On gravitational Stefan-Boltzmann law and Casimir effect in FRW universe


Both Stefan-Boltzmann law and the Casimir effect, in a universe described by the FRW metric with zero curvature, are calculated. These effects are described by Thermo Field Dynamics (TFD). The gravitational energy-momentum tensor is defined in the context of Teleparallel Equivalent to General Relativity (TEGR). Each of the two effects gives a consistent prediction with what is observed on a cosmological scale. One of the effect establishes a minimum range for the deceleration parameter. While another leads to the conclusion that a possible cosmological constant has a very small order of magnitude.

This is a preview of subscription content, access via your institution.


  1. 1.

    Ulhoa, S.C., Santos, A.F., Furtado, T.F., Khanna, F.C.: Adv. High Energy Phys. 2019, 1–6 (2019)

    Article  Google Scholar 

  2. 2.

    Matsubara, T.: Prog. Theor. Phys. 14, 351 (1955)

    ADS  Article  Google Scholar 

  3. 3.

    Ulhoa, S.C., Santos, A.F., Khanna, F.C.: Gen. Relativ. Gravit. 49, 54 (2017)

    ADS  Article  Google Scholar 

  4. 4.

    Acquaviva, G., Iorio, A., Scholtz, M.: Ann. Phys. 387, 317–333 (2017)

    ADS  Article  Google Scholar 

  5. 5.

    Acquaviva, G., Iorio, A., Smaldone, L.: Phys. Rev. D 102, 106002 (2020)

    ADS  MathSciNet  Article  Google Scholar 

  6. 6.

    Maluf, J.W.: Ann. Phys. 525(5), 339–357 (2013)

    MathSciNet  Article  Google Scholar 

  7. 7.

    Ulhoa, S.C., Spaniol, E.P., Gomes, R., Santos, A.F., Santana, A.E.: Adv. High Energy Phys. 2020, 1–9 (2020)

    Article  Google Scholar 

  8. 8.

    Krssak, M., van den Hoogen, R.J., Pereira, J.G., Böhmer, C.G., Coley, A.A.: Class. Quantum Gravit. 36(18), 183001 (2019)

    ADS  Article  Google Scholar 

  9. 9.

    Emtsova, E.D., Petrov, A.N., Toporensky, A.V.: Class. Quantum Gravit. 37(9), 095006 (2020)

    ADS  Article  Google Scholar 

  10. 10.

    Hohmann, M., Jarv, L., Krssak, M., Pfeifer, C.: Phys. Rev. D 100(8), 084002 (2019)

    ADS  MathSciNet  Article  Google Scholar 

  11. 11.

    Maluf, J.W., Ulhoa, S.C., da Rocha-Neto, J.F., Carneiro, F.L.: Class. Quantum Gravit. 37, 067003 (2020)

    ADS  Article  Google Scholar 

  12. 12.

    Schwinger, J.: J. Math. Phys. 2, 407 (1961)

    ADS  MathSciNet  Article  Google Scholar 

  13. 13.

    Schwinger, J.: Lecture Notes Of Brandeis University Summer Institute (1960)

  14. 14.

    Takahashi, Y., Umezawa, H.: Coll. Phenom. 2, 55 (1975)

    Google Scholar 

  15. 15.

    Takahashi, Y., Umezawa, H.: Int. J. Mod. Phys. B 10, 1755 (1996)

    ADS  Article  Google Scholar 

  16. 16.

    Takahashi, Y., Umezawa, H., Matsumoto, H.: Thermofield Dynamics and Condensed States. North-Holland, Amsterdan (1982)

    Google Scholar 

  17. 17.

    Khanna, F.C., Malbouisson, A.P.C., Malboiusson, J.M.C., Santana, A.E.: Themal Quantum Field Theory: Algebraic Aspects and Applications. World Scientific, Singapore (2009)

    Google Scholar 

  18. 18.

    Umezawa, H.: Advanced Field Theory: Micro, Macro and Thermal Physics. AIP, New York (1993)

    Google Scholar 

  19. 19.

    Santana, A.E., Khanna, F.C.: Phys. Lett. A 203, 68 (1995)

    ADS  MathSciNet  Article  Google Scholar 

  20. 20.

    Santana, A.E., Khanna, F.C., Chu, H., Chang, C.: Ann. Phys. 249, 481 (1996)

    ADS  Article  Google Scholar 

  21. 21.

    Khanna, F.C., Malbouisson, A.P.C., Malbouisson, J.M.C., Santana, A.E.: Ann. Phys. 326, 2634 (2011)

    ADS  Article  Google Scholar 

  22. 22.

    Ulhoa, S.C., Santos, A.F., Khanna, F.C.: Int. J. Theor. Phys. p. 1995 (2017)

  23. 23.

    Feeney, S.M., Mortlock, D.J., Dalmasso, N.: Mon. Not. R. Astron. Soc. 476, 3861 (2018)

    ADS  Article  Google Scholar 

  24. 24.

    Pan, S., Sharov, G.S., Yang, W.: Phys. Rev. D 101, 103533 (2020)

    ADS  MathSciNet  Article  Google Scholar 

Download references


This work by A. F. S. is supported by CNPq projects 430194/2018-8 and 313400/2020-2.

Author information



Corresponding author

Correspondence to A. F. Santos.

Additional information

Publisher's Note

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

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Santos, A.F., Ulhoa, S.C., Spaniol, E.P. et al. On gravitational Stefan-Boltzmann law and Casimir effect in FRW universe. Gen Relativ Gravit 53, 54 (2021). https://doi.org/10.1007/s10714-021-02826-y

Download citation


  • Finite temperature
  • TFD
  • Teleparallel gravity
  • Casimir effect
  • FRW universe