The Rotating Metrics with the Mass Function ˆM(u; r) in reference to the Theory of General Relativity

Abstract

Background/Objectives: With reference to the theory of general relativity, the rotating metrics can be designed with mass function ˆM (u; r) . Methods/Statistical analysis: The methods/analysis adapted is the theoretical and mathematical analysis on the theory of general relativity. Findings: The line element can be found out with the help of mass function ˆM(u; r) following the find of the covariant complex null tetrad vectors for the rotating metrics. Then, the NP spin — coefficients, the Ricci scalars, and the Weyl scalars for the rotating metrics can be found out. The expanded form of the mass function ˆM(u; r) with (a=0) can be shown from Wang and Wu (1999). From the expended form of the mass function ˆM(u; r)with (a ̸= 0) , the NP coefficients can be derived. With the help of the scalar k, the surface gravity of the black hole is derived. Novelty/Applications: The findings of covariant complex null tetrad vectors for the rotating metrics, the NP spin — coefficients, the Ricci scalars, and the Weyl scalars are new analysis towards the theory of general relativity. Specific applications are the deep studies on the black hole and its surface gravity. It can be concluded that generally the rotating metric possesses a geodesic (k =e = 0), actually shear free (s = 0) , purely expanding (ˆq ̸ ( = 0) and a non-zero twistw2 ̸= 0) null vector la (Chandrasekhar, 1983). With the help of a scalar k, on a horizon of a Black hole, the surface gravity of the black hole is derived. Keywords: The Mass Function; NP Spin – Coefficients; The Ricci Scalars; The Weyl Scalars; The Surface Gravity of the Black Hole