# Topology of the Misner space and its \(g\)-boundary

## Abstract

The Misner space is a simplified 2-dimensional model of the 4-dimensional Taub-NUT space that reproduces some of its pathological behaviours. In this paper we provide an explicit base of the topology of the complete Misner space \(\mathbb {R}^{1,1}/boost\). Besides we prove that some parts of this space, that behave like topological boundaries, are equivalent to the \(g\)-boundaries of the Misner space.

## Keywords

Misner space g-Boundary Spacetime topology## Notes

### Acknowledgments

The authors are very grateful to Juan Margalef Roig and Miguel Sánchez Caja for their useful comments and support, and specially to Fernando Barbero and Robert Geroch for their patience, comments and priceless help. This work has been supported by the Spanish MINECO research Grant FIS2012-34379 and the Consolider-Ingenio 2010 Program CPAN (CSD2007-00042).

## References

- 1.Durin Bruno, B., Pioline, B.: Closed strings in misner space: a toy model for a big bounce?, String theory: from gauge interactions to cosmology. arXiv:hep-th/0501145v2, Springer, pp. 177–200 (2006)
- 2.Flores, J.L., Herrera, J., Sánchez, M.: Hausdorff separability of the boundaries for spacetimes and sequential spaces, Preprint (2014)Google Scholar
- 3.Geroch, R.: Local characterization of singularities in general relativity. J. Math. Phys.
**9**, 450 (1968)ADSCrossRefzbMATHMathSciNetGoogle Scholar - 4.Geroch, R.: What is a singularity in general relativity? Ann. Phys.
**48**(3), 526–540 (1968)ADSCrossRefzbMATHGoogle Scholar - 5.Geroch, R., Can-bin, L., Wald, R.M.: Singular boundaries of space–times. J. Math. Phys.
**23**, 432 (1982)ADSCrossRefzbMATHMathSciNetGoogle Scholar - 6.Hajicek, P.: Embedding of singularities. Gen. Relativ. Gravit.
**1**(1), 27–29 (1970)ADSCrossRefGoogle Scholar - 7.Hawking, S.W.: Singularities and the Geometry of Space–Time. Unpublished Essay Submitted for the Adams Prize. Cambridge University, Cambridge (1966)Google Scholar
- 8.Hawking, S.W., Ellis, G.F.R.: The Large Scale Structure of Space–Time. Cambrigde University Press, Cambrigde (1973)CrossRefzbMATHGoogle Scholar
- 9.Hikida, Y., Nayak, R.R., Panigrahi, K.L.: D-branes in a big bang/big crunch universe: Misner space. J. High Energy Phys. arXiv:hep-th/0508003v2 2005(09),023 (2005)
- 10.Javaloyes Victoria, M.A., Sánchez Caja, M.: An Introduction to Lorentzian Geometry and its Applications, Ed. Universidad de Sao Paulo, (2010)Google Scholar
- 11.Jonsson, R.M.: Visualizing curved spacetime. Am. J. Phys. arXiv:0708.2483v1
- 12.Lévy-Leblond, J.M.: Speed(s). Am. J. Phys.
**48**, 345–347 (1980)ADSCrossRefGoogle Scholar - 13.Margalef Roig, J., Outerelo Domínguez, E.: Introducción a la topología, Ed. Complutense, (1993)Google Scholar
- 14.Misner, C.W.: Taub-NUT as a counterexample to almost anything. Tech. Rep. Uni. Maryland
**529**, 1–22 (1965)Google Scholar - 15.Munkres, J.R.: Topology, vol. 2. Prentice Hall, Upper Saddle River (2000)Google Scholar
- 16.O’Neill, B.: Semi-Riemannian Geometry: With Applications to Relativity. Academic Press, New York (1983)zbMATHGoogle Scholar
- 17.Thorne, K.S.: Misner space as a prototype for almost any pathology, directions in general relativity: Papers in Honor of Charles Misner, 1, vol. 1, p. 333 (1993)Google Scholar
- 18.Willard, S.: General Topology. Courier Dover Publications, New York (2004)zbMATHGoogle Scholar