Abstract
The group of rigid motions is considered to guide the search for a natural system of space-time coordinates in General Relativity. This search leads us to a natural extension of the space-times that support Painlevé–Gullstrand synchronization. As an interesting example, here we describe a system of rigid coordinates for the cross mode of gravitational linear plane waves.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
Notes
Throughout the paper we will use the following notation: Latin indices \(i,j,k=1,2,3\); \(dx\,dy=\frac{1}{2}(dx\otimes dy+dy\otimes dx)\); \(T_{(i}Q_{j)}=\frac{1}{2}(T_{i}Q_{j}+T_{j}Q_{i})\); \(\delta _{ij}\) is the three-dimensional identity; \(f_{,i}=\frac{\partial f}{\partial x^{i}}\), where \(x^{i}\) are the space coordinates; \(f_{,\lambda }=\frac{\partial f}{\partial \lambda }\); \(\bar{d}\) is the restriction of the differential to \(d\lambda =0\), i.e. \(\bar{d} f(x,\lambda )= \frac{\partial f}{\partial x^{i}} dx^i\).
In this section we will use a prime, \(f'\), to indicate the derivation of a function with respect to its argument.
References
Einstein, A.: The relativity principle. Jahrbuch der Radioaktivitfit and Elektronik 4, 411–462 (1907)
Kretschmann, E.: Über den physikalischen Sinn der Relativitätspostulate, A. Einsteins neue und seine ursprüngliche Relativitätstheorie. Annalen der Physik 358, 575–614 (1918)
Fock, V.: The Theory of Space, Time and Gravitation. Macmillan, New York (1964). (1964.2 d rev. ed. Translated from the Russian by N. Kemmer)
Antoci, S., Liebscher, D.-E.: The Group Aspect in the Physical Interpretation of General Relativity Theory (2009). arXiv preprint arXiv:0910.2073
Ellis, G., Matravers, D.: General covariance in general relativity? Gen. Relativ. Gravit. 27, 777–788 (1995)
Bel, L.: Born’s group and Generalized isometries. In: Diaz, J., Lorente, M. (eds.) Relativity in General: Proceedings of the Relativity Meeting’93, Editions Frontieres (1994)
Llosa, J.: An Extension of Poincaré Group Abiding Arbitrary Acceleration (2015). arXiv preprint arXiv:1512.07465
Bel, L.: Static elastic deformations in general relativity (1996). arXiv preprint arXiv:gr-qc/9609045
Coll, B.: About deformation and rigidity in relativity. J. Phys.: Conf. Ser. 66(1), 012001 (2007)
Painlevé, P.: Le Mecanique Classique et la Theorie de la Relativite. L’Astronomie 36, 6–9 (1922)
Gullstrand, A.: Allgemeine Lösung des Statischen Einkörperproblems in der Einsteinschen Gravitationstheorie. Almqvist & Wiksell, Stockholm (1922)
Barceló, C., Liberati, S., Visser, M.: Analogue gravity. Living Rev. Rel 8, 214 (2005)
Jaén, X., Molina, A.: Gen. Relativ. Gravit. 45, 1531–1546 (2013)
Jaén, X., Molina, A.: Gen. Relativ. Gravit. 46, 1–14 (2014)
Jaén, X., Molina, A.: Gen. Relativ. Gravit. 47, 1–16 (2015)
Misner, C.W., Thorne, K.S., Wheeler, J.A.: Gravitation. Macmillan, London (1973)
Manasse, F.K., Misner, C.W.: Fermi normal coordinates and some basic concepts in differential geometry. J. Math. Phys. 4, 735 (1963). doi:10.1063/1.1724316
Acknowledgements
We thank the reviewer for his/her interesting comments that has lead us to improve the paper. Support for this work to AM was provided by FIS2015-65140-P (MINECO/FEDER).
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Jaén, X., Molina, A. Rigid covariance as a natural extension of Painlevé–Gullstrand space-times: gravitational waves. Gen Relativ Gravit 49, 108 (2017). https://doi.org/10.1007/s10714-017-2272-1
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/s10714-017-2272-1