Self Sustained Wormholes Remo Garattini Università di Bergamo I.N.F.N. - Sezione di Milano MG 11 Berlin,
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 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 Effective Einstein Equations G is the Einstein tensor, 8G, 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 (Preprint gr-qc/ )] 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 (Preprint hep-th/ )]
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 Where g is a perturbation series in terms of g
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 (Preprint gr-qc/ )
10 Time-like unit vector On the constant time hypersurface Integrating on the constant time hypersurface Integrating on the constant time hypersurface
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 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 Integration rules on Gaussian wave functionals 123
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15 Graviton Contribution W.K.B. method and graviton contribution to the classical part
16 Self-Consistent Equation The value of the wormhole energy in the chosen background is One-loop self consistent equation
17 Regularization Riemann zeta function Equivalent to the Zero Point Energy subtraction procedure of the Casimir effect Total regularized one loop energy
18 Isolating the divergence
19 Renormalization Bare gravitational coupling constant changed into The finite part becomes
20 Renormalization Group Equation Eliminate the dependance on and impose G must be treated as running
21 Finding the wormhole radius At the scale One solution for
22 Two Possibilities we identify G 0 ( ) with the squared Planck length. we identify with the Planck scale
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,..)