The Study of the Energy Conditions of the Universe and Unique Solutions of the Einstein’s Field Equations in the Lights of the Theory of General Relativity

Abstract

Objectives: To study the energy conditions of the universe and to find unique solutions of the Einstein’s field equations. Methods: A mathematical formulation developed to study the energy conditions of the universe and a new way of unique solutions of the famous Einstein’s field equations with appropriate theoretical and mathematical analysis of the theory of general relativity. Findings: With the reference to the power series expansion of the mass function ˆM(u; r) developed by Wang and Wu (1999), we have derived two new ways to solve Einstein’s field equations by introducing new values of n as n=2 and n=-2. Novelty: With the reference to the power series expansion of the mass function ˆM(u; r), we have found new ways to solve Einstein’s field equations with n=2 and n=-2. And we have derived a total of four new solutions of the Einstein’s field equations. And the solutions are very innovative and different from (i) Schwarzschild solution and (ii) de Sitter solution. The solutions own (i) the first solution with the line element in the equation (11) describes a stationary solution. It has coordinate singularity at r = (2m)􀀀1 . (ii) The second solution with the line element in the equation (14) will be reduced to those Schwarzschild black holes when m=0 with singularities at r=2M. Also, it will be that dark energy when M=0 with singularities at r = (2m)􀀀1 . (iii) The third solution with the line element in the equation (17) describes a stationary solution. It has coordinate singularity at r = (2m)13 (iv) The fourth solution with the line element in the equation (20) describes a stationary solution. It has coordinate singularity at r = (2m)13. Keywords: The Mass Function; Schwarzschild Solution; Einstein’s Field Equation; The Theory of General Relativity; Space - time Curvature