Advertisement

Conformal transformation route to gravity’s rainbow

  • Miao He
  • Ping Li
  • Zi-Liang Wang
  • Jia-Cheng Ding
  • Jian-Bo Deng
Research Article

Abstract

Conformal transformation as a mathematical tool has been used in many areas of gravitational physics. In this paper, we consider gravity’s rainbow, in which the metric can be treated as a conformal rescaling of the original metric. By using the conformal transformation technique, we get a specific form of a modified Newton’s constant and cosmological constant in gravity’s rainbow, which implies that the total vacuum energy is dependent on probe energy. Moreover, the result shows that Einstein gravity’s rainbow can be described by energy-dependent \(f(E,\tilde{R})\) gravity. At last, we study the f(R) gravity, when gravity’s rainbow is considered, which can also be described as energy-dependent \(\tilde{f}(E,\tilde{R})\) gravity.

Keywords

Conformal transformation Gravity’s rainbow f(R) Gravity 

Notes

Acknowledgements

We would like to thank the National Natural Science Foundation of China (Grant No. 11571342) for supporting us on this work.

References

  1. 1.
    Colladay, D., Kostelecký, V.A.: Lorentz-violating extension of the standard model. Phys. Rev. D 58, 116002 (1998)ADSCrossRefGoogle Scholar
  2. 2.
    Coleman, S., Glashow, S.L.: High-energy tests of Lorentz invariance. Phys. Rev. D 59, 116008 (1999)ADSCrossRefGoogle Scholar
  3. 3.
    Amelino-Camelia, G., Piran, T.: Planck-scale deformation of Lorentz symmetry as a solution to the ultrahigh energy cosmic ray and the TeV-photon paradoxes. Phys. Rev. D 64, 036005 (2001)ADSCrossRefGoogle Scholar
  4. 4.
    Jacobson, T., Liberati, S., Mattingly, D.: TeV astrophysics constraints on Planck scale Lorentz violation. Phys. Rev. D 66, 081302 (2002)ADSCrossRefGoogle Scholar
  5. 5.
    Myers, R.C., Pospelov, M.: Ultraviolet modifications of dispersion relations in effective field theory. Phys. Rev. Lett. 90, 211601 (2003)ADSMathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    Jacobson, T., Liberati, S., Mattingly, D., Stecker, F.W.: New limits on planck scale lorentz violation in QED. Phys. Rev. Lett. 93, 021101 (2004)ADSCrossRefGoogle Scholar
  7. 7.
    Magueijo, J., Smolin, L.: Lorentz invariance with an invariant energy scale. Phys. Rev. Lett. 88, 190403 (2002)ADSCrossRefGoogle Scholar
  8. 8.
    Magueijo, J., Smolin, L.: Generalized Lorentz invariance with an invariant energy scale. Phys. Rev. D 67, 044017 (2003)ADSMathSciNetCrossRefGoogle Scholar
  9. 9.
    Amelino-Camelia, G.: Testable scenario for relativity with minimum length. Phys. Lett. B 510(1), 255–263 (2001)ADSCrossRefMATHGoogle Scholar
  10. 10.
    Amelino-camelia, G.: Relativity in spacetimes with short-distance structure governed by an observer-independent (Planckian) length scale. Int. J. Mod. Phys. D 11(01), 35–59 (2002)ADSMathSciNetCrossRefMATHGoogle Scholar
  11. 11.
    Magueijo, J., Smolin, L.: Gravity’s rainbow. Class. Quantum. Gravit. 21(7), 1725 (2004)ADSMathSciNetCrossRefMATHGoogle Scholar
  12. 12.
    Garattini, R., Mandanici, G.: Modified dispersion relations lead to a finite zero point gravitational energy. Phys. Rev. D 83, 084021 (2011)ADSCrossRefGoogle Scholar
  13. 13.
    Li, H., Ling, Y., Han, X.: Modified (A)dS Schwarzschild black holes in rainbow spacetime. Class. Quantum Gravit. 26(6), 065004 (2009)ADSMathSciNetCrossRefMATHGoogle Scholar
  14. 14.
    Gangopadhyay, S., Dutta, A., Faizal, M.: Constraints on the generalized uncertainty principle from black-hole thermodynamics. EPL 112(2), 20006 (2015)ADSCrossRefGoogle Scholar
  15. 15.
    Amelino-Camelia, G., Arzano, M., Ling, Y., Mandanici, G.: Black-hole thermodynamics with modified dispersion relations and generalized uncertainty principles. Class. Quantum Gravit. 23(7), 2585 (2006)ADSMathSciNetCrossRefMATHGoogle Scholar
  16. 16.
    Ling, Y., Li, X., Zhang, H.: Thermodynamics of modified black holes from gravity’s rainbow. Mod. Phys. Lett. A 22(36), 2749–2756 (2007)ADSMathSciNetCrossRefMATHGoogle Scholar
  17. 17.
    Galán, P., Marugán, G.A.M.: Entropy and temperature of black holes in a gravity’s rainbow. Phys. Rev. D 74, 044035 (2006)ADSMathSciNetCrossRefGoogle Scholar
  18. 18.
    Hendi, S.H., Faizal, M., Panah, B.E., Panahiyan, S.: Charged dilatonic black holes in gravity’s rainbow. Eur. Phys. J. C 76(5), 296 (2016)ADSCrossRefGoogle Scholar
  19. 19.
    Ali, A.F.: Black hole remnant from gravity’s rainbow. Phys. Rev. D 89, 104040 (2014)ADSCrossRefGoogle Scholar
  20. 20.
    Adler, Ronald J., Chen, Pisin, Santiago, David I.: The generalized uncertainty principle and black hole remnants. Gen. Relativ. Gravit. 33(12), 2101–2108 (2001)ADSMathSciNetCrossRefMATHGoogle Scholar
  21. 21.
    Ling, Y.: Rainbow universe. JCAP 2007(08), 017 (2007)ADSCrossRefGoogle Scholar
  22. 22.
    Awad, A., Ali, A.F., Majumder, B.: Nonsingular rainbow universes. JCAP 2013(10), 052 (2013)CrossRefGoogle Scholar
  23. 23.
    Hendi, S.H., Momennia, M., Panah, B.E., Faizal, M.: Nonsingular universes in Gauss–Bonnet gravity’s rainbow. ApJ 827(2), 153 (2016)ADSCrossRefGoogle Scholar
  24. 24.
    Majumder, B.: Singularity free rainbow universe. Int. J. Mod. Phys. D 22(12), 1342021 (2013)ADSMathSciNetCrossRefMATHGoogle Scholar
  25. 25.
    Olmo, G.J.: Palatini actions and quantum gravity phenomenology. JCAP 2011(10), 018 (2011)CrossRefGoogle Scholar
  26. 26.
    Ling, Y., Wu, Q.: The big bounce in rainbow universe. Phys. Lett. B 687(2), 103–109 (2010)ADSCrossRefGoogle Scholar
  27. 27.
    Hendi, S.H., Faizal, M.: Black holes in Gauss–Bonnet gravity’s rainbow. Phys. Rev. D 92, 044027 (2015)ADSMathSciNetCrossRefGoogle Scholar
  28. 28.
    Hendi, S.H., Panahiyan, S., Panah, B.E., Faizal, M., Momennia, M.: Critical behavior of charged black holes in Gauss–Bonnet gravity’s rainbow. Phys. Rev. D 94, 024028 (2016)ADSMathSciNetCrossRefGoogle Scholar
  29. 29.
    Hendi, S.H., Panah, B.E., Panahiyan, S.: Topological charged black holes in massive gravity’s rainbow and their thermodynamical analysis through various approaches. Phys. Lett. B B769, 191–201 (2017)ADSCrossRefMATHGoogle Scholar
  30. 30.
    Hendi, S.H., Panahiyan, S., Upadhyay, S., Panah, B.E.: Charged BTZ black holes in the context of massive gravity’s rainbow. Phys. Rev. D 95, 084036 (2017)ADSCrossRefGoogle Scholar
  31. 31.
    Hendi, S. H., Panah, B. E., Panahiyan, S., Momennial, M.: \({F(R)}\) gravity’s rainbow and its Einstein counterpart. Adv. High Energy Phys., 2016:9813582, (2016)Google Scholar
  32. 32.
    Garattini, R.: Distorting general relativity: gravity’s rainbow and \(f({R})\) theories at work. JCAP 2013(06), 017 (2013)CrossRefGoogle Scholar
  33. 33.
    Garattini, R., Saridakis, E.N.: Gravity’s rainbow: a bridge towards Hořava–Lifshitz gravity. Eur. Phys. J. C 75(7), 343 (2015)ADSCrossRefGoogle Scholar
  34. 34.
    Faraoni, V., Gunzig, E., Nardone, P.: Conformal transformations in classical gravitational theories and in cosmology. Fund. Cosmic Phys. 20, 121 (1999)ADSGoogle Scholar
  35. 35.
    Dicke, R.H.: Mach’s principle and invariance under transformation of units. Phys. Rev. 125, 2163–2167 (1962)ADSMathSciNetCrossRefMATHGoogle Scholar
  36. 36.
    Dabrowski, M.P., Garecki, J., Blaschke, D.B.: Conformal transformations and conformal invariance in gravitation. Ann. der Phys. 18(1), 13–32 (2009)ADSMathSciNetCrossRefMATHGoogle Scholar
  37. 37.
    Hendi, S.H., Talezadeh, M.S.: Nonlinearly charged dilatonic black holes and their Brans–Dicke counterpart: energy dependent spacetime. Gen. Relativ. Gravit 49(1), 12 (2016)ADSMathSciNetCrossRefMATHGoogle Scholar
  38. 38.
    Magnano, G., Sokołowski, L.M.: Physical equivalence between nonlinear gravity theories and a general-relativistic self-gravitating scalar field. Phys. Rev. D 50, 5039–5059 (1994)ADSMathSciNetCrossRefGoogle Scholar
  39. 39.
    Kimberly, D., Magueijo, J., Medeiros, J.: Nonlinear relativity in position space. Phys. Rev. D 70, 084007 (2004)ADSMathSciNetCrossRefGoogle Scholar
  40. 40.
    De Felice, A., Tsujikawa, S.: \(f({R})\) Theories. Living Rev. Rel. 13(1), 3 (2010)CrossRefMATHGoogle Scholar
  41. 41.
    Capozziello, S., De Felice, A.: \(f({R})\) cosmology from Noether’s symmetry. JCAP 2008(08), 016 (2008)ADSMathSciNetCrossRefGoogle Scholar
  42. 42.
    Sotiriou, T.P., Faraoni, V.: \(f({R})\) theories of gravity. Rev. Mod. Phys. 82, 451–497 (2010)ADSCrossRefMATHGoogle Scholar
  43. 43.
    Alexander, S., Magueijo, J.: Non-commutative geometry as a realization of varying speed of light cosmology. arXiv preprint hep-th/0104093 (2001)Google Scholar
  44. 44.
    Amendola, L., Polarski, D., Tsujikawa, S.: Are \(f({R})\) dark energy models cosmologically viable? Phys. Rev. Lett. 98, 131302 (2007)ADSMathSciNetCrossRefMATHGoogle Scholar
  45. 45.
    Amendola, L., Gannouji, R., Polarski, D., Tsujikawa, S.: Conditions for the cosmological viability of \(f({R})\) dark energy models. Phys. Rev. D 75, 083504 (2007)ADSCrossRefGoogle Scholar
  46. 46.
    Carroll, S.M., Duvvuri, V., Trodden, M., Turner, M.S.: Is cosmic speed-up due to new gravitational physics? Phys. Rev. D 70, 043528 (2004)ADSCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  • Miao He
    • 1
  • Ping Li
    • 1
  • Zi-Liang Wang
    • 1
  • Jia-Cheng Ding
    • 1
  • Jian-Bo Deng
    • 1
  1. 1.Institute of Theoretical PhysicsLanZhou UniversityLanzhouPeople’s Republic of China

Personalised recommendations