1. from Black Hole Entropy to the Cosmological Constant Modified Dispersion Relations: from Black Hole Entropy to the Cosmological Constant Remo Garattini Università di Bergamo I.N.F.N. - Sezione di Milano Benasque 19 September 2011

2. Introduction Part I Black Hole Entropy Part I Black Hole Entropy Part II The Cosmological Constant Part II The Cosmological Constant

3. Part I: Black Hole Entropy 3

4. Black Hole Entropy [J. D. Bekenstein, Phys. Rev. D 7, 949 (1973). S. W. Hawking, Comm. Math. Phys. 43, 199 (1975).] The prescription for assigning a Bekenstein-Hawking entropy to a black hole of surface area A was first inferred in the mid '70s from the formal similarities between black hole dynamics and thermodynamics, combined with Hawking's discovery that black hole radiates thermally with a characteristic (Hawking) temperature 4

5. 5  b(r) is the shape function    (r) is the redshift function Gerard 't Hooft,1 in 1985 considered the statistical thermodynamics of quantum fields in the Hartle-Hawking state (i.e. having the Hawking temperature T H at large radii) propagating on a fixed Schwarzschild background of mass M. To control divergences, 't Hooft introduced a brick wall" with radius slightly larger than the gravitational radius 2MG. He found, in addition to the expected volume-dependent thermodynamical quantities describing hot fields in a nearly at space, additional contributions proportional to the area. These contributions are, however, also proportional to  , where  is the proper distance from the horizon, and thus diverge in the limit  0. For a specic choice of , he recovered the Bekenstein- Hawking formula [G. 't Hooft, Nucl. Phys. B256, 727 (1985).] How it works?? Consider a massless scalar field in the following background

6. 6 From the equation of motion, we can define an r-dependent radial wave number

7. 7 In proximity of r 0

8. 8 Eliminating the brick wall from QFT Procedures Renormalization of Newton's constant [L. Susskind and J. Uglum, Phys. Rev. D50, 2700 (1994). J. L. F. Barbon and R. Emparan, Phys. Rev. D52, 4527 (1995) 1995), hep-th/9502155. E. Winstanley, Phys. Rev. D63, 084013 (2001) 2001), hep-th/0011176.] Pauli-Villars regularization [J.-G. Demers, R. Lafrance and R. C. Myers, Phys. Rev. D52, 2245 (1995), gr- qc/9503003. D. V. Fursaev and S. N. Solodukhin, Phys. Lett. B365, 51 (1996), hep-th/9412020. S. P. Kim, S. K. Kim, K.-S. Soh and J. H. Yee, Int. J. Mod. Phys. A12, 5223 (1997) gr-qc/9607019.]

9. 9 Eliminating the brick wall using Generalized Uncertainty Principle Counting of Quantum States Modified by GUP X. Li, Phys. Lett. B 540, 9 (2002), gr-qc/0204029. Z. Ren, W. Yue-Qin and Z. Li-Chun, Class. Quant. Grav. 20 (2003), 4885. G. Amelino-Camelia, Class.Quant.Grav. 23, 2585 (2006), gr-qc/0506110. G. Amelino-Camelia, Gen.Rel.Grav. 33, 2101 (2001), gr-qc/0106080.

10. 10 Eliminating the brick wall using Modified Dispersion Relations [ R.Garattini P.L.B. B685 (2010) 329 e-Print: arXiv:0902.3927 [gr-qc]] Doubly Special Relativity G. Amelino-Camelia, Int.J.Mod.Phys. D 11, 35 (2002); gr-qc/001205. G. Amelino-Camelia, Phys.Lett. B 510, 255 (2001); hep-th/0012238. Curved Space Proposal  Gravity’s Rainbow [J. Magueijo and L. Smolin, Class. Quant. Grav. 21, 1725 (2004) arXiv:gr-qc/0305055].

11. 11 From the equation of motion, we can define an r-dependent radial wave number

12. 12 In proximity of the throat, we consider the approximate free energy Assumption on  (E/E P )

13. 13 Case a) In the limit  E P >>1 To recover the area law, we have to set This corresponds to a changing of the time variable with respect to the Schwarzschild time. Discrepancy of a factor of 3/2 with the ’t Hooft result

14. In the limit  E P >>1 To recover the area law, we have to set Discrepancy of a factor of 3/2 with the ’t Hooft result In the limit  E P >>1 Case b) with  =3 Case b) with  ≠3 14

15. Part II: The Cosmological Constant 15

16. 16 Wheeler-De Witt Equation B. S. DeWitt, Phys. Rev.160, 1113 (1967).  G ijkl is the super-metric, 8G and  is the cosmological constant  R is the scalar curvature in 3-dim.   can be seen as an eigenvalue   [g ij ] can be considered as an eigenfunction

17. 17 Re-writing the WDW equation Where

18. 18 Solve this infinite dimensional PDE with a Variational Approach  is a trial wave functional of the gaussian type Schrödinger Picture Spectrum of  depending on the metric Energy (Density) Levels Wheeler-De Witt Equation B. S. DeWitt, Phys. Rev.160, 1113 (1967).

19. 19 Canonical Decomposition  h is the trace (spin 0)  (L ij is the gauge part [spin 1 (transverse) + spin 0 (long.)]. [spin 1 (transverse) + spin 0 (long.)]. F.P determinant (ghosts) F.P determinant (ghosts)  h  ij transverse-traceless  graviton (spin 2) M. Berger and D. Ebin, J. Diff. Geom.3, 379 (1969). J. W. York Jr., J. Math. Phys., 14, 4 (1973); Ann. Inst. Henri Poincaré A 21, 319 (1974).

20. 20 We can define an r-dependent radial wave number Standard Regularization

21. 21 Gravity’s Rainbow and the Cosmological Constant R.G. and G.Mandanici, Phys. Rev. D 83, 084021 (2011), arXiv:1102.3803 [gr-qc] ,  parameters Failure of Convergence

22. 22  =1,  =1/3  =1,  =-1/4 A double limit on  

23. 23 Gravity’s Rainbow and the Cosmological Constant

24. 24 Gravity’s Rainbow and the Cosmological Constant

25. 25 Gravity’s Rainbow and the Cosmological Constant

26. 26 Connection with the Noncommutative Approach Connection with the Noncommutative Approach [R.G. & P. Nicolini 1006.4518 [gr-qc] ]

27. 27 Conclusions, Problems and Outlook  Black Hole Entropy can be computed without a brick wall with the help of Rainbow’s Gravity  The same application of Rainbow’s functions can be considered on the Cosmological Constant with the help of the Wheeler-De Witt Equation  Sturm-Liouville Problem.  Neither Standard Regularization nor Renormalization are required. This also happens in NonCommutative geometries  Regularization substituted by a Modified Geometry Finite Results!!!  Effect on particles motion  In preparation