Abstract
Given a semi-Riemannian 4-manifold (M, g) with two distinguished vector fields satisfying properties determined by their shear, twist and various Lie bracket relations, a family of Kähler metrics gK is constructed, defined on an open set in M, which coincides with M in many typical examples. Under certain conditions g and gK share various properties, such as a Killing vector field or a vector field with a geodesic flow. In some cases the Kähler metrics are complete. The Ricci and scalar curvatures of gK are computed under certain assumptions in terms of data associated to g. Many examples are described, including classical spacetimes in warped products, for instance de Sitter spacetime, as well as gravitational plane waves, metrics of Petrov type D such as Kerr and NUT metrics, and metrics for which gK is an SKR metric. For the latter an inverse ansatz is described, constructing g from the SKR metric.
References
[1] A. B. Aazami, Symplectic 4-manifolds via Lorentzian geometry, Proceedings of the American Mathematical Society, 145 (2017), pp. 387–394.Search in Google Scholar
[2] A. B. Aazami and G. Maschler, Canonical Kähler metrics on classes of Lorentzian 4-manifolds, arXiv:1811.08999.Search in Google Scholar
[3] D. Alekseevsky and F. Zuddas, Cohomogeneity one Kähler and Kähler-Einstein manifolds with one singular orbit I, Annals of Global Analysis and Geometry, 52 (2017), pp. 99—-128.Search in Google Scholar
[4] A. N. Aliev and C. Saçlıoğlu, Self-dual fields harbored by a Kerr–Taub-BOLT instanton, Physics Letters B, 632 (2006), pp. 725–727.Search in Google Scholar
[5] V. Apostolov, D. M. J. Calderbank and P. Gauduchon, Ambitoric geometry I: Einstein metrics and extremal ambikähler structures, J. Reine Angew. Math. 721 (2016), 109-—147.10.1515/crelle-2014-0060Search in Google Scholar
[6] V. Apostolov, Generalized Goldberg-Sachs theorems for pseudo-Riemannian four-manifolds, Journal of Geometry and Physics, 27 (1998), pp. 185–198.Search in Google Scholar
[7] J. K. Beem, P. Ehrlich, and K. Easley, Global Lorentzian Geometry, vol. 202 of Pure and Applied Mathematics, Marcel Dekker, Inc., 1996.Search in Google Scholar
[8] D. M. J. Calderbank, Selfdual Einstein metrics and conformal submersions, arXiv math/0001041, (2000).Search in Google Scholar
[9] D. M. J. Calderbank and H. Pedersen, Selfdual spaces with complex structures, Einstein-Weyl geometry and geodesics, in Annales de L’institut Fourier, vol. 50, Chartres: L’Institut, 1950-, 2000, pp. 921–964.10.5802/aif.1779Search in Google Scholar
[10] A. M. Candela and M. Sánchez, Geodesics in semi-Riemannian manifolds: geometric properties and variational tools, Recent developments in pseudo-Riemannian geometry, (2008), pp. 359–418.Search in Google Scholar
[11] J. Cheeger and D. G. Ebin, Comparison theorems in Riemannian geometry, vol. 9, North-Holland Publishing Company Amsterdam, 1975.Search in Google Scholar
[12] W. A. de Graaf, Classification of solvable Lie algebras, Experimental mathematics, 14 (2005), pp. 15–25.Search in Google Scholar
[13] A. Derdzinski and G. Maschler, Local classification of conformally-Einstein Kähler metrics in higher dimensions, Proceedings of the London Mathematical Society, 87 (2003), pp. 779–819.Search in Google Scholar
[14] A. Derdzinski and G. Maschler, Special Kähler-Ricci potentials on compact Kähler manifolds, Journal für die reine und angewandte Mathematik, 593 (2006), pp. 73–116.Search in Google Scholar
[15] K. Dixon, Regular ambitoric 4-manifolds: from Riemannian Kerr to a complete classification, arXiv:1604.03156, (2016).Search in Google Scholar
[16] E. J. Flaherty, Hermitian and Kählerian geometry in relativity, vol. 46 of Lecture Notes in Physics, Springer, 1976.10.1007/3-540-07540-2Search in Google Scholar
[17] J. L. Flores and M. Sánchez, On the geometry of pp-wave type spacetimes, in Analytical and Numerical Approaches to Mathematical Relativity, Springer, 2006, pp. 79–98.10.1007/3-540-33484-X_4Search in Google Scholar
[18] P. Gauduchon, Variétés Lorentziennes de type hermitien à tenseur de Ricci nul, preprint, (1993).Search in Google Scholar
[19] J. Goldberg and R. Sachs, Republication of: A theorem on Petrov types, General Relativity and Gravitation, 41 (2009), pp. 433–444.Search in Google Scholar
[20] A. Harris and G. Paternain, Conformal great circle flows on the 3-sphere, Proceedings of the American Mathematical Society, (2015).10.1090/proc/12819Search in Google Scholar
[21] W. M. Kinnersley, Type D gravitational fields, PhD thesis, California Institute of Technology, 1969.Search in Google Scholar
[22] W. M. Kinnersley, Type D vacuum metrics, Journal of Mathematical Physics, 10 (1969), pp. 1195–1203.Search in Google Scholar
[23] T. Leistner and D. Schliebner, Completeness of compact Lorentzian manifolds with abelian holonomy, Mathematische Annalen, 364 (2016), pp. 1469–1503.Search in Google Scholar
[24] G. Maschler, Central Kähler metrics, Transactions of the American Mathematical Society, 355 (2003), pp. 2161—-2182.Search in Google Scholar
[25] P. Nurowski, Einstein equations and Cauchy-Riemann geometry, PhD thesis, SISSA, 1993.Search in Google Scholar
[26] P. Nurowski and A. Trautman, Robinson manifolds as the Lorentzian analogs of Hermite manifolds, Differential Geometry and its Applications, 17 (2002), pp. 175–195.Search in Google Scholar
[27] B. O’Neill, Semi-Riemannian geometry. With applications to relativity, vol. 103 of Pure and Applied Mathematics, Academic Press, 1983.Search in Google Scholar
[28] B. O’Neill, The geometry of Kerr black holes, Wellesley, Mass.: AK Peters, 1995.Search in Google Scholar
[29] R. Sachs, Gravitational waves in general relativity. vi. the outgoing radiation condition, in Proceedings of the Royal Society of London A: Mathematical, Physical and Engineering Sciences, vol. 264, The Royal Society, 1961, pp. 309–338.10.1098/rspa.1961.0202Search in Google Scholar
[30] I. P. Silva, J. L. Flores, and J. Herrera, Rigidity of geodesic completeness in the Brinkmann class of gravitational wave space-times, arXiv:1605.03619, (2016).Search in Google Scholar
[31] W. P. Thurston and J. W. Milnor, The geometry and topology of three-manifolds, Princeton University Princeton, 1979.Search in Google Scholar
[32] A. Trautman, On complex structures in physics, in On Einstein’s path, Springer, 1999, pp. 487–501.10.1007/978-1-4612-1422-9_34Search in Google Scholar
[33] D. Varolin, Three variations on a theme in complex analytic geometry, Analytic and Algebraic Geometry, IAS/Park City Math. Ser., 17 (2008), pp. 183–294.Search in Google Scholar
[34] S.-T. Yau, Open problems in geometry, in Proceedings of Symposia in Pure Mathematics, vol. 54, 1993, pp. 1–28.10.1090/pspum/054.1/1216573Search in Google Scholar
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