Quantum formulation of the Einstein equivalence principle

Abstract

The validity of just a few physical conditions comprising the Einstein equivalence principle (EEP) suffices to ensure that gravity can be understood as spacetime geometry. The EEP is therefore subject to ongoing experimental verification, with present-day tests reaching the regime in which quantum mechanics becomes relevant. Here we show that the classical expression of the EEP does not apply in such a regime. The EEP requires equivalence between the rest mass-energy of a system, the mass-energy that constitutes its inertia, and the mass-energy that constitutes its weight. In quantum mechanics, the energy contributing to the mass is given by a Hamiltonian operator of the internal degrees of freedom. Therefore, we introduce a quantum expression of the EEP—equivalence between the rest, inertial and gravitational internal energy operators. Validity of the classical EEP does not imply the validity of its quantum formulation, which thus requires independent experimental verification. We propose new tests as well as re-analysing existing experiments, and we discuss to what extent they allow quantum aspects of the EEP to be tested.

Access options

Rent or Buy article

Get time limited or full article access on ReadCube.

from$8.99

All prices are NET prices.

Fig. 1: Testing quantum formulation of the WEP.
Fig. 2: Internal state oscillations for testing quantum formulation of LLI and LPI.
Fig. 3: Interferometric test of the quantum formulation of LPI.

References

  1. 1.

    Einstein, A. Über das Relativitätsprinzip und die aus demselben gezogenen Folgerungen. Jb. Radioakt. 4, 411–462 (1907).

    Google Scholar 

  2. 2.

    Philoponus, J. Corollaries on Place and Void (transl. Furley, D.) (Cornell Univ. Press, Ithaca, NY, 1987). .

  3. 3.

    Will, C. M. The confrontation between general relativity and experiment. Living Rev. Relativ. 17, 4 (2014).

    ADS  Article  Google Scholar 

  4. 4.

    Rosi, G. et al. Quantum test of the equivalence principle for atoms in coherent superposition of internal energy states. Nat. Commun. 8, 15529 (2017).

    ADS  Article  Google Scholar 

  5. 5.

    Zych, M., Costa, F., Pikovski, I. & Brukner, Č. Quantum interferometric visibility as a witness of general relativistic proper time. Nat. Commun. 2, 505 (2011).

    ADS  Article  Google Scholar 

  6. 6.

    Pikovski, I., Zych, M., Costa, F. & Brukner, Č. Universal decoherence due to gravitational time dilation. Nat. Phys. 11, 668–672 (2015).

    Article  Google Scholar 

  7. 7.

    Zych, M. Quantum Systems under Gravitational Time Dilation (Springer, Cham, 2017).

    Google Scholar 

  8. 8.

    Reinhardt, S. et al. Test of relativistic time dilation with fast optical atomic clocks at different velocities. Nat. Phys. 3, 861–864 (2007).

    Article  Google Scholar 

  9. 9.

    Chou, C.-W., Hume, D., Rosenband, T. & Wineland, D. Optical clocks and relativity. Science 329, 1630–1633 (2010).

    ADS  Article  Google Scholar 

  10. 10.

    Zych, M., Costa, F., Pikovski, I., Ralph, T. C. & Brukner, Č. General relativistic effects in quantum interference of photons, classical and quantum. Gravity 29, 224010 (2012).

    MathSciNet  Article  Google Scholar 

  11. 11.

    Zych, M., Pikovski, I., Costa, F. & Brukner, Č. General relativistic effects in quantum interference of ‘clocks’. J. Phys. Conf. Ser. 723, 012044 (2016).

    Article  Google Scholar 

  12. 12.

    Bushev, P. A., Cole, J. H., Sholokhov, D., Kukharchyk, N. & Zych, M. Single electron relativistic clock interferometer. New J. Phys. 18, 093050 (2016).

    ADS  Article  Google Scholar 

  13. 13.

    Pikovski, I., Zych, M., Costa, F. & Brukner, Č. Time dilation in quantum systems and decoherence. New J. Phys. 19, 025011 (2017).

    ADS  Article  Google Scholar 

  14. 14.

    Einstein, A. Ist die Trägheit eines Körpers von seinem Energieinhalt abhängig? Ann. Phys. 323, 639–641 (1905).

    Article  Google Scholar 

  15. 15.

    Einstein, A. Über den Einfluss der Schwerkraft auf die Ausbreitung des Lichtes. Ann. Phys. 35, 898–908 (1911).

    Article  Google Scholar 

  16. 16.

    Rainville, S. et al. World Year of Physics: A direct test of E = mc 2. Nature 438, 1096–1097 (2005).

    ADS  Article  Google Scholar 

  17. 17.

    Aguilera, D. et al. STE–QUEST—test of the universality of free fall using cold atom interferometry. Class. Quantum Gravity 31, 115010 (2014).

    ADS  Article  Google Scholar 

  18. 18.

    Schlippert, D. et al. Quantum test of the universality of free fall. Phys. Rev. Lett. 112, 203002 (2014).

    ADS  Article  Google Scholar 

  19. 19.

    Altschul, B. et al. Quantum tests of the Einstein equivalence principle with the STE–QUEST space mission. Adv. Space Res. 55, 501–524 (2015).

    ADS  Article  Google Scholar 

  20. 20.

    Williams, J., wey Chiow, S., Yu, N. & Müller, H. Quantum test of the equivalence principle and space-time aboard the International Space Station. New J. Phys. 18, 025018 (2016).

    ADS  Article  Google Scholar 

  21. 21.

    Fray, S., Diez, C. A., Hänsch, T. W. & Weitz, M. Atomic interferometer with amplitude gratings of light and its applications to atom based tests of the equivalence principle. Phys. Rev. Lett. 93, 240404 (2004).

    ADS  Article  Google Scholar 

  22. 22.

    Charman, A. et al. Description and first application of a new technique to measure the gravitational mass of antihydrogen. Nat. Commun. 4, 1785 (2013).

    Article  Google Scholar 

  23. 23.

    Colladay, D. & Kostelecký, V. A. CPT violation and the standard model. Phys. Rev. D 55, 6760–6774 (1997).

    ADS  Article  Google Scholar 

  24. 24.

    Drummond, I. T. Quantum field theory in a multimetric background. Phys. Rev. D 88, 025009 (2013).

    ADS  Article  Google Scholar 

  25. 25.

    Lightman, A. P. & Lee, D. L. Restricted proof that the weak equivalence principle implies the Einstein equivalence principle. Phys. Rev. D 8, 364–376 (1973).

    ADS  Article  Google Scholar 

  26. 26.

    Haugan, M. P. Energy conservation and the principle of equivalence. Ann. Phys. 118, 156–186 (1979).

    ADS  Article  Google Scholar 

  27. 27.

    Lämmerzahl, C. Quantum tests of the foundations of general relativity. Class. Quantum Grav. 15, 13–27 (1998).

    ADS  MathSciNet  Article  Google Scholar 

  28. 28.

    Kostelecky, V. & Lane, C. Nonrelativistic quantum Hamiltonian for Lorentz violation. J. Math. Phys. 40, 6245–6253 (1999).

    ADS  MathSciNet  Article  Google Scholar 

  29. 29.

    Hohensee, M. A., Müller, H. & Wiringa, R. Equivalence principle and bound kinetic energy. Phys. Rev. Lett. 111, 151102 (2013).

    ADS  Article  Google Scholar 

  30. 30.

    Kostelecký, V. A. & Vargas, A. J. Lorentz and CPT tests with hydrogen, antihydrogen, and related systems. Phys. Rev. D 92, 056002 (2015).

    ADS  Article  Google Scholar 

  31. 31.

    Gasperini, M. Testing the principle of equivalence with neutrino oscillations. Phys. Rev. D 38, 2635–2657 (1988).

    ADS  Article  Google Scholar 

  32. 32.

    Mann, R. & Sarkar, U. Test of the equivalence principle from neutrino oscillation experiments. Phys. Rev. Lett. 76, 865–868 (1996).

    ADS  Article  Google Scholar 

  33. 33.

    Alan Kostelecký, V. & Mewes, M. Lorentz and CPT violation in neutrinos. Phys. Rev. D 69, 016005 (2004).

    ADS  Article  Google Scholar 

  34. 34.

    Bonder, Y. et al. Testing the equivalence principle with unstable particles. Phys. Rev. D 87, 125021 (2013).

    ADS  Article  Google Scholar 

  35. 35.

    Nordtvedt, K. Jr. Quantitative relationship between clock gravitational ‘red-shift’ violations and nonuniversality of free-fall rates in nonmetric theories of gravity. Phys. Rev. D 11, 245–247 (1975).

    ADS  Article  Google Scholar 

  36. 36.

    Storey, P. & Cohen-Tannoudji, C. The Feynman path integral approach to atomic interferometry. A tutorial. J. Phys. II 4, 1999–2027 (1994).

    Google Scholar 

  37. 37.

    Müntinga, H. et al. Interferometry with Bose–Einstein condensates in microgravity. Phys. Rev. Lett. 110, 093602 (2013).

    ADS  Article  Google Scholar 

  38. 38.

    Greenberger, D. M. The role of equivalence in quantum mechanics. Ann. Phys. 47, 116–126 (1968).

    ADS  Article  Google Scholar 

  39. 39.

    Davies, P. C. W. & Fang, J. Quantum theory and the equivalence principle. Proc. R. Soc. Lond. A 381, 469–478 (1982).

    ADS  Article  Google Scholar 

  40. 40.

    Viola, L. & Onofrio, R. Testing the equivalence principle through freely falling quantum objects. Phys. Rev. D 55, 455–462 (1997).

    ADS  Article  Google Scholar 

  41. 41.

    Lämmerzahl, C. On the equivalence principle in quantum theory. General Relativ. Gravit. 28, 1043–1070 (1996).

    ADS  MathSciNet  Article  Google Scholar 

  42. 42.

    Alvarez, C. & Mann, R. Testing the equivalence principle in the quantum regime. General Relativ. Gravit. 29, 245–250 (1997).

    ADS  MathSciNet  Article  Google Scholar 

  43. 43.

    Adunas, G., Rodriguez-Milla, E. & Ahluwalia, D. V. Probing quantum violations of the equivalence principle. General Relativ. Gravit. 33, 183–194 (2001).

    ADS  MathSciNet  Article  Google Scholar 

  44. 44.

    Davies, P. C. W. Quantum mechanics and the equivalence principle. Class. Quantum Grav. 21, 2761–2772 (2004).

    ADS  MathSciNet  Article  Google Scholar 

  45. 45.

    Kajari, E. & et al. Inertial and gravitational mass in quantum mechanics. Appl. Phys. B 100, 43–60 (2010).

    ADS  Article  Google Scholar 

  46. 46.

    Okon, E. & Callender, C. Does quantum mechanics clash with the equivalence principle and does it matter? Eur. J. Phil. Sci. 1, 133–145 (2011).

    MathSciNet  Article  Google Scholar 

  47. 47.

    Herrmann, S. et al. Testing the equivalence principle with atomic interferometry. Class. Quantum Grav. 29, 184003 (2012).

    ADS  Article  Google Scholar 

  48. 48.

    Nobili, A. M. et al. On the universality of free fall, the equivalence principle, and the gravitational redshift. Am. J. Phys. 81, 527–536 (2013).

    ADS  Article  Google Scholar 

  49. 49.

    Amelino-Camelia, G., Ellis, J., Mavromatos, N., Nanopoulos, D. V. & Sarkar, S. Tests of quantum gravity from observations of γ-ray bursts. Nature 393, 763–765 (1998).

    ADS  Article  Google Scholar 

  50. 50.

    Maartens, R. & Koyama, K. Brane-world gravity. Living Rev. Relativ. 13, 5 (2010).

    ADS  Article  Google Scholar 

  51. 51.

    Damour, T. Theoretical aspects of the equivalence principle. Class. Quantum Grav. 29, 184001 (2012).

    ADS  MathSciNet  Article  Google Scholar 

  52. 52.

    Lebed, A. G. Inequivalence between gravitational mass and energy due to quantum effects at microscopic and macroscopic levels. Int. J. Mod. Phys. D 26, 1730022 (2017).

    ADS  MathSciNet  Article  Google Scholar 

  53. 53.

    Birrell, N. D. & Davies, P. C. W. Quantum Fields in Curved Space (Cambridge Univ. Press, Cambridge, 1982).

    Google Scholar 

  54. 54.

    Schlamminger, S., Choi, K.-Y., Wagner, T., Gundlach, J. & Adelberger, E. Test of the equivalence principle using a rotating torsion balance. Phys. Rev. Lett. 100, 041101 (2008).

    ADS  Article  Google Scholar 

  55. 55.

    Orlando, P. J., Mann, R. B., Modi, K. & Pollock, F. A. A test of the equivalence principle(s) for quantum superpositions. Class. Quantum Grav. 33, 19LT01 (2016).

    MathSciNet  Article  Google Scholar 

  56. 56.

    Geiger, R. & Trupke, M. Proposal for a quantum test of the weak equivalence principle with entangled atomic species. Phys. Rev. Lett. 120, 043602 (2018).

    ADS  Article  Google Scholar 

Download references

Acknowledgements

We thank I. Pikovski and F. Costa for comments on early drafts of this manuscript; and C. Kiefer, D. Giulini and G. Tino for discussions. We acknowledge support from the ARC Centre EQuS CE110001013, the University of Queensland (UQ Fellowship grant no. 2016000089), the Templeton World Charity Foundation (TWCF 0064/AB38), the ÖAW Innovationsfonds ‘Quantum Regime of Gravitational Source Masses’, the Doctoral Programme CoQuS and the research platform TURIS. This publication was made possible through the support of a grant from the John Templeton Foundation. The opinions expressed in this publication are those of the authors and do not necessarily reflect the views of the John Templeton Foundation. M.Z. acknowledges the traditional owners of the land on which the University of Queensland is situated, the Turrbal and Jagera people.

Author information

Affiliations

Authors

Contributions

M.Z. and Č.B. contributed to all aspects of the research, with the leading input from M.Z.

Corresponding author

Correspondence to Magdalena Zych.

Ethics declarations

Competing interests

The authors declare no competing interests.

Additional information

Publisher’s note: Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Supplementary information

Supplementary Information

Supplementary Notes and References

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Zych, M., Brukner, Č. Quantum formulation of the Einstein equivalence principle. Nature Phys 14, 1027–1031 (2018). https://doi.org/10.1038/s41567-018-0197-6

Download citation

Further reading

Search

Quick links

Sign up for the Nature Briefing newsletter for a daily update on COVID-19 science.
Get the most important science stories of the day, free in your inbox. Sign up for Nature Briefing