Abstract
In this work we introduce a new family of non-relativistic spacetimes: standard stationary Galilean spacetimes, which constitute the local geometric model of stationary Galilean spacetimes. We also study the geodesic completeness of stationary Galilean spacetimes as well as the the geometric conditions for these spacetimes that guarantee the existence of a global splitting as a standard stationary Galilean spacetime.
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Albujer, A.L., Herrera, J., Rubio, R.M.: New examples of static spacetimes admitting a unique standard decomposition. Gen. Relat. Gravit. 51, 39 (2019)
Aledo, J.A., Romero, A., Rubio, R.M.: The existence and uniqueness of standard static splitting. Class. Quantum Gravity 32, 105004 (2015)
Bernal, A.N., López, M., Sánchez, M.: Fundamental units of length and time. Found. Phys. 32, 77–108 (2002)
Bernal, A.N., Sánchez, M.: Leibnizian, Galilean and Newtonian structures of space-time. J. Math. Phys. 44, 1129–1149 (2003)
Candela, A.M., Romero, A., Sánchez, M.: Completeness of the trajectories of particles coupled to a general force field. Arch. Ration. Mech. Anal. 208, 255–274 (2013)
Cartan, E.: Les variètés a conexion affine et la théorie de la relativité généralisée. Ann. Ec. Norm. Sup. 40, 1–25 (1923)
Cartan, E.: Les variètés a conexion affine et la théorie de la relativité généralisée (suite). Ann. Ec. Norm. Sup. 41, 325–412 (1924)
Christian, J.: Exactly soluble sector of quantum gravity. Phys. Rev. D 56, 4844–4877 (1997)
de la Fuente, D., Rubio, R.M.: Galilean generalized Robertson-Walker spacetimes: a new family of Galilean geometrical models. J. Math. Phys. 59, 022903 (2018)
Dixon, W.G.: On the uniqueness of the Newtonian theory as a geometric theory of gravitation. Commun. Math. Phys. 45, 167–182 (1975)
Duggal, K., Sharma, R.: Symmetries of Spacetimes and Riemannian Manifolds. Springer, Berlin (2013)
Flores, J.L.: The Riemannian and Lorentzian splitting theorems. Atlantis Trans. Geom. 1, 1–20 (2017)
Friedrichs, K.: Eine invariante Formulierung des Newtonschen Gravitationsgesetzes und des Grenzüberganges vom Einsteinschen zum Newtonschen Gesetz. Math. Ann. 98, 566–575 (1928)
Javaloyes, M.A., Sánchez, M.: A note on the existence of standard splittings for conformally stationary spacetimes. Class. Quantum Gravity 25, 168001 (2008)
Havas, P.: Four-dimensional formulations of Newtonian mechanics and their relation to the special and the general theory of relativity. Rev. Mod. Phys. 36, 938–965 (1964)
Künzle, H.P.: Galilei and Lorentz structures on space-time: comparison of the corresponding geometry and physics. Ann. I. H. Poincaré A 17, 337–362 (1972)
Künzle, H.P.: Covariant Newtonian limts of Lorentz space-times. Gen. Relat. Gravit. 7, 445–457 (1976)
Malament, D.B.: Topic in the Formulations of General Relativity and Newtonian Gravitation Theory. Chicago Lectures in Physics. University of Chicago Press, Chicago (2012)
O’Neill, B.: Semi-Riemannian Geometry with Applications to Relativity, vol. 103. Academic Press, New York (1983)
Penrose, R.: On gravity’s role in quantum state reduction. Gen. Relat. Gravit. 8, 581–600 (1996)
Rodrigues, W.A., Souza, Q., Bozhkov, Y.: The mathematical structure of Newtonian spacetime: classical dynamics and gravitation. Found. Phys. 25, 871–924 (1995)
Sachs, R.K., Wu, H.: General Relativity for Mathematicians. Springer, New York (1977)
Sánchez, M., Senovilla, J.M.M.: A note on the uniqueness of global static decompositions. Class. Quantum Gravity 24, 6121 (2007)
Spivak, M.: A Comprehensive Introduction to Differential Geometry. Publish or Perish, Houston (1999)
Trautman, A.: Comparison of Newtonian and Relativistic Theories of Space-Time, Perspectives in Geometry and Relativity. Indiana University Press, Bloomington (1966)
Warner, F.W.: Foundations of Differentiable Manifolds and Lie Groups. Springer, Berlin (1983)
Acknowledgements
The authors would like to thank the referee for his deep reading and valuable suggestions. This article has been partially supported by Spanish MINECO and ERDF Project MTM2016-78807-C2-1-P.
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de la Fuente, D., Pelegrín, J.A.S. & Rubio, R.M. On the geometry of stationary Galilean spacetimes. Gen Relativ Gravit 53, 8 (2021). https://doi.org/10.1007/s10714-020-02772-1
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DOI: https://doi.org/10.1007/s10714-020-02772-1
Keywords
- Leibnizian and Galilean structures
- Stationary Galilean spacetime
- Geodesic completeness
- Global splitting theorems