# Time Contraction Within Lightweight Reference Frames

## Abstract

The special theory of relativity teaches us that, although distinct inertial frames perceive the same dynamical laws, space and time intervals differ in value. We revisit the problem of time contraction using the paradigmatic model of a fast-moving laboratory within which a photon is emitted and posteriorly absorbed. In our model, however, the laboratory is composed of two independent parallel plates, each of which allowed to be sufficiently light so as to get kickbacks upon emission and absorption of light. We show that the lightness of the laboratory accentuates the time contraction. We also discuss how the photon frequency shifts upon reflection in a light moving mirror. Although often imperceptible, these effects will inevitably exist whenever realistic finite-mass bodies are involved. More fundamentally, they should necessarily permeate any eventual approach to the problem of relativistic quantum frames of reference.

## Keywords

Time contraction Lightweight reference frames Doppler effect Special relativity## Notes

### Acknowledgements

We gratefully acknowledge A. D. Ribeiro, G. M. Kremer, and C. A. Duarte for discussions. M.F.S. and R.M.A. acknowledge the financial support from the CAPES and the National Institute for Science and Technology of Quantum Information (INCT-IQ, CNPq/Brazil), respectively.

## References

- 1.H. Goldstein, C. Poole, J. Safko.
*Classical Mechanics*, 3rd edn (Addison-Wesley, New York, 2001)MATHGoogle Scholar - 2.J.B. Marion, S.T. Thornton.
*Classical Dynamics of Particles and Systems*, 5th edn (Brooks/Cole – Thomson Learning, Belmont, 2003)Google Scholar - 3.Y. Aharonov, T. Kaufherr, Quantum frames of reference. Phys. Rev. D.
**30**, 368 (1984)ADSMathSciNetCrossRefGoogle Scholar - 4.S.D. Bartlett, T. Rudolph, R.W. Spekkens, Reference frames, superselection rules, and quantum information. Rev. Mod. Phys.
**79**, 555 (2007)ADSMathSciNetCrossRefMATHGoogle Scholar - 5.R.M. Angelo, N. Brunner, S. Popescu, A.J. Short, P. Skrzypczyk, Physics within a quantum reference frame. J. Phys. A Math. Theor.
**44**, 145304 (2011)ADSMathSciNetCrossRefMATHGoogle Scholar - 6.R.M. Angelo, A.D. Ribeiro, Kinematics and dynamics in noninertial quantum frames of reference. J. Phys. A Math. Theor.
**45**, 465306 (2012)ADSMathSciNetCrossRefMATHGoogle Scholar - 7.S.T. Pereira, R.M. Angelo, Galilei covariance and Einstein’s equivalence principle in quantum reference frames. Phys. Rev. A.
**91**, 022107 (2015)ADSMathSciNetCrossRefGoogle Scholar - 8.D. Halliday, R. Resnick.
*Fundamentals of Physics*, 3rd edn (John Wiley & Sons, New York, 1998)MATHGoogle Scholar - 9.P.A. Tipler, G. Mosca.
*Physics for Scientists and Engineers*, vol. 3, 6th edn (W. H. Freeman, New York, 2007)Google Scholar - 10.W. Künding, Measurement of the transverse doppler effect in an accelerated system. Phys. Rev.
**129**, 2371 (1963)ADSCrossRefGoogle Scholar - 11.J.C. Hafele, R.E. Keating, Around-the-world Atomic Clocks: Predicted Relativistic Time Gains. Science.
**177**(4044), 168 (1972)ADSCrossRefGoogle Scholar - 12.J. Bailey, K. Borer, F. Combley, H. Drumm, F. Krienen, F. Lange, E. Picasso, W. von Ruden, F.J.M. Farley, J.H. Field, W. Flegel, P.M. Hattersley, Measurements of relativistic time dilatation for positive and negative muons in a circular orbit. Nature.
**268**, 301 (1977)ADSCrossRefGoogle Scholar - 13.R.W. McGowan, D.M. Giltner, S.J. Sternberg, S.A. Lee, New measurement of the relativistic Doppler shift in neon. Phys. Rev. Lett.
**70**, 251 (1993)ADSCrossRefGoogle Scholar - 14.G. Saathoff, S. Karpuk, U. Eisenbarth, G. Huber, S. Krohn, R.M. Horta, S. Reinhardt, D. Schwalm, A. Wolf, G. Gwinner, Improved test of time dilation in special relativity. Phys. Rev. Lett.
**91**, 190403 (2003)ADSCrossRefGoogle Scholar - 15.S. Reinhardt, G. Saathoff, H. Buhr, L.A. Carlson, A. Wolf, D. Schwalm, S. Karpuk, C. Novotny, G. Huber, M. Zimmermann, R. Holzwarth, T. Udem, T.W. Hänsch, G. Gwinner, Test of relativistic time dilation with fast optical atomic clocks at different velocities. Nat. Phys.
**3**, 861 (2007)CrossRefGoogle Scholar - 16.C. Lämmerzah, Special relativity: a matter of time. Nat. Phys.
**3**, 831 (2007)CrossRefGoogle Scholar - 17.B. Najjari, A. Surzhykov, A.B. Voitkiv, Relativistic time dilation and the spectrum of electrons emitted by 33-TeV lead ions penetrating thin foils. Phys. Rev. A.
**77**, 042714 (2008)ADSCrossRefGoogle Scholar - 18.P. Dubé, A.A. Madej, M. Tibbo, J.E. Bernard, High-accuracy measurement of the differential scalar polarizability of a
^{88}Sr^{+}clock using the time-dilation effect. Phys. Rev. Lett.**112**, 173002 (2014)ADSCrossRefGoogle Scholar - 19.B. Botermann, D. Bing, C. Geppert, G. Gwinner, T.W. Hänsch, G. Huber, S. Karpuk, A. Krieger, T. Kühl, W. Nörtershäuser, C. Novotny, S. Reinhardt, R. Sánchez, D. Schwalm, T. Stöhlker, A. Wolf, G. Saathoff, Test of time dilation using stored Li
^{+}ions as clocks at relativistic speed. Phys Rev. Lett.**113**, 120405 (2014)ADSCrossRefGoogle Scholar - 20.D.M. Greenberger, Inadequacy of the usual galilean transformation in quantum mechanics. Phys. Rev. Lett.
**87**, 100405 (2011)MathSciNetCrossRefGoogle Scholar - 21.A. Gjurchinovski, Reflection of light from a uniformly moving mirror. Am. J. Phys.
**72**, 1316 (2004)ADSMathSciNetCrossRefGoogle Scholar