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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.
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