Sciresol Sciresol https://indjst.org/author-guidelines Indian Journal of Science and Technology 0974-5645 10.17485/IJST/v14i15.307 The Rotating Metrics with the Mass Function M^(u,r) in reference to the Theory of General Relativity Debnath Prasenjit prasenjit1001@yahoo.com 1 Ishwarchandra Ngangbam 1 The Department of Physics , National Institute of Technology Agartala Jirania, Tripura, 799046, +91 - 7085709227 India 14 15 2021 Abstract

Background/Objectives: With reference to the theory of general relativity, the rotating metrics can be designed with mass function M^(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 M^(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 M^(u,r) with (a=0) can be shown from Wang and Wu (1999). From the expended form of the mass function M^(u,r)with a0 , 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*=ε=0 , actually shear free (σ=0) , purely expanding (θ^0) and a non-zero twist ω*20 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 None
Introduction

The rotating metrics with mass function M^(u,r) can be presented with respect to the theory of general relativity. The line element can be of the form of given below (Ibohal, 2005a) 1, 2.

ds2=1-2rM^(u,r)R-2du2+2dudr+4arM^(u,r)R-2sin2θdudϕ-2asin2θdrdϕ-R2dθ2-r2+a22-Δ*a2sin2θR-2sin2θdϕ2

Here, R2=r2+a2cos2θ and Δ*=r2-2rM^(u,r)+a2.

And the covariant complex null tetrad vectors for the rotating metrics can be written as follows below 3, 4.

la=δa1-asin2θδa4na=Δ2R2δa1+δa2-Δ2R2asin2θδa4m¯a=-12R¯iasinθδa1+R2δa3-ir2+a2sinθδa4ma=-12R-iasinθδa1+R2δa3+ir2+a2sinθδa4

The NP spin — coefficients, the Ricci scalars, Weyl scalars for the given rotating metrics a written as follows 5, 6:

The spin — coefficients:

k*=σ=λ=ε=0ρ*=-1R¯,μ*=-Δ2R¯R2α=(2ai-Rcosθ)22R¯R¯sinθ,β=cotθ22Rπ=iasinθ2R¯R¯,τ=-iasinθ2R2γ=12R¯R2r-M^-rM^,rR¯-Δ*v=12R¯R2iar sin θM^,u

The notations k*,ρ*,μ* carry the usual NP spin coefficients k,ρ,μ . The notation ρ,μ can be treated as energy density and null density respectively in energy – momentum tensor for the rest of the paper. In Riemannian geometry, the scalar curvatures or the Ricci scalars are the simplest invariant in Riemannian manifold. It actually assigns a single real number determined by the so called intrinsic geometry of that manifold near that very point. Now, the Ricci scalars are as follows 7, 8 :

ϕ00=ϕ01=ϕ10=ϕ02=ϕ20=0ϕ11=14R2R24r2M^,r+R2-2M^,-rM^,mrϕ12=122R2R2iasinθRM^,u-rM^,ruR¯ϕ21=-122R2R2iasinθR¯M^,-rM^,ruRϕ22=-12R2R22r2M^,u+a2rM^,sin2θΛ*=112R22M^,r+rM^,mr

The Weyl scalars are

φ0=φ1=0φ2=1R¯R¯R2-RM^+R¯6M^r(4r+2iacosθ)-r6R¯R¯M^mrφ3=-iasinθ22R¯R¯R2(4r+R¯)M^,u+rR¯M^,urφ4=a2rsin2θ2R¯R¯R2R2R2M^,uu-2rM^,u

From all these NP spin — coefficients we find that generally the rotating metric possesses a geodesic k*=ε=0, actually shear free σ=0, purely expanding (θ^0) and a non-zero twist ω*20 null vector la (Chandrasekhar, 1983) 9, 10, where

θ^-12(ρ+ρ¯)=rR2,ω*2-14(ρ-ρ¯)2=-a2cos2θR2R2

Again, then, the energy momentum tensor for the rotating metric will look like as follows 11, 12:

Tab=μlalb+2ρlaηb+2pmcm¯b+2ωlcm¯b+2ω¯lamb

And with the following quantities as follows:

μ=-1KR2R22r2M^u+a2rsin2θM^uutρ=2r2KR2R2M^,rp=-1K2a2cos2θR2R2M^rr+rR2M^mω=-iasinθ2KR2R2RM^u-rR¯M^,ur

All the above quantities have the relations with Ricci scalars as follows 13, 14

Kμ=2ϕ22,Kω=-2ϕ12Kρ=2ϕ11+6Λ,Kp=2ϕ11-6Λ

The result implies that once we obtain Ricci scalars ϕ11,ϕ12,ϕ22 and Λ as in equation 4 for a given particular space – time metric, we can always find μ,ρ and ρ which describes energy momentum tensor.

Wang and Wu (1999) actually have expanded the rotating mass function M^(u,r) for the non – rotating solution (a=0) in the power of as follows

M^(u,r)=n=-qn(u)rn

qn(u) is an arbitrary function of u. Wang and Wu consider the above summation as in integral form when the spectrum index 'n' is actually continuous in nature. Using the expression in equation 8, we can generate rotating metrics with (a0) as follows below by replacing mass function M^(u,r)of equation 10 with the arbitrary function qn(u). Thus we can rewrite equation 8 as follows 15, 16 .

μ=-1KR2R2n=-2qn(u)urn+2+a2sin2θqn(u)nurn+1ρ=2KR2R2n=-(n+2)qn(u)rn+1p=-1KR2n=-nqn(u)rn-12a2cos2θR2+(n-1)ω=-iasinθ2KR2R2n=-(R-nR¯)qn(u),urn

The same way, the other NP quantities can be found with the arbitrary function qn(u). The derivatives of the mass function M^(u,r)and NP quantities can be easily related by doing derivatives of the equation (10).

In general relativity, the Newtonian acceleration (surface gravity) is not so clear cut concept. For a black hole, the surface gravity should be treated relativistically, one can not define surface gravity as acceleration of a test body at its surface because there is no surface exists in a black hole. It is because the surface gravity of a body at event horizon turns out to be infinity according to general relativity. After renormalization, the surface gravity of a black hole can be analogous to Newtonian surface gravity, but they are not the same thing. In fact, the surface gravity of a black hole is not well-defined. But, one can define the surface gravity of a black hole whose event horizon is a killing horizon. The surface gravity k of a static killing horizon is the acceleration. If ka is normalized killing vector, then,

kabkb=kkb

According to Carter (1968, 1973) and also York (1984), we can introduce a scalar k which can be defined by the relation nbbna=kna , here the null vector nb is parameterized by the coordinate u such that d/du=naa . And again this scalar k, generally, can be expressed as NP spin coefficient γ as follows:

k=nbbnala=-(γ+γ¯)

Where γ is a spin — coefficient cited in equation 3. On a horizon of a Black hole, the scalar k is called the surface gravity of the black hole.

Conclusion

With a start of the rotating metrics with Mass Function M^(u,r), the line element is designed following the covariant complex null tetrad vectors for the rotating metrics with mass function M^(u,r). Then, it follows, the derivations of NP Spin — Coefficients, the Ricci Scalars, the Weyl Scalars. From all this NP spin — coefficients we found that generally the rotating metric possesses a geodesic k*=ε=0 , actually shear free σ=0, purely expanding (θ^0) and a non-zero twist ω*20 null vector la . Once we found the Ricci Scalars, we can always find the energy momentum tensors. Wang and Wu actually have expanded the rotating mass function M^(u,r)for the non — rotating solution a=0 as the power of r . With help of the expanded form of mass function M^(u,r), we have generated the rotating metrics with a0 . With the help of a scalar k, on a horizon of a Black hole, the surface gravity of the black hole is derived.

Acknowledgment

We truly acknowledge Dr. Aparna Nath, Associate Professor, The Department of Physics, National Institute of technology Agartala for providing great inspiration and tremendous motivation to complete this study. Also, we are extremely thankful to the department of physics National Institute of Technology Agartala for proper coordination and cooperation.

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