Skip to main content
SpringerLink
Log in

A note on the computation of geometrically defined relative velocities

  • Research Article
  • Published:
General Relativity and Gravitation Aims and scope Submit manuscript

Abstract

We discuss some aspects about the computation of kinematic, spectroscopic, Fermi and astrometric relative velocities that are geometrically defined in general relativity. Mainly, we state that kinematic and spectroscopic relative velocities only depend on the 4-velocities of the observer and the test particle, unlike Fermi and astrometric relative velocities, that also depend on the acceleration of the observer and the corresponding relative position of the test particle, but only at the event of observation and not around it, as it would be deduced, in principle, from the definition of these velocities. Finally, we propose an open problem in general relativity that consists on finding intrinsic expressions for Fermi and astrometric relative velocities avoiding terms that involve the evolution of the relative position of the test particle. For this purpose, the proofs given in this paper can serve as inspiration.

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

Access this article

Log in via an institution

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. Soffel M. et al.: The IAU 2000 resolutions for astrometry, celestial mechanics and metrology in the relativistic framework: explanatory supplement. Astron. J. 126, 687–2706 (2003) (arXiv:astro-ph/0303376)

    Article  Google Scholar 

  2. Lindegren L., Dravins D.: The fundamental definition of ‘radial velocity’. Astron. Astrophys. 401, 1185–1202 (2003) (arXiv:astro-ph/0302522)

    Article  ADS  Google Scholar 

  3. Bolós V.J.: Intrinsic definitions of “relative velocity” in general relativity. Commun. Math. Phys. 273, 217–236 (2007) (arXiv:gr-qc/0506032)

    Article  ADS  MATH  Google Scholar 

  4. Klein D., Collas P.: Recessional velocities and Hubble’s law in Schwarzschild-de Sitter space. Phys. Rev. D 81, 063518 (2010) (arXiv:1001.1875)

    Article  ADS  Google Scholar 

  5. Klein D., Randles E.: Fermi coordinates, simultaneity, and expanding space in Robertson–Walker cosmologies. Ann. Henri Poincaré 12, 303–328 (2011) (arXiv:1010.0588)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  6. Bolós, V.J., Klein, D.: Relative velocities for radial motion in expanding Robertson–Walker spacetimes. Preprint (2011). (arXiv:1106.3859)

  7. Carrera M., Giulini D.: On Doppler tracking in cosmological spacetimes. Class. Quantum Grav. 23, 7483–7492 (2006) (arXiv:gr-qc/0605078)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  8. O’Neill B.: Semi-Riemannian Geometry With Applications to Relativity. Pure and Applied Mathematics, vol. 103. Elsevier Science, San Diego (1983)

    Google Scholar 

  9. Cheeger J., Ebin D.G.: Comparison Theorems in Riemannian Geometry. North-Holland Publishing Company, Amsterdam (1975)

    MATH  Google Scholar 

  10. do Carmo M.P.: Riemannian Geometry. Birkhäuser, Boston (1992)

    MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Dpto. Matemáticas para la Economía y la Empresa, Facultad de Economía, Universidad de Valencia, Avda. Tarongers s/n, 46022, Valencia, Spain

    Vicente J. Bolós

Authors
  1. Vicente J. Bolós
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Vicente J. Bolós.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Bolós, V.J. A note on the computation of geometrically defined relative velocities. Gen Relativ Gravit 44, 391–400 (2012). https://doi.org/10.1007/s10714-011-1278-3

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10714-011-1278-3

Keywords

Search

Navigation