Advertisement

Minimal length, Friedmann equations and maximum density

  • Adel Awad
  • Ahmed Farag AliEmail author
Open Access
Article

Abstract

Inspired by Jacobson’s thermodynamic approach [4], Cai et al. [5, 6] have shown the emergence of Friedmann equations from the first law of thermodynamics. We extend Akbar-Cai derivation [6] of Friedmann equations to accommodate a general entrop-yarea law. Studying the resulted Friedmann equations using a specific entropy-area law, which is motivated by the generalized uncertainty principle (GUP), reveals the existence of a maximum energy density closed to Planck density. Allowing for a general continuous pressure p(ρ, a) leads to bounded curvature invariants and a general nonsingular evolution. In this case, the maximum energy density is reached in a finite time and there is no cosmological evolution beyond this point which leaves the big bang singularity inaccessible from a spacetime prospective. The existence of maximum energy density and a general nonsingular evolution is independent of the equation of state and the spacial curvature k. As an example we study the evolution of the equation of state p = ωρ through its phase-space diagram to show the existence of a maximum energy which is reachable in a finite time.

Keywords

Spacetime Singularities Cosmology of Theories beyond the SM Models of Quantum Gravity 

Notes

Open Access

This article is distributed under the terms of the Creative Commons Attribution License (CC-BY 4.0), which permits any use, distribution and reproduction in any medium, provided the original author(s) and source are credited.

References

  1. [1]
    J.M. Bardeen, B. Carter and S.W. Hawking, The Four laws of black hole mechanics, Commun. Math. Phys. 31 (1973) 161 [INSPIRE].ADSCrossRefzbMATHMathSciNetGoogle Scholar
  2. [2]
    S.W. Hawking, Particle Creation by Black Holes, Commun. Math. Phys. 43 (1975) 199 [Erratum ibid. 46 (1976) 206] [INSPIRE].
  3. [3]
    J.D. Bekenstein, Black holes and entropy, Phys. Rev. D 7 (1973) 2333 [INSPIRE].ADSMathSciNetGoogle Scholar
  4. [4]
    T. Jacobson, Thermodynamics of space-time: The Einstein equation of state, Phys. Rev. Lett. 75 (1995) 1260 [gr-qc/9504004] [INSPIRE].ADSCrossRefzbMATHMathSciNetGoogle Scholar
  5. [5]
    R.-G. Cai and S.P. Kim, First law of thermodynamics and Friedmann equations of Friedmann-Robertson-Walker universe, JHEP 02 (2005) 050 [hep-th/0501055] [INSPIRE].ADSCrossRefMathSciNetGoogle Scholar
  6. [6]
    M. Akbar and R.-G. Cai, Thermodynamic Behavior of Friedmann Equations at Apparent Horizon of FRW Universe, Phys. Rev. D 75 (2007) 084003 [hep-th/0609128] [INSPIRE].ADSGoogle Scholar
  7. [7]
    A.J.M. Medved and E.C. Vagenas, When conceptual worlds collide: The GUP and the BH entropy, Phys. Rev. D 70 (2004) 124021 [hep-th/0411022] [INSPIRE].ADSMathSciNetGoogle Scholar
  8. [8]
    R.J. Adler, P. Chen and D.I. Santiago, The Generalized uncertainty principle and black hole remnants, Gen. Rel. Grav. 33 (2001) 2101 [gr-qc/0106080] [INSPIRE].ADSCrossRefzbMATHMathSciNetGoogle Scholar
  9. [9]
    M. Cavaglia, S. Das and R. Maartens, Will we observe black holes at LHC?, Class. Quant. Grav. 20 (2003) L205 [hep-ph/0305223] [INSPIRE].ADSCrossRefzbMATHMathSciNetGoogle Scholar
  10. [10]
    M. Cavaglia and S. Das, How classical are TeV scale black holes?, Class. Quant. Grav. 21 (2004) 4511 [hep-th/0404050] [INSPIRE].ADSCrossRefzbMATHMathSciNetGoogle Scholar
  11. [11]
    B. Majumder, Black Hole Entropy and the Modified Uncertainty Principle: A heuristic analysis, Phys. Lett. B 703 (2011) 402 [arXiv:1106.0715] [INSPIRE].ADSCrossRefMathSciNetGoogle Scholar
  12. [12]
    A.F. Ali, No Existence of Black Holes at LHC Due to Minimal Length in Quantum Gravity, JHEP 09 (2012) 067 [arXiv:1208.6584] [INSPIRE].ADSCrossRefGoogle Scholar
  13. [13]
    A.F. Ali, H. Nafie and M. Shalaby, Minimal length, maximal energy and black-hole remnants, Europhys. Lett. 100 (2012) 20004.CrossRefGoogle Scholar
  14. [14]
    D. Amati, M. Ciafaloni and G. Veneziano, Can Space-Time Be Probed Below the String Size?, Phys. Lett. B 216 (1989) 41 [INSPIRE].ADSCrossRefGoogle Scholar
  15. [15]
    M. Maggiore, A Generalized uncertainty principle in quantum gravity, Phys. Lett. B 304 (1993) 65 [hep-th/9301067] [INSPIRE].ADSCrossRefGoogle Scholar
  16. [16]
    M. Maggiore, Quantum groups, gravity and the generalized uncertainty principle, Phys. Rev. D 49 (1994) 5182 [hep-th/9305163] [INSPIRE].ADSMathSciNetGoogle Scholar
  17. [17]
    M. Maggiore, The Algebraic structure of the generalized uncertainty principle, Phys. Lett. B 319 (1993) 83 [hep-th/9309034] [INSPIRE].ADSCrossRefMathSciNetGoogle Scholar
  18. [18]
    L.J. Garay, Quantum gravity and minimum length, Int. J. Mod. Phys. A 10 (1995) 145 [gr-qc/9403008] [INSPIRE].ADSCrossRefGoogle Scholar
  19. [19]
    S. Hossenfelder, M. Bleicher, S. Hofmann, J. Ruppert, S. Scherer et al., Collider signatures in the Planck regime, Phys. Lett. B 575 (2003) 85 [hep-th/0305262] [INSPIRE].ADSCrossRefGoogle Scholar
  20. [20]
    C. Bambi and F.R. Urban, Natural extension of the Generalised Uncertainty Principle, Class. Quant. Grav. 25 (2008) 095006 [arXiv:0709.1965] [INSPIRE].ADSCrossRefMathSciNetGoogle Scholar
  21. [21]
    A.F. Ali, S. Das and E.C. Vagenas, Discreteness of Space from the Generalized Uncertainty Principle, Phys. Lett. B 678 (2009) 497 [arXiv:0906.5396] [INSPIRE].ADSCrossRefMathSciNetGoogle Scholar
  22. [22]
    F. Scardigli, Generalized uncertainty principle in quantum gravity from micro-black hole Gedanken experiment, Phys. Lett. B 452 (1999) 39 [hep-th/9904025] [INSPIRE].ADSCrossRefGoogle Scholar
  23. [23]
    A. Kempf, G. Mangano and R.B. Mann, Hilbert space representation of the minimal length uncertainty relation, Phys. Rev. D 52 (1995) 1108 [hep-th/9412167] [INSPIRE].ADSMathSciNetGoogle Scholar
  24. [24]
    A. Kempf, Nonpointlike particles in harmonic oscillators, J. Phys. A 30 (1997) 2093 [hep-th/9604045] [INSPIRE].ADSMathSciNetGoogle Scholar
  25. [25]
    F. Brau, Minimal length uncertainty relation and hydrogen atom, J. Phys. A 32 (1999) 7691 [quant-ph/9905033] [INSPIRE].ADSMathSciNetGoogle Scholar
  26. [26]
    B. Majumder, The Generalized Uncertainty Principle and the Friedmann equations, Astrophys. Space Sci. 336 (2011) 331 [arXiv:1105.2425] [INSPIRE].ADSCrossRefGoogle Scholar
  27. [27]
    J.E. Lidsey, Holographic Cosmology from the First Law of Thermodynamics and the Generalized Uncertainty Principle, Phys. Rev. D 88 (2013) 103519 [arXiv:0911.3286] [INSPIRE].ADSGoogle Scholar
  28. [28]
    R.B. Mann and S.N. Solodukhin, Quantum scalar field on three-dimensional (BTZ) black hole instanton: Heat kernel, effective action and thermodynamics, Phys. Rev. D 55 (1997) 3622 [hep-th/9609085] [INSPIRE].ADSMathSciNetGoogle Scholar
  29. [29]
    R.K. Kaul and P. Majumdar, Logarithmic correction to the Bekenstein-Hawking entropy, Phys. Rev. Lett. 84 (2000) 5255 [gr-qc/0002040] [INSPIRE].ADSCrossRefMathSciNetGoogle Scholar
  30. [30]
    S. Das, P. Majumdar and R.K. Bhaduri, General logarithmic corrections to black hole entropy, Class. Quant. Grav. 19 (2002) 2355 [hep-th/0111001] [INSPIRE].ADSCrossRefzbMATHMathSciNetGoogle Scholar
  31. [31]
    A. Awad, Fixed points and FLRW cosmologies: Flat case, Phys. Rev. D 87 (2013) 103001 [arXiv:1303.2014] [INSPIRE].ADSGoogle Scholar
  32. [32]
    S.A. Hayward, Unified first law of black hole dynamics and relativistic thermodynamics, Class. Quant. Grav. 15 (1998) 3147 [gr-qc/9710089] [INSPIRE].ADSCrossRefzbMATHMathSciNetGoogle Scholar
  33. [33]
    G. Amelino-Camelia, M. Arzano and A. Procaccini, Severe constraints on loop-quantum-gravity energy-momentum dispersion relation from black-hole area-entropy law, Phys. Rev. D 70 (2004) 107501 [gr-qc/0405084] [INSPIRE].ADSMathSciNetGoogle Scholar
  34. [34]
    E.M. Lifshitz, L.P. Pitaevskii and V.B. Berestetskii, Landau-Lifshitz Course of Theoretical Physics, Volume 4: Quantum Electrodynamics, Reed Educational and Professional Publishing (1982).Google Scholar
  35. [35]
    D. Christodoulou, Reversible and irreversible transforations in black hole physics, Phys. Rev. Lett. 25 (1970) 1596 [INSPIRE].ADSCrossRefGoogle Scholar
  36. [36]
    D. Christodoulou and R. Ruffini, Reversible transformations of a charged black hole, Phys. Rev. D 4 (1971) 3552 [INSPIRE].ADSGoogle Scholar
  37. [37]
    C. Adami, The Physics of information, quant-ph/0405005 [INSPIRE].
  38. [38]
    R.-G. Cai, L.-M. Cao and Y.-P. Hu, Corrected Entropy-Area Relation and Modified Friedmann Equations, JHEP 08 (2008) 090 [arXiv:0807.1232] [INSPIRE].ADSCrossRefMathSciNetGoogle Scholar
  39. [39]
    S.H. Strogatz, Nonlinear Dynamics and Chaos, Preseus Books (1994).Google Scholar

Copyright information

© The Author(s) 2014

Open AccessThis article is distributed under the terms of the Creative Commons Attribution 4.0 International License (https://creativecommons.org/licenses/by/4.0), which permits use, duplication, adaptation, distribution, and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

Authors and Affiliations

  1. 1.Center for Theoretical PhysicsBritish University of EgyptSherouk CityEgypt
  2. 2.Centre for Fundamental PhysicsZewail City of Science and TechnologyGizaEgypt
  3. 3.Department of Physics, Faculty of ScienceAin Shams UniversityCairoEgypt
  4. 4.Department of Physics, Faculty of ScienceBenha UniversityBenhaEgypt

Personalised recommendations