Abstract
This paper reviews the concept of pp-waves and plane waves in classical general relativity theory. The first four sections give discussions of some algebraic constructions and symmetry concepts which will be needed in what is to follow. The final sections deal with the definitions of such wave solutions, their associated geometrical tensors in space–time and their Killing, homothetic, conformal and wave surface symmetries. Some unusual geometrical features of these solutions (compared with standard positive definite geometry) are described.
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Alexeevski, D.: Self similar Lorentzian manifolds. Ann. Glob. Anal. Geom. 3, 59–84 (1985)
Beem, J.K.: Proper homothetic maps and fixed points. Lett. Math. Phys. 2, 317–320 (1978)
Bel, L.: Les états de radiation et le problème de l’énergie en relativité générale. Cah. de. Phys. 16, 59–80 (1962)
Bel, L.: Radiation states and the problem of energy in general relativity. Gen. Relat. Gravit. 32, 2047–2078 (2000)
Brinkmann, H.W.: Einstein spaces which are mapped conformally on each other. Math. Ann. 94, 119–145 (1925)
Defrise, L.: Groupes d’isotropie et groupes de stabilite conforme dans les espaces Lorentziens. Thesis, Universite Libre de Bruxelles (1969)
Ehlers, J.; Kundt, W.: Gravitation: an introduction to current research. In: Witten, L. (Ed.), Wiley, New York, pp 49–101 (1962)
Goldberg, J.N.; Kerr, R.P.: Asymptotic properties of the electromagnetic field. J. Math. Phys. 5, 172–175 (1964)
Hall, G.S.: Sectional curvature and the determination of the metric in space-time. Gen. Relat. Gravit. 16, 79–87 (1984)
Hall, G.S.: Conformal symmetries and fixed points in space-time. J. Math. Phys. 31, 1198–1207 (1990)
Hall, G.S.: Covariantly constant tensors and holonomy structure in general relativity. J. Math. Phys. 32, 181–187 (1991)
Hall, G.S.; Lonie, D.P.: Holonomy groups and spacetimes. Class. Quant. Gravit. 17, 1369–1382 (2000)
Hall, G.S.: On the theory of Killing orbits in spacetime. Class. Quant. Gravit. 20, 4067–4084 (2003)
Hall, G.S.: Symmetries and Curvature Structure in General Relativity. World Scientific, Singapore (2004)
Hall, G.S.; Lonie, D.P.: Projective equivalence of Einstein spaces in general relativity class. Quant. Gravit. 26, 125009 (2009)
Hall, G.S.: The geometry of 4-dimensional Ricci-flat manifolds which admit a metric. J. Geom. Phys. 89, 50–59 (2015)
Hall, G.S.: Symmetries, Orbits and Isotropy in General Relativity Theory. In: Proceedings of the International Conference on Relativistic Astrophysics (ICRA-2015), Lahore (2015)
Hall, G.S.; Kırık, B.: Symmetries in 4-Dimensional Manifolds With Metric of Neutral Signature. University of Aberdeen, Aberdeen (2018)
Hermann, R.: On the accessibility problem in control theory. International Symposium on Nonlinear Differential Equations and Nonlinear Mechanics, Academic Press, New York, pp 325–332 (1963)
Kobayashi, S.; Nomizu, K.: Foundations of Differential Geometry, vol. 1. Interscience, New York (1963)
Ruh, B.: Krummungstreue Diffeomorphismen Riemannscher und pseudo-Riemannscher Mannigfaltigkeiten. Math. Z. 189, 371–391 (1985)
Kulkarni, R.S.: Curvature and metric. Ann. Math. 91, 311–331 (1970)
Petrov, A.Z.: Einstein Spaces. Pergamon, Berlin (1969)
Pirani, F.A.E.: Invariant formulation of gravitational radiation theory. Phys. Rev. 105, 1089–1099 (1957)
Pirani, F.A.E.: Lectures on General Relativity. Brandeis Summer Institute in Theoretical Physics 1, 249–373 (1964)
Sachs, R.K.: The outgoing radiation condition in general relativity. Proc. R. Soc. A A264, 309 (1961)
Schell, J.F.: Classification of four-dimensional Riemannian spaces. J. Math. Phys. 2(1961), 202–206 (2018)
Stephani, H.; Kramer, D.; MacCallum, M.; Hoenselaers, C.; Herlt, E.: Exact Solutions of Einstein’s Field Equations. Cambridge University Press, Cambridge (2003)
Stefan, P.: Accessible sets, orbits, and foliations with singularities. Proc. Lond. Math. Soc. 29, 699–713 (1974)
Sussmann, H.J.: Orbits of families of vector fields and integrability of distributions. Trans. Am. Math. Soc. 180, 171–188 (1973)
Trautman, A.: Lectures on General Relativity. Lectures Given at King’s College, London (1958)
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