Mercury’s Perihelion, the Gravitational Deflection of Light, the Shapiro time delay, the Gravitational Red Shift, etc., are explained with the use of SR only. The techniques used in this paper can be applied to any GR spacetime metric, Teleparallel Gravity, etc., in order to obtain the corresponding SR gravitational scalar generalized potential and vice versa. Thus, the case study of Newtonian Gravitational Potential according to SR leads to the corresponding non-Riemannian metric of GR. Finally, it is shown that the mainstream consideration of the Gravitational Red Shift contains two approximations, which are valid in weak gravitational fields only.