Foundations of Physics Letters

, Volume 18, Issue 1, pp 1–19 | Cite as

An Analytical Treatment of the Clock Paradox in the Framework of the Special and General Theories of Relativity

  • Lorenzo IorioEmail author

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In this paper we treat the so called clock paradox in an analytical way by assuming that a constant and uniform force F of finite magnitude acts continuously on the moving clock along the direction of its motion assumed to be rectilinear (in space). No inertial motion steps are considered. The rest clock is denoted as (1), the to and fro moving clock is (2), the inertial frame in which (1) is at rest in its origin and (2) is seen moving is I and, finally, the accelerated frame in which (2) is at rest in its origin and (1) moves forward and backward is A. We deal with the following questions: (1) What is the effect of the finite force acting on (2) on the proper time interval Δτ(2) measured by the two clocks when they reunite? Does a differential aging between the two clocks occur, as it happens when inertial motion and infinite values of the accelerating force is considered? The special theory of relativity is used in order to describe the hyperbolic (in spacetime) motion of (2) in the frame I. (II) Is this effect an absolute one, i.e., does the accelerated observer A comoving with (2) obtain the same results as that obtained by the observer in I, both qualitatively and quantitatively, as it is expected? We use the general theory of relativity in order to answer this question. It turns out that ΔτI = ΔτA for both the clocks, Δτ(2) does depend on g = F/m, and = Δτ(2)/Δτ(1) = (√1 − β2atanhβj)/β < 1. In it ; = V/c and V is the velocity acquired by (2) when the force is inverted.

Key words:

twin paradox special theory of relativity general theory of relativity accelerated motion 


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  1. 1.
    1. E. Erisksen and Ø. Grøn, “Relativistic dynamics in uniformly accelerated reference frames with application to the clock paradox,” Eur. J. Phys. 39, 39–44 (1990).CrossRefGoogle Scholar
  2. 2.
    2. H. Nikolié, “The role of acceleration and locality in the twin paradox,” Found. Phys. Lett. 13, 595–601 (2000).MathSciNetCrossRefGoogle Scholar
  3. 3.
    3. M. Pauri and M. Vallisneri, “Märzke-Wheeler coordinates for accelerated observers in special relativity,” Found. Phys. Lett. 13, 401 (2000).MathSciNetCrossRefGoogle Scholar
  4. 4.
    4. O. Wucknitz, “Sagnae effect, twin paradox and spacetime topology. Time and length in rotating systems and closed Minkowski spacetimes,” preprint gr-qe/0403111 (2004).Google Scholar
  5. 5.
    5. J. L. Synge, Relativity: The General Theory (North-Holland, Amsterdam, 1960).zbMATHGoogle Scholar
  6. 6.
    6. C. W. Misner, K. S. Thorne, and J. A. Wheeler, Gravitation (Freeman, San Francisco, 1973).Google Scholar
  7. 7.
    7. J. Bailey, et al., “Measurements of relativistic time dilatation for positive and negative muons in a circular orbit,” Nature 268, 301–305 (1977). J. Bailey, K. Borer, F. Combley, H. Drumm, C. Eck, F. J. M. Farley, J. H. Field, W. Flegel, P. M. Hattersley, F. Krienen, et al., “Final report on the CERN muon storage ring including the anomalous magnetic moment and the electric dipole moment of the muon, and a direct test of relativistic time dilation,” Nucl. Phys. B 150, 1–75 (1979).ADSCrossRefGoogle Scholar
  8. 8.
    8. R. P. Durbin, H. H. Loar, and W. W. Havens Jr., “The lifetimes of the pi + and pi − mesons.” Phys. Rev. 88, 179 183 (1952).ADSCrossRefGoogle Scholar
  9. 9.
    9. B. Mashhoon, “The hypothesis of locality in relativistic physics,” Phys. Lett. A 145 147–153 (1990); “Limitations of spacetime measurements,” Phys. Lett. A 143, 176–182 (1990).ADSCrossRefGoogle Scholar
  10. 10.
    10. H. Nikolié, “Relativistic contraction of an accelerated rod,” Am. J. Phys. 67, 1007–1012 (1999).ADSCrossRefGoogle Scholar
  11. 11.
    11. A. Tartaglia and M.L. Ruggiero, “Lorentz contraction and accelerated systems,” Eur. J. Phys. 24, 215–220 (2003).CrossRefzbMATHGoogle Scholar
  12. 12.
    12. C. Møller, The Theory of Relativity, 2nd edn. (Clarendon, Oxford, 1972).Google Scholar

Copyright information

© Springer Science+Business Media, Inc. 2005

Authors and Affiliations

  1. 1.Dipartimento Interateneo di Fisica dell’Università di Bari INFN-Sezione di BariBariItaly

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