Abstract
Khan and Penrose, and separately Szekeres, presented solutions of colliding impulsive and sandwich plane gravitational waves, respectively. They only considered equal strength waves, which would impart no momentum to test particles in their path on the plane of collision at the time. We had earlier probed both spacetimes using the pseudo-Newtonian (\( \psi \)N) formalism. In neither case was it really clear what parameter represented the “strength” of the wave. In this paper we identify the required parameter and extend the colliding sandwich plane gravitational wave solution to unequal strengths. Since the impulsive plane waves of Khan and Penrose correspond to a limit of the sandwich waves with the pulse duration tending to zero, we have used the procedure to obtain the generalization of the Khan–Penrose solution to unequal strengths. The resulting spacetimes are exact solutions of vacuum field equations.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
Data availibility
Not applicable.
References
Einstein, A.: Näherungsweise Integration der Feldgleichungender Gravitation. Sitzungsber. K. Preuss. Akad. Wiss. 1, 688–696 (1916)
Scheidegger, A.E.: Gravitational motion. Rev. Mod. Phys. 25(2), 451–452 (1953)
Weber, J., Wheeler, J.A.: Reality of the cylindrical gravitational waves of Einstein and Rosen. Rev. Mod. Phys. 29, 509–515 (1957)
Abbott, B.P., et al.: Observations of gravitational waves from a binary black hole merger. Phys. Rev. Lett. 116, 061102 (2016)
Einstein, A., Rosen, N.: On gravitational waves. J. Frankl. Inst. 223, 43–54 (1937)
Bondi, H., Pirani, F.A.E., Robinson, I.: Gravitational waves in general relativity. III. Exact plane waves. Proc. R. Soc. Lond. A 251, 519–533 (1959)
Szekeres, P.: Colliding gravitational waves. Nature 228, 1183–1184 (1970)
Szekeres, P.: Colliding plane gravitational waves. J. Math. Phys. 13, 286–294 (1972)
Khan, K.A., Penrose, R.: Scattering of two impulsive gravitational plane waves. Nature 229, 185–186 (1971)
Szabados, L.B.: On canonical pseudotensors, Sparling’s form and Noether currents. Class. Quant. Grav. 11, 2521 (1992)
Landau, L.D., Lifshitz, E.M.: The Classical Theory of Fields. Pergamon Press, Oxford (1987)
Papapetrou, A.: Einstein’s theory of gravitation and flat space. Proc. R. Irish A52, 11–23 (1948)
Qadir, A.: Einstein’s General Theory of Relativity. Cambridge Scholars Publishing, Cambridge (2020)
Ehlers, J., Kundt, W.: Exact solutions of the gravitational field equations. In: Witten, L. (ed.) Gravitation: An Introduction to Current Problems. Wiley, New York (1962)
Qadir, A., Sharif, M.: General formula for the momentum imparted to test particles in arbitrary spacetimes. Phys. Lett. A 167, 331–334 (1992)
Mahajan, S.M., Qadir, A., Valanju, V.M.: Reintroducing the concept of force into relativity theory. Nuovo Cim. 65, 404–418 (1981)
Qadir, A., Quamar, J.: Relativistic generalization of Newtonian forces. In: Ning, H. (ed.) Proceedings of the Third Marcel Grossmann Meeting, pp. 189–220. North Holland, Amsterdam (1983)
Quamar, J.: Relativistic Gravitodynamics and Forces. Ph.D. thesis, Quaid-i-Azam University (1984)
Qadir, A., Sharif, M.: The relativistic generalization of the gravitational force for arbitrary spacetimes. Nuovo Cim. B 107, 1071–1083 (1992)
Sharif, M.: Forces in Non-static Spacetimes. Ph.D. thesis, Quaid-i-Azam University (1991)
Abbasi, K.Q., Qadir, A.: Probing the Khan-Penrose colliding plane impulsive gravitational waves solution. J. Phys. Commun. 2, 025021 (2018)
Griffith, J.B.: Colliding Plane Waves in General Relativity. Oxford University Press, Oxford (1991)
Abbasi, K.Q., Hussain, I., Qadir, A.: Probing Szekeres’ colliding sandwich gravitational waves. Eur. Phys. J. Plus 136(5), 1–16 (2021)
Penrose, R.: Private communication (2023)
Barrabès, C., Hogan, P.A.: Colliding impulsive gravitational waves and a cosmological constant. Phys. Rev. D 92, 044032 (2015)
Acknowledgements
We are most grateful to Prof. Sir Roger Penrose (NL) for extremely useful and insightful comments on an earlier draft of this paper in his usual kind way. One of us (K. Q. Abbasi) is grateful to Prof. Ibrar Hussain for his useful discussions and guidance on the work.
Funding
In this study, there was no funding received or financial support provided for the research. The authors conducted the study independently without any external funding sources.
Author information
Authors and Affiliations
Contributions
Both authors contributed equally to the research and preparation of this manuscript.
Corresponding author
Ethics declarations
Conflict of interest
In this study, there are no competing interests to declare. The authors, affiliated institutions, or any other related bodies do not have any financial or personal relationships that could be perceived as potentially influencing the research or its publication. We affirm that there are no conflicts of interest to disclose.
Ethical approval
We would like to highlight that our study involves neither human participants nor animals. Therefore, there is no potential for ethical conflicts or harm to either humans or animals in this study.
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.
About this article
Cite this article
Abbasi, K.Q., Qadir, A. Colliding plane gravitational waves of unequal strength. Gen Relativ Gravit 55, 117 (2023). https://doi.org/10.1007/s10714-023-03162-z
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/s10714-023-03162-z