Advertisement

Setting the Stage: Review of Previous Results

  • Sumanta ChakrabortyEmail author
Chapter
Part of the Springer Theses book series (Springer Theses)

Abstract

In this chapter we will review some earlier results, which will be helpful for the later parts of the thesis. This includes some interesting results in general relativity, definition of Noether current and gravitational momentum etc. We have also reviewed Lanczos-Lovelock models of gravity as it will find extensive use in subsequent chapters. We have also provided a discussion on a particular coordinate system adapted to null surfaces, known as Gaussian Null Coordinates, which will be very useful for our thermodynamic considerations.

References

  1. 1.
    T. Padmanabhan, Gravitation: Foundations and Frontiers (Cambridge University Press, Cambridge, UK, 2010)CrossRefzbMATHGoogle Scholar
  2. 2.
    M.V. Ostrogradsky, Memoires de lAcademie Imperiale des Science de Saint-Petersbourg, 4, 385 (1850)Google Scholar
  3. 3.
    J. York, Role of conformal three geometry in the dynamics of gravitation. Phys. Rev. Lett. 28, 1082–1085 (1972)ADSCrossRefGoogle Scholar
  4. 4.
    G. Gibbons, S. Hawking, Action integrals and partition functions in quantum gravity. Phys. Rev. D 15, 2752–2756 (1977)ADSCrossRefGoogle Scholar
  5. 5.
    J. Charap, J. Nelson, Surface integrals and the gravitational action. J. Phys. A: Math. Gen. 16 1661 (1983)Google Scholar
  6. 6.
    C. Lanczos, Z. Phys. 73, 147 (1932)Google Scholar
  7. 7.
    C. Lanczos, Electricity as a natural property of Riemannian geometry. Rev. Mod. Phys. 39, 716–736 (1932)CrossRefzbMATHGoogle Scholar
  8. 8.
    C. Lanczos, Z. Phys. 39, 842 (1938)Google Scholar
  9. 9.
    C. Lanczos, A remarkable property of the Riemann-Christoffel tensor in four dimensions. Annals Math. 39, 842–850 (1938)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    D. Lovelock, The Einstein tensor and its generalizations. J. Math. Phys. 12, 498–501 (1971)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    K. Parattu, B.R. Majhi, T. Padmanabhan, Structure of the gravitational action and its relation with horizon thermodynamics and emergent gravity paradigm. Phys. Rev. D 87, 124011 (Jun, 2013). arXiv:gr-qc/1303.1535 [gr-qc], doi: 10.1103/PhysRevD.87.124011
  12. 12.
    A. Eddington, The Mathematical Theory of Relativity, 2nd edn. (Cambridge University Press, Cambridge, UK, 1924)zbMATHGoogle Scholar
  13. 13.
    E. Schrodinger, Space-Time Structure (Cambridge Science Classics. Cambridge University Press, Cambridge, UK, 1950)zbMATHGoogle Scholar
  14. 14.
    A. Einstein, B. Kaufman, A new form of the general relativistic field equations. Annals Math. 62, 128–138 (1955)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    A. Einstein, B. Kaufman, A new form of the general relativistic field equations. Annals Math. 62, 128–138 (1955), http://www.jstor.org/stable/2007103
  16. 16.
    T. Padmanabhan, Momentum density of spacetime and the gravitational dynamics. arXiv:1506.03814 [gr-qc]
  17. 17.
    T. Padmanabhan, Holographic gravity and the surface term in the Einstein-Hilbert action. Braz. J. Phys. 35, 362–372 (2005). arXiv:gr-qc/0412068 [gr-qc]
  18. 18.
    A. Mukhopadhyay, T. Padmanabhan, Holography of gravitational action functionals. Phys. Rev. D 74, 124023 (2006). arXiv:hep-th/0608120 ADSMathSciNetCrossRefGoogle Scholar
  19. 19.
    S. Kolekar, T. Padmanabhan, Holography in action. Phys. Rev. D 82, 024036 (2010). arXiv:1005.0619 [gr-qc]ADSCrossRefGoogle Scholar
  20. 20.
    C.W. Misner, K.S. Thorne, J.A. Wheeler, Gravitation, 3rd edn. (W. H. Freeman and Company, 1973)Google Scholar
  21. 21.
    H. Goldstein, C. Poole, J. Safko, Classical Mechanics. 3rd edn. (Pearson Education, 2007)Google Scholar
  22. 22.
    A. Palatini, Deduzione invariantiva delle equazioni gravitazionali dal principio di hamilton. Rend. Circ. Mat. Palermo 43, 203–212 (1919)CrossRefzbMATHGoogle Scholar
  23. 23.
    T. Padmanabhan, General relativity from a thermodynamic perspective. Gen. Rel. Grav. 46, 1673 (2014). arXiv:1312.3253 [gr-qc]ADSMathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    N. Dadhich, Characterization of the Lovelock gravity by Bianchi derivative. Pramana. 74, 875–882 (2010). arXiv:0802.3034 [gr-qc]
  25. 25.
    D.G. Boulware, S. Deser, String generated gravity models. Phys. Rev. Lett. 55, 2656 (1985)ADSCrossRefGoogle Scholar
  26. 26.
    L. Randall, R. Sundrum, A Large mass hierarchy from a small extra dimension. Phys. Rev. Lett. 83, 3370–3373 (1999). arXiv:hep-ph/9905221 [hep-ph]
  27. 27.
    P. Horava, E. Witten, Eleven-dimensional supergravity on a manifold with boundary. Nucl. Phys. B 475, 94–114 (1996). arXiv:hep-th/9603142 [hep-th]
  28. 28.
    N. Arkani-Hamed, S. Dimopoulos, G. Dvali, The hierarchy problem and new dimensions at a millimeter. Phys. Lett. B 429, 263–272 (1998). arXiv:hep-ph/9803315 [hep-ph]
  29. 29.
    T. Padmanabhan, D. Kothawala, Lanczos-Lovelock models of gravity. Phys. Rept. 531, 115–171 (2013). arXiv:1302.2151 [gr-qc]
  30. 30.
    A. Paranjape, S. Sarkar, T. Padmanabhan, Thermodynamic route to field equations in Lancos-Lovelock gravity. Phys. Rev. D 74, 104015 (2006). arXiv:hep-th/0607240 [hep-th]ADSMathSciNetCrossRefGoogle Scholar
  31. 31.
    A. Yale, T. Padmanabhan, Structure of Lanczos-Lovelock Lagrangians in Critical Dimensions. Gen. Rel. Grav. 43, 1549–1570 (2011). arXiv:1008.5154 [gr-qc]
  32. 32.
    N. Kiriushcheva, S.V. Kuzmin, On Hamiltonian formulation of the Einstein-Hilbert action in two dimensions. Mod. Phys. Lett. A 21, 899–906 (2006). arXiv:hep-th/0510260 [hep-th]
  33. 33.
    T. Padmanabhan, Some aspects of field equations in generalised theories of gravity. Phys. Rev. D 84, 124041 (2011). arXiv:1109.3846 [gr-qc]ADSCrossRefGoogle Scholar
  34. 34.
    T. Padmanabhan, A physical interpretation of gravitational field equations. AIP Conf. Proc. 1241, 93–108 (2010). arXiv:0911.1403 [gr-qc]
  35. 35.
    R.M. Wald, Black hole entropy is the Noether charge. Phys. Rev. D 48, 3427–3431 (1993). arXiv:gr-qc/9307038 [gr-qc]
  36. 36.
    T. Padmanabhan, Thermodynamical aspects of gravity: new insights. Rept. Prog. Phys. 73, 046901 (2010). arXiv:0911.5004 [gr-qc]ADSCrossRefGoogle Scholar
  37. 37.
    B.R. Majhi, T. Padmanabhan, Noether current, horizon virasoro algebra and entropy. Phys. Rev. D 85, 084040 (2012). arXiv:1111.1809 [gr-qc]ADSCrossRefGoogle Scholar
  38. 38.
    V. Iyer, R.M. Wald, Some properties of Noether charge and a proposal for dynamical black hole entropy. Phys. Rev. D 50, 846–864 (1994). arXiv:gr-qc/9403028 [gr-qc]
  39. 39.
    R.M. Wald, A. Zoupas, A general definition of ’conserved quantities’ in general relativity and other theories of gravity. Phys. Rev. D 61, 084027 (2000). arXiv:gr-qc/9911095 [gr-qc]ADSMathSciNetCrossRefzbMATHGoogle Scholar
  40. 40.
    A. Strominger, C. Vafa, Microscopic origin of the Bekenstein-Hawking entropy. Phys. Lett. B 379, 99–104 (1996). arXiv:hep-th/9601029 [hep-th]
  41. 41.
    A. Ashtekar, J. Baez, A. Corichi, K. Krasnov, Quantum geometry and black hole entropy. Phys. Rev. Lett. 80, 904–907 (1998). arXiv:gr-qc/9710007 [gr-qc]
  42. 42.
    J.M. Garcia-Islas, BTZ black hole entropy: a spin foam model description. Class. Quant. Grav. 25, 245001 (2008). arXiv:0804.2082 [gr-qc]ADSMathSciNetCrossRefzbMATHGoogle Scholar
  43. 43.
    L. Bombelli, R.K. Koul, J. Lee, R.D. Sorkin, A quantum source of entropy for black holes. Phys. Rev. D 34, 373–383 (1986)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  44. 44.
    B.R. Majhi, T. Padmanabhan, Thermality and heat content of horizons from infinitesimal coordinate transformations. arXiv:1302.1206 [gr-qc]
  45. 45.
    V. Moncrief, J. Isenberg, Symmetries of cosmological cauchy horizons. Commun. Math. Phys. 89(3) 387–413 (1983). doi: 10.1007/BF01214662
  46. 46.
    E.M. Morales, On a second law of black hole mechanics in a higher derivative theory of gravity (2008), http://www.theorie.physik.uni-goettingen.de/forschung/qft/theses/dipl/Morfa-Morales.pdf
  47. 47.
    P. Davies, Scalar particle production in Schwarzschild and Rindler metrics. J. Phys. A 8, 609–616 (1975)ADSCrossRefGoogle Scholar
  48. 48.
    W. Unruh, Notes on black hole evaporation. Phys. Rev. D 14, 870 (1976)ADSCrossRefGoogle Scholar

Copyright information

© Springer International Publishing AG 2017

Authors and Affiliations

  1. 1.Department of Theoretical PhysicsIndian Association for the Cultivation of ScienceJadavpur, KolkataIndia

Personalised recommendations