Extension Rules of Newman–Janis Algorithm for Rotation Metrics in General Relativity

Main Article Content

Yu-Ching, Chou

Abstract

Aims: The aim of this study is to extend the formula of Newman–Janis algorithm (NJA) and introduce the rules of the complexifying seed metric. The extension of NJA can help determine more generalized axisymmetric solutions in general relativity.
Methodology: We perform the extended NJA in two parts: the tensor structure and the seed metric function. Regarding the tensor structure, there are two prescriptions, the Newman–Penrose null tetrad and the Giampieri prescription. Both are mathematically equivalent; however, the latter is more concise. Regarding the seed metric function, we propose the extended rules of a complex transformation by r2/Σ and combine the mass, charge, and cosmologic constant into a polynomial function of r.

Results: We obtain a family of axisymmetric exact solutions to Einstein’s field equations, including the Kerr metric, Kerr–Newman metric, rotating–de Sitter, rotating Hayward metric, Kerr–de Sitter metric and Kerr–Newman–de Sitter metric. All the above solutions are embedded in ellipsoid- symmetric spacetime, and the energy-momentum tensors of all the above metrics satisfy the energy conservation equations.

Conclusion: The extension rules of the NJA in this research avoid ambiguity during complexifying the transformation and successfully generate a family of axisymmetric exact solutions to Einsteins field equations in general relativity, which deserves further study.

Keywords:
Cosmological constant, ellipsoid coordinate transformation, Newman–Janis algorithm, Kerr metric, Kerr–de Sitter metric, Kerr–Newman metric, Kerr–Newman–de Sitter metric, rotating Hayward metric.

Article Details

How to Cite
Chou, Y.-C. (2020). Extension Rules of Newman–Janis Algorithm for Rotation Metrics in General Relativity. Physical Science International Journal, 24(6), 1-14. https://doi.org/10.9734/psij/2020/v24i630194
Section
Original Research Article

References

Newman ET, Janis AI. Note on the Kerr spinning particle metric. J. Math. Phys.
;6:915.
DOI: 10.1063/1.1704350

Drake SP, Szekeres P. An explanation of the Newman-Janis algorithm; 1998. arXiv:gr- qc/9807001
DOI: 10.1023/A:1001920232180

Ibohal N. Rotating metrics admitting non- perfect fluids in General Relativity; 2004.
arXiv:gr-qc/0403098
DOI: 10.1007/s10714-005-0002-6

Giampieri G. Introducing angular momentum into a black hole using complex variables. Gravity Research Foundation; Ibohal N. Non-stationary de Sitter cosmological models. Int. J. Mod. Phys. D.
;18:853-863.
DOI:10.1142/S0218271809014807

Erbin, H. Janis-Newman algorithm:

Generating rotating and NUT charged black holes. Universe. 2017;3(1):19.
DOI: 10.3390/universe3010019

Erbin H. Janis–Newman algorithm: Simplifications and gauge field transformation; 2018. arXiv:1410.2602
[gr-qc]
DOI: 10.1007/s10714-015-1860-1

Chou YC. A derivation of the Kerr metric by ellipsoid coordinate transformation. Int. J.

Phy. Sci. 2017;12(11):130-136.
DOI: 10.5897/IJPS2017.4605

Chou YC. A derivation of the Kerr– Newman metric using ellipsoid coordinate transformation. Journal of Applied Physical

Science International. 2018;10(3):144.
Available: http://www.ikprress.org/index.- php/JAPSI/article/view/3189

Chou YC. A radiating Kerr black hole and Hawking radiation. Heliyon.
;6(1):e03336.
DOI: 10.1016/j.heliyon.2020.e03336

Chou YC. A derivation of the Kerr metric by ellipsoid coordinate transformation. Theory and Applications of Physical Science.
;1.
DOI:10.9734/bpi/taps/v1

Canonico R, Parisi L, Vilasi G. Theoretical models for astrophysical objects and the Newman–Janis algorithm. Eleventh International Conference on Geometry, Integrablllty and Quantization; 2009.

Hayward SA. Formation and evaporation of nonsingular black holes. Physical review letters, 2006;96(3):031103.
DOI: 10.1103/PhysRevLett.96.031103

Keane AJ. An extension of the Newman– Janis algorithm; 2014. arXiv:1407.4478v1 [gr-qc]
DOI: 10.1088/0264-9381/31/15/155003

Gonzalez EJ. Extended Newman-Janis algorithm for rotating and Kerr-Newman de Sitter and anti de Sitter metrics; 2015.
arXiv:1504.01728 [gr-qc]

Carter B. in Les Astres Occlus ed. by B. DeWitt, C. M. De Witt, Gordon and Breach, New York. 1973;57-124.

Gibbons GW, Hawking SW. Cosmological event horizons, thermodynamics, and particle creation. Phys. Rev. D.
;15:2738-2751.
DOI: 10.1103/PhysRevD.15.2738

Mallett RL. Metric of a rotating radiating charged mass in a de Sitter space.

Phys. Lett. A 1988;126:226-228. DOI: 1016/0375-9601(88)90750-5 Koberlain BD. Rotating, radiating black holes, inflation, and cosmic censorship.
Phys. Rev. D 1995;51:6783-6787.
DOI:10.1103/physrevd.51.6783