2001/05/24 by Y. Nutku, Nutku, Y., M. B. Sheftel +1
Mathematics · Physics and Astronomy · #35Q75 #83C15 #Advanced Differential Geometry Research #Black Holes and Theoretical Physics #Exactly Solvable and Integrable Systems (nlin.SI) #FOS: Physical sciences #General Relativity and Quantum Cosmology (gr-qc) #Geometric Analysis and Curvature Flows #Mathematical Physics (math-ph)
paper · pdf · doi:10.48550/arxiv.gr-qc/0105088
openalex publication_date 2001/05/24 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
This is a corrected and essentially extended version of the previous preprint arXiv:gr-qc/0105088 v4 (2002) by Y. Nutku and M. Sheftel containing new results. It is being published now in honor of Y. Nutku's memory. All responsibility for the additions and changes must be attributed to M. Sheftel. We present new anti-self-dual exact solutions of the Einstein field equations with Euclidean and neutral (ultra-hyperbolic) signatures that admit only one rotational Killing vector. Such solutions of the Einstein field equations are determined by non-invariant solutions of Boyer-Finley (BF) equation. For the case of Euclidean signature such a solution of the BF equation was first constructed by Calderbank and Tod. Two years later, Martina, Sheftel and Winternitz applied the method of group foliation to the Boyer-Finley equation and reproduced the Calderbank-Tod solution together with new solutions for the neutral signature. In the case of Euclidean signature we obtain new metrics which asymptotically locally look like a flat space and have a non-removable singular point at the origin. In the case of ultra-hyperbolic signature there exist three inequivalent forms of metric. Only one of these can be obtained by analytic continuation from the Calderbank-Tod solution whereas the other two are new.