The Bach-flat and conformally Einstein equations for Siklos spacetimes

Zaeim, Amirhesam

doi:10.22103/jmmr.2024.23460.1652

The Bach-flat and conformally Einstein equations for Siklos spacetimes

Document Type : Research Paper

Author

Amirhesam Zaeim

Department of Mathematics, Payame noor University, P.O. Box 19395-4697, Tehran, Iran

10.22103/jmmr.2024.23460.1652

Abstract

Within the large class of Siklos spacetimes, we completely classify Bach-flat metrics, which turn out to be related to a bi-harmonicity property of the defining function. Using this classification, we tackle the conformally Einstein property and several classes of conformally Einstein Siklos metrics are then determined, including all the homogeneous examples.

Keywords

Siklos spacetimes

Bach-flat metrics

conformally Einstein metrics

Subjects

Geometry and Topology

[1] Abbena, E., Garbiero, S. and Salamon, S. (2013). Bach-Flat Lie groups in dimension 4. C. R. Math. Acad. Sci. Paris 351, 303{306. https://doi.org/10.48550/arXiv.1303.4527.

[2] Almansi, E. (1898). sull'integrazione dell'equazione di erenziale 2n = 0. Annali di Matematica, tomo II, serie III.

[3] Brinkmann, HW. (1924). Riemann spaces conformal to Einstein spaces. Math. Ann. 91, 269{278. https://doi.org/10.1007/BF01556083.

[4] Calvaruso, G. (2019). Siklos spacetimes as homogeneous Ricci solitons. Class. Quantum Grav., 36, 095011, 13pp. https://doi.org/10.1088/1361-6382/ab10f6.

[5] Calvaruso, G. (2020). Conformally flat Siklos metrics are Ricci solitons. Axioms, 9(64). https://doi.org/10.3390/axioms9020064.

[6] Calvaruso, G. (2021). Solutions of the Ricci soliton equation for a large class of Siklos spacetimes. Int. J. Geom. Meth. Mod. Phys., 18, 2150052, 19pp. https://doi.org/10.1142/S0219887821500523.

[7] Calvino-Louzao, E., Garcia-Rio, E., Gutierrez-Rodriguez, I., and Vazquez-Lorenzo, R. (2017). Conformal geometry of non-reductive four-dimensional homogeneous spaces. Math. Nachr., 290, 1470{1490. https://doi.org/10.1002/mana.201600099.

[8] Calvino-Louzao, E., Garcia-Martinez, X., Garcia-Rio, E., Gutierrez-Rodriguez, I., and Vazquez-Lorenzo, R. (2019). Conformally Einstein and Bach-flat four-dimensional homogeneous manifolds. J. Math. Pures Appl., 130, 347{374.
https://doi.org/10.1016/j.matpur.2019.01.005.

[9] Gover, AR., and Nurowski, P. (2006). Obstructions to conformally Einstein metrics in n dimensions. Jour. Geom. Phys. 56(3), 450{484, https://doi.org/10.1016/j.geomphys.2005.03.001.

[10] Kozameh, CN., Newman, ET., and Tod, KP. (1985). Conformal Einstein spaces. Gen. Rel. Grav. 17, 343{352, https://doi.org/10.1007/BF00759678.

[11] Mohseni, M. (2018). Vacuum polarization in Siklos spacetimes. Phys. Rev. D 97, 024006, 6 pp, https://doi.org/10.1103/PhysRevD.97.024006.

[12] Ozsvath, I., Robinson, I., and Rozga, K. (1985). Plane-fronted gravitational and electromagnetic waves in spaces with cosmological constant. J. Math. Phys., 26, 1755{1761.

[13] Podolski, J. (1998). Interpretation of the Siklos solutions as exact gravitational waves in the anti-de Sitter universe. Class. Quantum Grav. 15,719{733, https://iopscience.iop.org/article/10.1088/0264-9381/15/3/019.

[14] Siklos, STC. (1985). Lobatchevski plane gravitational waves, in Galaxies, axisymmetric systems and relativity. Ed. M.A.H. MacCallum, page 247, Cambridge University Press, Cambridge.