Skip to main content
Log in

Infinitesimal relative position vector fields for observers in a reference frame and applications to conformally stationary spacetimes

  • Published:
Analysis and Mathematical Physics Aims and scope Submit manuscript

Abstract

We introduce and analyze the concept of infinitesimal relative position vector field between “infinitesimally nearby” observers, showing the equivalence between different definitions. Through the Fermi–Walker derivative of infinitesimal relative position vector fields along an observer in a reference frame, we characterize spacetimes admitting an umbilic foliation. Sufficient and necessary conditions for those spacetimes to be a conformally stationary spacetime are given. Finally, the important class of cosmological models known as generalized Robertson–Walker spacetimes is characterized.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Alías, L.J., Romero, A., Sánchez, M.: Uniqueness of complete spacelike hypersurfaces of constant mean curvature in generalized Robertson–Walker spacetimes. Gen. Relativ. Gravitat. 27(1), 71–84 (1995)

    Article  MathSciNet  Google Scholar 

  2. Beem, J.K., Ehrlich, P.E., Easley, K.L.: Global Lorentzian Geometry. Pure and Applied Mathematics, vol. 202. Marcel Dekker, New York (1996)

    MATH  Google Scholar 

  3. Caballero, M., Romero, A., Rubio, R.M.: Constant mean curvature spacelike hypersurfaces in lorentzian manifolds with a timelike gradient conformal vector field. Class. Quantum Gravity 28, 145009 (2011)

    Article  MathSciNet  Google Scholar 

  4. Chen, B.-Y.: A simple characterization of generalized Robertson–Walker spacetimes. Gen. Relativ. Gravitat. 46(12), 1833 (2014)

    Article  MathSciNet  Google Scholar 

  5. Coley, A.A., Tupper, B.O.J.: Zero-curvature Friedmann–Robertson–Walker models as exact viscous magnetohydrodynamic cosmologies. Astrophys. J. 271, 1–8, 07 (1983)

    Article  Google Scholar 

  6. Daftardar, V., Dadhich, N.: Gradient conformal killing vectors and exact solutions. Gen. Relativ. Gravitat. 26(9), 859–868 (1994)

    Article  MathSciNet  Google Scholar 

  7. Duggal, K.L., Sharma, R.: Connection and Curvature Symmetries, pp. 134–155. Springer, Boston (1999)

    Google Scholar 

  8. Eardley, D., Isenberg, J., Marsden, J., Moncrief, V.: Homothetic and conformal symmetries of solutions to Einstein’s equations. Commun. Math. Phys. 106(1), 137–158 (1986)

    Article  MathSciNet  Google Scholar 

  9. Ferrando, J.J., Morales, J.A., Portilla, M.: Inhomogeneous space-times admitting isotropic radiation: vorticity-free case. Phys. Rev. D 46, 578–584 (1992)

    Article  MathSciNet  Google Scholar 

  10. Gutiérrez, M., Olea, B.: Global decomposition of a Lorentzian manifold as a GRW spacetime. Differ. Geom. Appl. 27, 146–156,02 (2009)

    Article  Google Scholar 

  11. Hawking, S.W.: The existence of cosmic time functions. Proc. R. Soc. Lond. Ser. A Math. Phys. Sci. 308(1494), 433–435 (1969)

    Article  Google Scholar 

  12. Mantica, C.A., Molinari, L.G.: Generalized Robertson-Walker space times, a survey. Int. J. Geom. Methods Mod. Phys. 14, 1730001,03 (2017)

    Google Scholar 

  13. Molinari, L.G., Mantica, C.A.: A simple property of the weyl tensor for a shear, vorticity and acceleration-free velocity field. Gen. Relativ. Gravitat. 50(7), 81 (2018)

    Article  MathSciNet  Google Scholar 

  14. O’Neill, B.: Semi-Riemannian Geometry with Applications to Relativity (Pure and Applied Mathematics), vol. 103. Academic Press, London (1983)

    MATH  Google Scholar 

  15. Rainer, M., Schmidt, H.-J.: Inhomogeneous cosmological models with homogeneous inner hypersurface geometry. Gen. Relativ. Gravitat. 27(12), 1265–1293 (1995)

    Article  MathSciNet  Google Scholar 

  16. Rebouças, M.J., Skea, J.E.F., Tavakol, R.K.: Cosmological models expressible as gradient vector fields. J. Math. Phys. 37(2), 858–873 (1996)

    Article  MathSciNet  Google Scholar 

  17. Sachs, R.K., Wu, H.: General Relativity for Mathematicians. Graduate Texts in Mathematics, vol. 48. Springer, Berlin (1977)

    Book  Google Scholar 

  18. Sánchez, M.: On the geometry of generalized Robertson–Walker spacetimes: geodesics. Gen. Relativ. Gravitat. 30(6), 915–932 (1998)

    Article  MathSciNet  Google Scholar 

  19. Stephani, H., Kramer, D., MacCallum, M., Hoenselaers, C., Herlt, E.: Exact Solutions of Einstein’s Field Equations. Cambridge Monographs on Mathematical Physics, 2nd edn. Cambridge University Press, Cambridge (2003)

    Book  Google Scholar 

  20. Treciokas, R., Ellis, G.F.R.: Isotropic solutions of the Einstein–Boltzmann equations. Commun. Math. Phys. 23(1), 1–22 (1971)

    Article  MathSciNet  Google Scholar 

  21. Warner, R.W.: Foundations of Differentiable Manifolds and Lie Groups. Springer, Berlin (1983)

    Book  Google Scholar 

  22. Zafiris, E.: Irreducible decomposition of Einstein’s equations in spacetimes with symmetries. Ann. Phys. 263, 155–178 (1998)

    Article  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Magdalena Caballero.

Ethics declarations

Conflict of interest

The authors declare that they have no conflict of interest.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

The authors are partially supported by Spanish MINECO and FEDER Project MTM2016-78807-C2-1-P.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Caballero, M., de la Fuente, D. & Rubio, R.M. Infinitesimal relative position vector fields for observers in a reference frame and applications to conformally stationary spacetimes. Anal.Math.Phys. 9, 1977–1990 (2019). https://doi.org/10.1007/s13324-019-00293-y

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s13324-019-00293-y

Keywords

Mathematics Subject Classification

Navigation