On the Field Equations of General Relativity

Vu B Ho

doi:10.4236/jamp.2022.101005

Abstract

In this work, we examine the geometric character of the field equations of general relativity and propose to formulate relativistic field equations in terms of the Riemann curvature tensor. The resulted relativistic field equations are also integrated into the general framework that we have presented in our previous works that all known classical fields can be expressed in the same dynamical form. We also discuss a possibility to reformulate the field equations of general relativity so that the Ricci curvature tensor and the energy-momentum tensor can appear symmetrically in the field equations without violating the conservation law stated by the covariant derivative.

Highlights

IntroductionOne of the most insightful features that emerges from Einstein field equations of general relativity, Rαβ
Perhaps, one of the most insightful features that emerges from Einstein field equations of general relativity, Rαβ − 1 2 gαβ R= kTαβ, is that matter and the spacetime continuum should be presented with an intrinsic geometric structure of a differentiable manifold [1] [2]
The resulted relativistic field equations are integrated into the general framework that we have presented in our previous works that all known classical fields can be expressed in the same dynamical form

Summary

Introduction

One of the most insightful features that emerges from Einstein field equations of general relativity, Rαβ. The Einstein general relativistic field equations do not fully describe the assumed mathematical structure of matter and the spacetime continuum, because the mathematical object that involves in the equations is the Ricci curvature tensor Rαβ instead of the Riemann curvature tensor Rβαμν. In this work, we will discuss a possibility to formulate relativistic field equations using the Riemann curvature tensor instead of the Ricci curvature tensor. As shown, we still need Einstein field equations of general relativity, formulated in terms of the Ricci curvature tensor, to establish the required relationship between the Riemann curvature tensor and the energy-momentum tensor. We will show that relativistic field equations formulated in terms of the Riemann curvature tensor take the form presented by the general equation given in Equation (2)

Formulation of Relativistic Field Equations Using Riemann Curvature Tensor

A Reformulation of the Field Equations of General Relativity

Conclusion