Sestri Levante 8 6 2009 Self Sustained Traversable

Sestri Levante, 8 -6 -2009 Self Sustained Traversable Wormholes: from Phantom energy to noncommutative geometry Remo Garattini Università di Bergamo I. N. F. N. - Sezione di Milano

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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The traversable wormhole metric M. S. Morris and K. S. Thorne, Am. J. Phys. 56, 395 (1988). Condition Ø b(r) is the shape function Ø f (r) is the redshift function Proper radial distance 4

Einstein Field Equations Orthonormal frame 5

Effective Einstein Equations Consider a separation of the metric gmn into a background a perturbation The Einstein tensor Gmn 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/9610074. Gravitational geons revisited 6

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

u On the constant time hypersurface S u Time-like unit vector u Integrating on the constant time hypersurface S 8

u Gijkl is u R the super-metric is the scalar curvature in 3 -dim. Thus the fluctuations in the Einstein tensor are, in this context, the fluctuations of the hamiltonian. To compute the expectation value of the perturbed Einstein tensor in the transversetraceless sector, we use a variational procedure with gaussian wave functionals. Let us consider the 3 -dim. metric gij and perturb around a fixed background, 9

Canonical Decomposition 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). h is the trace u (Lx)ij is the longitudinal operator related to the F. P determinant (ghosts) u h^ij represents the transverse-traceless u component of the perturbation graviton 10

Integration rules on Gaussian wave functionals 1 2 3 4 5 11

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

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

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

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

Renormalization Group Equation u Eliminate the dependance on m and impose G 0 must be treated as running 16

Finding the wormhole radius with phantom energy [R. . G. Class. Quant. Grav. 24: 1189 -1210, 2007 gr-qc/0701019 ] Solution Asymptotic flatness 17

Inhomogeneous phantom energy [R. G. & F. S. N. Lobo C. Q. G. 24 2401 (2007) gr-qc/0701020] Solution 18

Noncommutative geometry [R. G. & F. S. N. Lobo P. L. B. 671 146 (2009) 0811. 0919 [gr-qc] ] Form of the Solution 19

Conclusions and Perspectives u u u u Semiclassical Einstein field equations: a source for selfconsistent 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? !? 20