Self Sustained Traversable Wormholes: from Phantom energy to noncommutative geometry Remo Garattini Università di Bergamo I.N.F.N. - Sezione di Milano Sestri Levante,

Introduction A wormhole can be represented by two asymptotically flat regions joined by a bridge. One very simple and at the same time fundamental example of wormhole is represented by the Schwarzschild solution of the Einstein's field equations. One of the prerogatives of a wormhole is its ability to connect two distant points in space-time. In this amazing perspective, it is immediate to recognize the possibility of traveling crossing wormholes as a short-cut in space and time. A Schwarzschild wormhole does not possess this property. Traversable wormholes

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4 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

5 Einstein Field Equations Orthonormal frame

6 Effective Einstein Equations Consider a separation of the metric  into a background and a perturbation 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 which is unperturbed related to the background geometry and a part related to quantum fluctuations P. R. Anderson and D. R. Brill, Phys.Rev. D 56 (1997) 4824, gr-qc/ Gravitational geons revisited

7 General setting for self sustained traversable wormholes renormalized expectation value of the stress-energy tensor operator of the quantized field where If the matter field source is absent

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

9 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,

10 Canonical Decomposition  h is the trace  (L ij is the longitudinal operator related to the F.P determinant (ghosts)  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).

11 Integration rules on Gaussian wave functionals 12345

12 Graviton Contribution W.K.B. method and graviton contribution to the classical part Ghosts contribution cancels out

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

14 Regularization Zeta function Regularization  Equivalent to the Zero Point Energy subtraction procedure of the Casimir effect Lichnerowicz Potentials

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

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

17 Finding the wormhole radius with phantom energy ] Finding the wormhole radius with phantom energy [R..G. Class.Quant.Grav.24: ,2007 gr-qc/ ] Solution Asymptotic flatness

18 Solution Inhomogeneous phantom energy [R.G. & F.S.N. Lobo C.Q.G (2007) gr-qc/ ]

19 Noncommutative geometry [R.G. & F.S.N. Lobo P.L.B (2009) [gr-qc] ] Form of the Solution

20 Conclusions and Perspectives  Semiclassical Einstein field equations: a source for self- consistent solutions.  Variational Approach to the problem.  Removing infinities with the zeta function Regularization  Casimir energy graviton contribution.  Renormalization and renormalization group equation.  The obtained "traversability" has to be regarded as in "principle" rather than in "practice" because of the wormhole radius size.  No ghosts!! Trace contribution?!?  Massive graviton!! Different background!!  Modified Gravity Theories – Modified Dispersion Relations?!?