Original Article  The Evaluation of the Rotating Metrics with a ̸= 0 from the Mass Function ˆM (u; r) with the reference to the Theory of General Relativity

Ngangbam Ishwarchandra,Prasenjit Debnath

doi:10.17485/ijst/v15i5.1560

Abstract

Objectives: With the reference to the Einstein’s theory of general relativity (1915), the evaluation of the rotating metrics such as Newman – Penrose Spin – Coefficients or NP Spin – Coefficients, the Ricci Scalars, the Weyl Scalars are designed with a ̸=0from the mass function or metric function ˆM (u; r). Methods: The methods / analysis adapted are the theoretical and mathematical analysis on the Einstein’s theory of general relativity. Findings: The Newman – Penrose Spin Coefficients (NP Spin – Coefficients), the Ricci Scalars, and the Weyl Scalars for the rotating metrics ˆM(u; r) with a ̸= 0 has been evaluated. Given by Wang and Wu (1999), the expanded form of the mass function or metric function with a ̸=0 has been used to evaluate the rotating metrics – NP Spin – Coefficients, the Ricci Scalars and the Weyl Scalars for a ̸= 0. The outcome is that the evaluation of rotating metrics with a ̸= 0 i.e. all the Newman – Penrose Spin Coefficients (NP Spin – Coefficients), the Ricci Scalars and the Weyl Scalars greatly simplifies the analysis of the theory of general relativity. Novelty: From the expended form of the mass function or metric function ˆM(u; r) with a ̸= 0 given by Wang and Wu (1999), all NP Spin – Coefficients, the Ricci Scalars, the Weyl Scalars has been derived for the option a ̸= 0. This paper evaluates the rotating metrics with all NP Spin – Coefficients, the Ricci Scalars and the Weyl Scalars with which greatly simplifies the analysis of the theory of general relativity. Also it is new way of formulation of the theory of general relativity with a ̸= 0. Keywords: The Einstein’s theory of general relativity; The mass function or metric function; The Newman – Penrose Spin - Coefficients (NP Spin – Coefficients); The Ricci Scalars; The Weyl Scalars.

Highlights

The Newman – Penrose Spin – Coefficients, the Ricci Scalars, the Weyl Scalars for the rotating metrics with mass function M (u, r)or metric function M (u, r) can be written as given below [1,2] : Debnath & Ishwarchandra / Indian Journal of Science and Technology 2022;15(5):216–220The Newman – Penrose Spin – Coefficients or NP Spin – Coefficients with mass function or metric function M (u, r) are: k∗ = σ = λ = ε = 0 ρ∗ = − 1 R, μ ∗∆ 2RR2 α (22a√i 2−RRRcsoins θ θ ), β c√ot θ 2 2R π i√a s2iRnRθ
From all the above Newman – Penrose Spin – Coefficients or the NP Spin – Coefficients, we have found that, in general case, tzheerorotwtaitsitn(gωm∗2et=ric0s)pnouslslevsescatcotrulaal(lCy haagnedordaesseikch(akr∗
Once we have found the Ricci Scalars, we can always find the energy momentum tensors from the Ricci Scalars

Summary

Introduction

The Newman – Penrose Spin – Coefficients (the NP Spin – Coefficients), the Ricci Scalars, the Weyl Scalars for the rotating metrics with mass function M (u, r)or metric function M (u, r) can be written as given below [1,2] : Debnath & Ishwarchandra / Indian Journal of Science and Technology 2022;15(5):216220. The Newman – Penrose Spin – Coefficients or NP Spin – Coefficients with mass function or metric function M (u, r) are: k∗ = σ = λ = ε = 0 ρ∗ = − 1 R , μ ∗. ∆ 2RR2 α (22a√i 2−RRRcsoins θ θ ).

Objectives

Conclusion