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

Spyridon Vossos,Christos G Massouros,Elias Vossos

doi:10.3390/particles4040039

Spyridon Vossos, Christos G Massouros + Show 1 more

Open Access

https://doi.org/10.3390/particles4040039

Copy DOI

Journal: Particles	Publication Date: Dec 8, 2021
Citations: 1	License type: CC BY 4.0

Affiliation: National and Kapodistrian University of Athens

Abstract

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 V=V(r,r.,ϕ.), where r is the distance from the center of gravity and r.,ϕ. 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.

Highlights

Throughout this paper, the weak Einstein’s Equivalence Principle (EEP) is adopted [1](p. 245): mG = m, (1)where the gravitational mass is equal to the inertial rest mass (m), as it appears in all classical mechanics (Newtonian Physics)
The approach described in this paper can be applied to any gravity theory (Riemannian and/or non-Riemannian) with the use of the corresponding time dilation
The gravitational field can be described well via either variable metrics of spacetime according to General Relativity (GR), or Gravitational Generalized Potential according to Special Relativity (SR)

Summary

Introduction

Throughout this paper, the weak Einstein’s Equivalence Principle (EEP) is adopted [1]. ) of φ are the radial and angular velocities, respectively This method is applied in the case of the Schwarzschild metric of GR and gives the same equations of motion, explaining the precession of Mercury’s perihelion, gravitational deflection of light, and Shapiro time delay via SR. The approach described in this paper can be extended to any theory of gravity (Riemannian and/or non-Riemannian), such as Lorentz Gauge Theory [9,10,11,12,13,14], Teleparallel Gravity [15,16,17,18,19,20,21], etc., with the use of the corresponding time dilation In this way, we can work with the gravitational fields in the same manner as in any other type of field (such as the electro-magnetic field) in Minkowski spacetime (M4) with the Lorentz metric and we avoid dealing with the motions of particles in the curved spacetime of GR.

Schwarzschild Metric

Gravitational Red Shift

The Reverse Procedure

Findings

Discussion

Conclusions