New Way to Calculate Ricci Tensor and Ricci Scalar

Abed El Karim S. Abou Layla

doi:10.4236/jhepgc.2019.53042

Abstract

In the theory of general relativity, the finding of the Einstein Field Equation happens in a complex mathematical operation, a process we don’t need any more. Through a new theory in vector analysis, we’ll see that we can calculate the components of the Ricci tensor, Ricci scalar, and Einstein Field Equation directly in an easy way without the need to use general relativity theory hypotheses, principles, and symbols. Formulating the general relativity theory through another theory will make it easier to understand this relativity theory and will help combining it with electromagnetic theory and quantum mechanics easily.

Highlights

Through a new theory in vector analysis, we’ll see that we can calculate the components of the Ricci tensor, Ricci scalar, and Einstein Field Equation directly in an easy way without the need to use general relativity theory hypotheses, principles, and symbols
We will calculate the components of the Ricci tensor and Ricci scalar for many metrics, such as The Robertson-Walker metric and Schwarzschild metric directly, means that we don’t need the hypotheses, principles, and symbols of general relativity theory to get stress energy tensor, and this is important to annul the difference between the electromagnetic and gravitational, to better understanding of the universe, gravity, and black holes
We will calculate the components of the Ricci tensor, Ricci scalar and Einstein field equation for many metrics directly by using our new way, we will see that all results will be correct and consistent with the calculations founded in General Relativity Books

Summary

Introduction

All the books that explain general relativity theory present it in a conventional and complex way, an issue made it impossible to unit this theory with the electromagnetic theory and the quantum mechanics. To define my new theory (called the extended field theory), I published my first research papers, Old Mechanics, Gravity, Electromagnetics and Relativity in One Theory: “Part I”, in “Journal of High Energy Physics, Gravitation and Cosmology (JHEPGC)”, I implemented some principles and derived some of physics equations that can apply to both electric and gravitational fields, without distinguishing in this theory. We will calculate the components of the Ricci tensor and Ricci scalar for many metrics, such as The Robertson-Walker metric and Schwarzschild metric directly, means that we don’t need the hypotheses, principles, and symbols of general relativity theory to get stress energy tensor, and this is important to annul the difference between the electromagnetic and gravitational, to better understanding of the universe, gravity, and black holes. ∂ ∂x j hi vij where vij is a complex orthogonal unit (explained in my first paper) dxi, j as a constant value

Discussion

Calculate the Components of the Ricci Tensor

Solution of the Robertson-Walker Metric

Solution of the Schwarzschild Metric