1. Self Sustained Wormholes Remo Garattini Università di Bergamo I.N.F.N. - Sezione di Milano MG 11 Berlin, 24-7-2006

2. 2 The traversable wormhole metric M. S. Morris and K. S. Thorne, Am. J. Phys. 56, 395 (1988).  b(r) is the shape function    (r) is the redshift function Proper radial distance Condition

3. 3 Special case Then the traversable wormhole metric in Schwarzschild coordinates becomes The new coordinate l covers the range -∞< l <+∞. The constant time hypersurface Σ is an Einstein-Rosen bridge with wormhole topology S²×R¹. The Einstein-Rosen bridge defines a bifurcation surface dividing Σ in two parts denoted by Σ + and Σ -.

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6. 6 Effective Einstein Equations  G  is the Einstein tensor,  8G,  T  is the stress-energy tensor. Hochberg, Popov and Sushkov considered a self-consistent solution of the semiclassical Einstein equations corresponding to a Lorentzian wormhole coupled with a quantum scalar field [Hochberg D, Popov A and Sushkov S V 1997 Phys. Rev. Lett. 78 2050 (Preprint gr-qc/9701064)] Khusnutdinov and Sushkov fixed their attention to the computation of the ground state of a massive scalar field in a wormhole background. They tried to see if a self-consistent solution restricted to the energy component appears in this configuration [ Khusnutdinov N R and Sushkov S V 2002 Phys. Rev. D 65 084028 (Preprint hep-th/0202068)]

7. 7  Consider a separation of the metric g  into a background and a perturbation G  The Einstein tensor G  can also be divided into a part describing the curvature due to the background geometry and that due to the perturbation

8. 8 Where g  is a perturbation series in terms of g 

9. 9 Semiclassical gravity renormalized expectation value of the stress-energy tensor operator of the quantized field where If the matter field source is absent Gravitational geon considered by Anderson and Brill Anderson P R and Brill D R 1997 Phys. Rev. D 56 4824 (Preprint gr-qc/9610074)

10. 10  Time-like unit vector  On the constant time hypersurface  Integrating on the constant time hypersurface   Integrating on the constant time hypersurface 

11. 11 G ijkl is the super-metric  G ijkl is the super-metric  R is the scalar curvature in 3-dim. To compute the expectation value of the perturbed Einstein tensor in the transverse- traceless sector, we use a variational procedure with gaussian wave functionals. Thus the fluctuations in the Einstein tensor are, in this context, the fluctuations of the hamiltonian. Let us consider the 3-dim. metric g ij and perturb around a fixed background,

12. 12 Canonical Decomposition  h is the trace  (L ij is the longitudinal operator  h  ij represents the transverse-traceless component of the perturbation  graviton 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).

13. 13 Integration rules on Gaussian wave functionals 123

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15. 15 Graviton Contribution W.K.B. method and graviton contribution to the classical part

16. 16 Self-Consistent Equation  The value of the wormhole energy in the chosen background is One-loop self consistent equation

17. 17 Regularization Riemann zeta function  Equivalent to the Zero Point Energy subtraction procedure of the Casimir effect Total regularized one loop energy

18. 18 Isolating the divergence

19. 19 Renormalization  Bare gravitational coupling constant changed into The finite part becomes

20. 20 Renormalization Group Equation  Eliminate the dependance on  and impose G  must be treated as running

21. 21 Finding the wormhole radius  At the scale   One solution for

22. 22 Two Possibilities  we identify G 0 (   ) with the squared Planck length.  we identify   with the Planck scale

23. 23 Conclusions and Outlooks  Semi-classical einstein equations  Self- consistent wormhole.  Variational Approach as aprocedure to average the graviton fluctuations.  Zeta function Regularization, Renormalization and renormalization group equation applied to the Newton constant.  On self-consistent solution of Planckian size.  Try to improve the result with a massive graviton.  Possible hints from phantom energy (work in progress,..)