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

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

Part I: Black Hole Entropy 3

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

7 In proximity of r 0

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/ E. Winstanley, Phys. Rev. D63, (2001) 2001), hep-th/ ] Pauli-Villars regularization [J.-G. Demers, R. Lafrance and R. C. Myers, Phys. Rev. D52, 2245 (1995), gr- qc/ D. V. Fursaev and S. N. Solodukhin, Phys. Lett. B365, 51 (1996), hep-th/ S. P. Kim, S. K. Kim, K.-S. Soh and J. H. Yee, Int. J. Mod. Phys. A12, 5223 (1997) gr-qc/ ]

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/ Z. Ren, W. Yue-Qin and Z. Li-Chun, Class. Quant. Grav. 20 (2003), G. Amelino-Camelia, Class.Quant.Grav. 23, 2585 (2006), gr-qc/ G. Amelino-Camelia, Gen.Rel.Grav. 33, 2101 (2001), gr-qc/

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

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

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

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

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

Part II: The Cosmological Constant 15

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 Re-writing the WDW equation Where

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

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

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

23 Gravity’s Rainbow and the Cosmological Constant

24 Gravity’s Rainbow and the Cosmological Constant

25 Gravity’s Rainbow and the Cosmological Constant

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

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