×

Spacetime groups. (English) Zbl 1455.83002

Summary: A spacetime group is a connected 4-dimensional Lie group \(G\) endowed with a left invariant Lorentz metric \(h\) such that the connected component of the isometry group of \(h\) is \(G\) itself. The Newman-Penrose formalism is used to give an algebraic classification of spacetime groups, that is, we determine a complete list of inequivalent spacetime Lie algebras, which are pairs \((\mathfrak{g}, \eta)\), with \(\mathfrak{g}\) being a 4-dimensional Lie algebra and \(\eta\) being a Lorentzian inner product on \(\mathfrak{g}\). A full analysis of the equivalence problem for spacetime Lie algebras is given, which leads to a completely algorithmic solution to the problem of determining when two spacetime Lie algebras are isomorphic. The utility of our classification is demonstrated by a number of applications. The results of a detailed study of the Einstein field equations for various matter fields on spacetime groups are given, which resolve a number of open cases in the literature. The possible Petrov types of spacetime groups that, generically, are algebraically special are completely characterized. Several examples of conformally Einstein spacetime groups are exhibited. Finally, we describe some novel features of a software package created to support the computations and applications of this paper.
©2020 American Institute of Physics

MSC:

83C05 Einstein’s equations (general structure, canonical formalism, Cauchy problems)
83C60 Spinor and twistor methods in general relativity and gravitational theory; Newman-Penrose formalism
83C20 Classes of solutions; algebraically special solutions, metrics with symmetries for problems in general relativity and gravitational theory
83-08 Computational methods for problems pertaining to relativity and gravitational theory
53Z05 Applications of differential geometry to physics
35Q75 PDEs in connection with relativity and gravitational theory
22E15 General properties and structure of real Lie groups
22E43 Structure and representation of the Lorentz group
PDFBibTeX XMLCite
Full Text: DOI arXiv

References:

[1] Petrov, A. Z., Einstein Spaces (1969), Pergamon Press · Zbl 0174.28305
[2] Farnsworth, D. L.; Kerr, R. P., Homogeneous dust filled cosmological solutions, J. Math. Phys., 7, 1625 (1966) · doi:10.1063/1.1705075
[3] Hiromoto, R. E.; Ozsvath, I., On homogeneous solutions of Einstein’s field equations, Gen. Relativ. Gravitation, 9, 299-327 (1978) · Zbl 0427.53012 · doi:10.1007/bf00760424
[4] Hall, G. S.; Morgan, T.; Perjés, Z., Three dimensional spacetimes, Gen. Relativ. Gravitation, 19, 1137-1146 (1987) · Zbl 0629.53022 · doi:10.1007/bf00759150
[5] Anderson, I. M. and Torre, C. G., The Differential Geometry software project; available at https://digitalcommons.usu.edu/dg_publications/.
[6] Kruchkovich, G. I., The classification of three-dimensional Riemannian spaces by groups of motions, Usp. Matem. Nauk SSSR9, part 1, 59, 3 (1954)
[7] MacCallum, M. A. H.; Harvey, A., On the classification of the real four-dimensional lie algebras, On Einstein’s Path (1999), Springer: Springer, New York · Zbl 0959.17003
[8] Calvaruso, G., Four-dimensional pseudo-Riemannian lie groups, Rendiconti Seminario Matematico Univ. Pol. Torino Workshop for Sergio Console, 31-43 (2016), Rendiconti Seminario Matematico Univ. Pol.: Rendiconti Seminario Matematico Univ. Pol., Torino · Zbl 1440.53087
[9] Patera, J.; Sharp, T.; Winternitz, P.; Zassenhaus, H., Invariants of real low dimensional Lie algebra, J. Math. Phys., 71, 986-994 (1976) · Zbl 0357.17004 · doi:10.1063/1.522992
[10] Snobl, L.; Winternitz, P., Classification and Identification of Lie Algebras (2014), American Mathematical Society · Zbl 1331.17001
[11] Fee, G. J., Homogeneous spacetimes (1979), University of Waterloo
[12] Newman, E.; Penrose, R., An approach to gravitational radiation by a method of spin coefficients, J. Math. Phys., 3, 566-578 (1962) · Zbl 0108.40905 · doi:10.1063/1.1724257
[13] Stephani, H.; Kramer, D.; MacCallum, M.; Hoenselaers, C.; Herlt, E., Exact Solutions of Einstein’s Field Equations (2003), Cambridge University Press · Zbl 1057.83004
[14] Stewart, J., Advanced General Relativity (1991), Cambridge University Press
[15] Karlhede, A., On a coordinate-invariant description of Riemannian manifolds, Gen. Relativ. Gravitation, 12, 963 (1980) · Zbl 0462.53011 · doi:10.1007/bf00757367
[16] Karlhede, A., Gravitational Field Geometry as the Geometry of Automorphisms (1962), Pergamon Press
[17] Kaigorodov, V., Einstein spaces of maximum mobility, Dokl. Akad. Nauk SSSR, 7, 893 (1962) · Zbl 0122.22004
[18] Ozsváth, I., New homogeneous solutions of Einstein’s field equations with incoherent matter obtained by a spinor technique, J. Math. Phys., 6, 590-609 (1965) · Zbl 0131.43203 · doi:10.1063/1.1704311
[19] 13. Note that the perfect fluid solutions (12.30-12.32) given in Ref. 13 are contained in one case here—the splitting into the three cases of Ref. 13 is only required for the explicit integration of the Newman-Penrose equations to find the coordinate form of the spacetime metric.
[20] Komrakov, B., Einstein-Maxwell equation on four-dimensional homogeneous spaces, Lobachevskii J. Math., 8, 33-165 (2001) · Zbl 1011.83005
[21] Ozsváth, I., Homogeneous solutions of the Einstein-Maxwell equations, J. Math. Phys., 6, 1255-1265 (1965) · doi:10.1063/1.1704767
[22] McLenaghan, R. G.; Tariq, N., A new solution of the Einstein-Maxwell equations, J. Math. Phys., 16, 2306-2312 (1975) · doi:10.1063/1.522461
[23] Tupper, B. O. J., A class of algebraically general solutions of the Einstein Maxwell equations for non-null electromagnetic fields II, Gen. Relativ. Gravitation, 7, 479-486 (1976) · doi:10.1007/bf00766405
[24] Henneaux, M., Electromagnetic fields invariant up to a duality rotation under a group of isometries, J. Math. Phys., 25, 2276-2283 (1984) · doi:10.1063/1.526432
[25] Cox, D.; Little, J.; O’Shea, D., Ideals, Varieties and Algorithms (2000), Springer
[26] Heck, A., Introduction to Maple (2003), Springer · Zbl 1020.65001
[27] Flanders, H., Differential Forms with Applications to the Physical Sciences (1983), Dover · Zbl 0112.32003
[28] Spivak, M., A Comprehensive Introduction to Differential Geometry (1979), Publish or Perish · Zbl 0439.53002
[29] Calvaruso, G.; Zaeim, A., Conformally flat homogeneous pseudo-riemannian four-manifolds, Tohoku Math. J., 66, 54 (2014) · Zbl 1296.53136 · doi:10.2748/tmj/1396875661
[30] Honda, K.; Tsukada, K.; Sanchez, M.; Ortega, M.; Romero, A., Conformally Flat Homogeneous Lorentzian Manifolds (2013), Springer · Zbl 1282.53061
[31] Gover, A. R.; Nurowski, P., Obstructions to conformally Einstein metrics in n-dimensions, J. Geom. Phys., 56, 450-484 (2006) · Zbl 1098.53014 · doi:10.1016/j.geomphys.2005.03.001
[32] Ashtekar, A.; Magnon-Ashtekar, A., A technique for analyzing the structure of isometries, J. Math. Phys., 19, 1567-1572 (1979) · Zbl 0443.53047 · doi:10.1063/1.523864
[33] Atkins, R., Existence of parallel sections of a vector bundle, J. Geom. Phys., 61, 309-311 (2011) · Zbl 1206.53024 · doi:10.1016/j.geomphys.2010.09.016
[34] Besse, A. L., Einstein Manifolds (1986), Springer
[35] Michel, J.-P.; Somberg, P.; Šilhan, J., Prolongation of symmetric Killing tensors and commuting symmetries of the Laplace operator, Rocky Mountain J. Math., 47, 587-619 (2017) · Zbl 1371.35034 · doi:10.1216/rmj-2017-47-2-587
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.