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Self Sustained Traversable Wormholes: from Phantom energy to noncommutative geometry Remo Garattini Università di Bergamo I.N.F.N. - Sezione di Milano Sestri Levante, 8-6-2009
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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
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5 Einstein Field Equations Orthonormal frame
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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/9610074. Gravitational geons revisited
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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
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8 Time-like unit vector On the constant time hypersurface Integrating on the constant time hypersurface Integrating on the constant time hypersurface
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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,
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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).
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11 Integration rules on Gaussian wave functionals 12345
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12 Graviton Contribution W.K.B. method and graviton contribution to the classical part Ghosts contribution cancels out
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13 Self-Consistent Equation The value of the wormhole energy in the chosen background is One-loop self consistent equation for the energy
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14 Regularization Zeta function Regularization Equivalent to the Zero Point Energy subtraction procedure of the Casimir effect Lichnerowicz Potentials
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15 Renormalization Bare gravitational coupling constant changed into The finite part becomes
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16 Renormalization Group Equation Eliminate the dependance on and impose G must be treated as running
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17 Finding the wormhole radius with phantom energy ] Finding the wormhole radius with phantom energy [R..G. Class.Quant.Grav.24:1189-1210,2007 gr-qc/0701019 ] Solution Asymptotic flatness
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18 Solution Inhomogeneous phantom energy [R.G. & F.S.N. Lobo C.Q.G.24 2401 (2007) gr-qc/0701020]
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19 Noncommutative geometry [R.G. & F.S.N. Lobo P.L.B. 671 146 (2009) 0811.0919 [gr-qc] ] Form of the Solution
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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?!?
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