# Gravity’s Rainbow induces topology change

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## Abstract

In this work, we explore the possibility that quantum fluctuations induce a topology change, in the context of Gravity’s Rainbow. A semiclassical approach is adopted, where the graviton one-loop contribution to a classical energy in a background spacetime is computed through a variational approach with Gaussian trial wave functionals. The energy density of the graviton one-loop contribution, or equivalently the background spacetime, is then let to evolve, and consequently the classical energy is determined. More specifically, the background metric is fixed to be Minkowskian in the equation governing the quantum fluctuations, which behaves essentially as a backreaction equation, and the quantum fluctuations are let to evolve; the classical energy, which depends on the evolved metric functions, is then evaluated. Analyzing this procedure, a natural ultraviolet cutoff is obtained, which forbids the presence of an interior spacetime region, and this may result in a multiply connected spacetime. Thus, in the context of Gravity’s Rainbow, this process may be interpreted as a change in topology, and in principle it results in the presence of a planckian wormhole.

## Keywords

Topology Change Quantum Fluctuation Classical Energy Casimir Energy Einstein Field Equation## 1 Introduction

It was John A. Wheeler [1, 2] who first conjectured that spacetime could be subjected to a topological fluctuation at the Planck scale, meaning that spacetime undergoes a deep and rapid transformation in its structure. The changing spacetime is best known as *spacetime foam*, which can be taken as a model for the quantum gravitational vacuum. Wheeler also considered wormhole-type solutions as objects of the spacetime quantum foam connecting different regions of spacetime at the Planck scale [2, 3]. These Wheeler wormholes were obtained from the coupled equations of electromagnetism and general relativity and were denoted “geons”, i.e., gravitational-electromagnetic entities. However, these solutions were singular and were not traversable [4]. In fact, the geon concept possesses interesting properties, such as the absence of charges or currents and the gravitational mass originates solely from the energy stored in the electromagnetic field, i.e., there are no material masses present. These characteristics gave rise to the terms “charge without charge” and “mass without mass”, respectively.

Paging through history, one finds that these entities were further explored by several authors in different contexts. Indeed, Ernst analyzed idealized spherical “geons” using a simple adaptation of the Ritz variational principle [5], and furthermore explored toroidal geons, where the electromagnetic vector potential is vanishingly small except within a toroidal region of space [6]. In fact, the electromagnetic field physically consists of light waves circling the torus in either direction, so that the torus of electromagnetic field energy was denoted a *toroidal geon*. It was shown that toroidal geons of large major radius to minor radius ratio may be studied using an approximation of linear geons, where the electromagnetic field energy is confined to an infinitely long circular cylinder rather than to a torus. Indeed, the electromagnetic field potentials of a toroidal geon or of a linear geon possess the same general nature as the electromagnetic field potentials encountered in the solution of classical toroidal and cylindrical wave guide problems. Thus, these results provided the foundation material for a proposed later treatment of toroidal geons.

Later, Brill and Hartle [7] extended the previous analysis to the case where gravitational waves are the source of the geon’s mass energy, where the background spherically symmetric metric describes the large-scale features of the geon. It was shown that the waves superimposed on this background have an amplitude small enough so that their dynamics can be analyzed in the linear approximation. However, their wavelength is so short, and their time dependence so rapid that their energy is appreciable and produces the strongly curved background metric in which they move. It was also found that the large-scale features of the spherical gravitational geons are identical to those of the spherical electromagnetic geons analyzed previously. In fact, later work by Anderson and Brill [8] showed that the geon solution is a self-consistent solution to Einstein’s equations and that, to leading order, the equations describing the geometry of the gravitational geon are identical to those derived by Wheeler for the electromagnetic geon.

Komar [9] showed that there exist solutions of the vacuum Einstein field equations with the property that exterior to the Schwarzschild radius, the solution appears to be that of a static spherically symmetric particle of mass \(m\), whereas interior to the Schwarzschild radius the topology remains Euclidean and the solutions have the property of a bundle of gravitational radiation so intense that the mutual gravitational attraction of the various parts of the bundle prevent the radiation from spreading beyond the Schwarzschild radius. Komar also argued that no singularity can ever be observed exterior to the Schwarzschild radius. However, it was shown that the Komar bootstrap gravitational geon solution does in fact display a singular behavior along portions of an axis in the regions in which the solution deviates from the standard Schwarzschild solution [10].

An interesting geon solution was explored by Kaup [11] in the context of the Klein–Gordon Einstein equations (Klein–Gordon geons), which reveal that these geons have properties that are different from the other gravitating systems studied previously. Indeed, the equilibrium states of these geons seem analogous to other gravitating systems, but it was shown that adiabatic perturbations are forbidden, when the stability is considered from a thermodynamical viewpoint. The reason for this is that the equations of state for the thermodynamical variables are not algebraic equations, but instead they are differential equations. Consequently, the usual concept of an equation of state breaks down when Klein–Gordon geons are considered. When the question of stability is reconsidered in terms of infinitesimal perturbations of the basic fields, it was then found that Klein–Gordon geons will not undergo spherically symmetric gravitational collapse. Thus, the Klein–Gordon geons considered by Kaup are counterexamples to the conjecture that gravitational collapse is inevitable.

In fact, much work was done over the decades, but due to the extremely ambitious program and the lack of experimental evidence the geon concept soon died out. However, it is interesting to note that Misner inspired by Wheeler’s geon representation, found wormhole solutions to the source-free Einstein equations [12], and with the introduction of multi-connected topologies in physics, the question of causality inevitably arose. Thus, Wheeler and Fuller examined this situation in the Schwarzschild solution and found that causality is preserved [13], as the Schwarzschild throat pinches off in a finite time, preventing the traversal of a signal from one region to another through the wormhole. Nevertheless, Graves and Brill [14], considering the Reissner–Nordström metric, also found wormhole-type solutions possessing an electric flux flowing through the wormhole. They found that the region of minimum radius, the “throat”, contracted, reaching a minimum and re-expanded after a finite proper time, rather than pinching off as in the Schwarzschild case. The throat, “cushioned” by the pressure of the electric field through the throat, pulsated periodically in time.

In the context of the quantum gravitational vacuum, some authors have investigated the effects of such a foamy space on the cosmological constant, for instance, one example is the celebrated Coleman mechanism, where wormhole contributions suppress the cosmological constant, explaining its small observed value [15]. Nevertheless, how to realize such a foam-like space and also whether this represents the real quantum gravitational vacuum is still unknown. However, it is interesting to observe that Ellis et al. considered a foam-like structure built in terms of D-branes to discuss phenomenological aspects [16, 17, 18, 19]. Wheeler when discussing the quantum fluctuations in the spacetime metric [2] considered that a typical fluctuation in a typical gravitational potential is of the order \(\Delta g\sim (hG/c^{3})^{1/2}/L\) which become appreciable for small length scales \(L\). A fundamental question is whether a change in topology may be induced by large metric fluctuations. In fact, Wheeler has argued in favor of a topology change and recently researchers in quantum gravity have accepted that the notion of spacetime foam leads to topology-changing quantum amplitudes and to interference effects between spacetimes of different topologies [20].

*classical*evolution of general relativistic spacetimes. These were summarized in two points by Visser [20]:

- 1.
In causally well-behaved classical spacetimes the topology of space does not change as a function of time.

- 2.
In causally ill-behaved classical spacetimes the topology of space can sometimes change.

*quantum*point of view we can separate the problem of topology change generated by a canonical quantization approach and a functional integral quantization approach. The Hawking topology change theorem is thus enough to show that the topology of space cannot change in canonically quantized gravity [22]. In the Feynman functional integral quantization of gravitation things are different. Indeed, in this formalism, it is possible to adopt an approach to spacetime foam where we know that fluctuations of topology become an important phenomenon at least at the Planck scale [23].

In the present paper, we are interested in the possibility that quantum fluctuations induce a topology change, in the context of Gravity’s Rainbow [35, 36, 37]. The latter is a distortion of the spacetime metric at energies comparable to the Planck energy, and a general formalism, denoted as deformed or doubly special relativity, was developed in order to preserve the relativity of inertial frames, maintain the Planck energy invariant and impose the requirement that in the limit \(E/E_{P}\rightarrow 0\), the speed of a massless particle tends to a universal and invariant constant, \(c\). Here, we adopt a semiclassical approach, where the graviton one-loop contribution to a classical energy in a background spacetime is computed through a variational approach with Gaussian trial wave functionals. In fact, it has been shown explicitly that the finite one-loop energy may be considered as a self-consistent source for a traversable wormhole [38]. In addition to this, a renormalization procedure was introduced and a zeta function regularization was involved to handle the divergences. The latter approach was also explored [39] in the context of phantom energy traversable wormholes [40, 41, 42]. It was shown that the latter semiclassical approach prohibits solutions with a constant equation of state parameter, which further motivates the imposition of a radial dependent parameter, \(\omega (r)\), and only permits solutions with a steep positive slope proportional to the radial derivative of the equation of state parameter, evaluated at the throat [39]. Using the semiclassical approach outlined above, exact wormhole solutions in the context of noncommutative geometry were also analyzed, and their physical properties and characteristics were explored [43]. Indeed, wormhole geometries have been obtained in a wide variety of contexts, namely, in modified theories of gravity [44, 45, 46, 47, 48, 49, 50, 51], electromagnetic signatures of accretion disks around wormhole spacetimes [52, 53], etc. (we refer the reader to [54] for a review). The semiclassical procedure followed in this work relies heavily on the formalism outlined in Ref. [38]. Rather than reproduce the formalism, we shall refer the reader to Ref. [38] for details, when necessary.

In this work, we explore an alternative approach to the semiclassical approach outlined above. Note that the traditional manner is to fix a background metric and obtain self-consistent solutions. Here, we let the quantum fluctuations evolve, and the classical energy, which depends on the evolved metric functions, is then evaluated. A natural ultraviolet (UV) cutoff is obtained which forbids an interior spacetime region, and which may result in a multiply connected spacetime. Thus, in the context of Gravity’s Rainbow, this process may be interpreted as a change in topology, and consequently results in the presence of a planckian wormhole.

This paper is organized in the following manner: In Sect. 2, the semiclassical approach is briefly outlined, and the graviton one-loop contribution to a classical energy is computed through a variational approach with Gaussian trial wave functionals. In Sect. 3, the self-sustained equation is interpreted in a novel way, where the quantum fluctuations are let to evolve and the classical energy is then computed, consequently one arrives at solutions which may be interpreted as a change in topology. Finally, in Sect. 4, we conclude.

## 2 The classical term and the one-loop energy in Gravity’s Rainbow

### 2.1 Effective field equations in a spherically symmetric background

*geon*considered by Anderson and Brill [8], where the relevant difference lies in the averaging procedure). More specifically, the metric may be separated into a background component, \(\bar{g}_{\mu \nu }\) and a perturbation \(h_{\mu \nu }\), i.e., \(g_{\mu \nu }=\bar{g}_{\mu \nu }+h_{\mu \nu }\). The Einstein tensor may also be split into a part describing the curvature due to the background geometry and that due to the perturbation, i.e., \(G_{\mu \nu }(g_{\alpha \beta })=G_{\mu \nu }(\bar{g}_{\alpha \beta })+\Delta G_{\mu \nu }(\bar{g}_{\alpha \beta },h_{\alpha \beta })\), where \(\Delta G_{\mu \nu }(\bar{g}_{\alpha \beta },h_{\alpha \beta })\) may be considered a perturbation series in terms of \(h_{\mu \nu }\). If the matter field source is absent, one may define an effective stress-energy tensor for the fluctuations as

*Rainbow’s functions*, with the only assumption that

### 2.2 The one-loop energy in Gravity’s Rainbow

## 3 Topology change

### 3.1 Specific example I: \(g_{1}\left( E/E_{P}\right) =\exp (-\alpha \frac{E^{2}}{E_{P}^{2}}),\qquad g_{2}\left( E/E_{P}\right) =1\)

### 3.2 Specific example II: \(g_{1}(E/E_{P})=g_{2}(E/E_{P})=g(E/E_{P})\)

### 3.3 Specific example III: \(g_{2}\left( E/E_{P}\right) =1+E/E_{P}\) and \(g_{1}\left( E/E_{P}\right) =g\left( E/E_{P}\right) \left( 1+E/E_{P}\right) ^{6}\)

## 4 Summary and discussion

Note that in principle, one can adopt other backgrounds including a positive cosmological constant, i.e., a de Sitter spacetime, or a negative cosmological constant, namely an anti-de Sitter spacetime concluding that a hole can be produced by Zero Point Energy quantum fluctuations. Finally using the Casimir energy indicator [24], one can conclude that the presence of holes in spacetime seems to be favored leading therefore to a multiply connected spacetime. It is also interesting to note that one could compute the transition from Minkowski to a de Sitter spacetime or anti-de Sitter spacetime. It is important to remark that once the topology has been changed nothing can be said on the classical/quantum stability of the new spacetime because the whole calculation has been performed without a time evolution. It would also be interesting to consider solutions with charge. Indeed in [78, 79], the Wheeler–DeWitt equation was considered as a device for finding eigenvalues of a Sturm–Liouville problem. In particular, the Maxwell charge was interpreted as an eigenvalue of the Wheeler–De Witt equation generated by the gravitational field fluctuations. More specifically, it was shown that electric/magnetic charges could be generated by quantum fluctuations of the pure gravitational field. It would also be interesting to consider the presence of electric/magnetic charges in the solutions outlined in this paper, and work along these lines is currently under way.

## Notes

### Acknowledgments

FSNL acknowledges financial support of the Fundação para a Ciência e Tecnologia through the grants CERN/FP/123615/2011 and CERN/FP/123618/2011.

## References

- 1.J.A. Wheeler, Ann. Phys.
**2**, 604 (1957)ADSCrossRefzbMATHGoogle Scholar - 2.J.A. Wheeler, Phys. Rev.
**97**, 511–536 (1955)ADSCrossRefzbMATHMathSciNetGoogle Scholar - 3.J.A. Wheeler,
*Geometrodynamics*(Academic Press, New York, 1962)zbMATHGoogle Scholar - 4.R.P. Geroch, J. Math. Phys.
**8**, 782 (1967)ADSCrossRefzbMATHMathSciNetGoogle Scholar - 5.F.J. Ernst Jr, Phys. Rev.
**105**, 1662–1664 (1957)ADSCrossRefzbMATHMathSciNetGoogle Scholar - 6.F.J. Ernst Jr, Phys. Rev.
**105**, 1665–1670 (1957)ADSCrossRefzbMATHMathSciNetGoogle Scholar - 7.D.R. Brill, J.B. Hartle, Phys. Rev.
**135**, B271–B278 (1964)ADSCrossRefMathSciNetGoogle Scholar - 8.P.R. Anderson, D.R. Brill, Phys. Rev. D
**56**, 4824–4833 (1997)ADSCrossRefMathSciNetGoogle Scholar - 9.A. Komar, Phys. Rev.
**137**, B462–B466 (1965)ADSCrossRefMathSciNetGoogle Scholar - 10.C.H. Brans, Phys. Rev.
**140**, B1174–B1176 (1965)ADSCrossRefMathSciNetGoogle Scholar - 11.D.J. Kaup, Phys. Rev.
**172**, 1331–1342 (1968)ADSCrossRefGoogle Scholar - 12.C.W. Misner, Phys. Rev.
**118**, 1110–1111 (1960)ADSCrossRefzbMATHMathSciNetGoogle Scholar - 13.R.W. Fuller, J.A. Wheeler, Phys. Rev.
**128**, 919–929 (1962)ADSCrossRefzbMATHMathSciNetGoogle Scholar - 14.J.C. Graves, D.R. Brill, Phys. Rev.
**120**, 1507–1513 (1960)ADSCrossRefzbMATHMathSciNetGoogle Scholar - 15.S.R. Coleman, Nucl. Phys. B
**310**, 643 (1988)ADSCrossRefzbMATHGoogle Scholar - 16.J. Ellis, N.E. Mavromatos, D.V. Nanopoulos, Gen. Rel. Grav.
**32**, 127 (2000)ADSCrossRefzbMATHMathSciNetGoogle Scholar - 17.J. Ellis, N.E. Mavromatos, D.V. Nanopoulos, Phys. Rev. D
**61**, 027503 (2000)ADSCrossRefMathSciNetGoogle Scholar - 18.J. Ellis, N.E. Mavromatos, D.V. Nanopoulos, Phys. Rev. D
**62**, 084019 (2000)ADSCrossRefMathSciNetGoogle Scholar - 19.J. Ellis, N.E. Mavromatos, D.V. Nanopoulos, Int. J. Mod. Phys. A
**26**, 2243 (2011)ADSCrossRefzbMATHGoogle Scholar - 20.M. Visser,
*Lorentzian Wormholes: From Einstein to Hawking*(American Institute of Physics, New York, 1995)Google Scholar - 21.P. K. F. Kuhfittig, Neutron star interiors and topology change. arXiv:1207.6602 [gr-qc]
- 22.S.W. Hawking, Phys. Rev. D
**46**, 603 (1992)ADSCrossRefMathSciNetGoogle Scholar - 23.S.W. Hawking, Phys. Rev. D
**18**, 1747 (1978)ADSCrossRefGoogle Scholar - 24.A. DeBenedictis, R. Garattini, F.S.N. Lobo, Phys. Rev. D
**78**, 104003 (2008)ADSCrossRefGoogle Scholar - 25.F.S.N. Lobo, Class. Quant. Grav.
**23**, 1525 (2006)ADSCrossRefzbMATHMathSciNetGoogle Scholar - 26.R. Garattini, Int. J. Mod. Phys. D
**4**, 635 (2002)Google Scholar - 27.R. Garattini, Mod. Phys. Lett. A
**13**, 159–164 (1998)ADSCrossRefMathSciNetGoogle Scholar - 28.R. Garattini, Phys. Lett. B
**446**, 135 (1999)ADSCrossRefzbMATHMathSciNetGoogle Scholar - 29.R. Garattini, Phys. Lett. B
**459**, 461 (1999)ADSCrossRefzbMATHMathSciNetGoogle Scholar - 30.R. Garattini, Entropy
**2**, 26 (2000)ADSCrossRefzbMATHGoogle Scholar - 31.R. Garattini, Class. Quant. Grav.
**17**, 3335 (2000)ADSCrossRefzbMATHMathSciNetGoogle Scholar - 32.R. Garattini, Class. Quant. Grav.
**18**, 571 (2001)ADSCrossRefzbMATHMathSciNetGoogle Scholar - 33.R. Garattini, Int. J. Theor. Phys.
**9**, 129 (2002)MathSciNetGoogle Scholar - 34.R. Garattini, Int. J. Mod. Phys. A
**17**, 1965 (2002)ADSCrossRefzbMATHMathSciNetGoogle Scholar - 35.G. Amelino-Camelia, Int. J. Mod. Phys. D
**11**, 35 (2002)ADSCrossRefzbMATHMathSciNetGoogle Scholar - 36.G. Amelino-Camelia, Phys. Lett. B
**510**, 255 (2001)ADSCrossRefzbMATHGoogle Scholar - 37.J. Magueijo, L. Smolin, Class. Quant. Grav.
**21**, 1725 (2004)ADSCrossRefzbMATHMathSciNetGoogle Scholar - 38.R. Garattini, Class. Quant. Grav.
**22**, 1105 (2005)ADSCrossRefzbMATHMathSciNetGoogle Scholar - 39.R. Garattini, F.S.N. Lobo, Class. Quant. Grav.
**24**, 2401 (2007)ADSCrossRefzbMATHMathSciNetGoogle Scholar - 40.S.V. Sushkov, Phys. Rev. D
**71**, 043520 (2005)ADSCrossRefGoogle Scholar - 41.F.S.N. Lobo, Phys. Rev. D
**71**, 084011 (2005)ADSCrossRefMathSciNetGoogle Scholar - 42.F.S.N. Lobo, Phys. Rev. D
**71**, 084011 (2005)ADSCrossRefMathSciNetGoogle Scholar - 43.R. Garattini, F.S.N. Lobo, Phys. Lett. B
**671**, 146 (2009)ADSCrossRefMathSciNetGoogle Scholar - 44.F.S.N. Lobo, Phys. Rev. D
**75**, 064027 (2007)ADSCrossRefMathSciNetGoogle Scholar - 45.F.S.N. Lobo, M.A. Oliveira, Phys. Rev. D
**80**, 104012 (2009) Google Scholar - 46.F.S.N. Lobo, Class. Quant. Grav.
**25**, 175006 (2008)Google Scholar - 47.N.M. Garcia, F.S.N. Lobo, Phys. Rev. D
**82**, 104018 (2010)ADSCrossRefGoogle Scholar - 48.N. Montelongo Garcia, F.S.N. Lobo, Class. Quant. Grav.
**28**, 085018 (2011)Google Scholar - 49.C.G. Boehmer, T. Harko, F.S.N. Lobo, Phys. Rev. D
**85**, 044033 (2012)ADSCrossRefGoogle Scholar - 50.S. Capozziello, T. Harko, T.S. Koivisto, F.S.N. Lobo, G.J. Olmo, Phys. Rev. D
**86**, 127504 (2012)ADSCrossRefGoogle Scholar - 51.T. Harko, F.S.N. Lobo, M.K. Mak, S.V. Sushkov. arXiv:1301.6878 [gr-qc]
- 52.T. Harko, Z. Kovacs, F.S.N. Lobo, Phys. Rev. D
**79**, 064001 (2009)ADSCrossRefGoogle Scholar - 53.T. Harko, Z. Kovacs, F.S.N. Lobo, Phys. Rev. D
**78**, 084005 (2008)ADSCrossRefGoogle Scholar - 54.F.S.N. Lobo. arXiv:0710.4474 [gr-qc]
- 55.R. Garattini, Int. J. Mod. Phys. A
**14**, 2905–2920 (1999)ADSCrossRefzbMATHMathSciNetGoogle Scholar - 56.B.S. DeWitt, Phys. Rev.
**160**, 1113 (1967)ADSCrossRefzbMATHGoogle Scholar - 57.R. Garattini, G. Mandanici, Phys. Rev. D
**85**, 023507 (2012)ADSCrossRefGoogle Scholar - 58.R. Garattini, G. Mandanici, Phys. Rev. D
**83**, 084021 (2011)ADSCrossRefGoogle Scholar - 59.R. Garattini, F.S.N. Lobo, Phys. Rev. D
**85**, 024043 (2012)ADSCrossRefGoogle Scholar - 60.M.S. Morris, K.S. Thorne, Am. J. Phys.
**56**, 395 (1988)ADSCrossRefzbMATHMathSciNetGoogle Scholar - 61.T. Regge, J.A. Wheeler, Phys. Rev.
**108**, 1063 (1957)ADSCrossRefzbMATHMathSciNetGoogle Scholar - 62.G. ’t Hooft, Nucl. Phys. B
**256**, 727 (1985)Google Scholar - 63.V. Dzhunushaliev, Am. J. Mod. Phys.
**2**(3), 132 (2013). arXiv:0912.5326 [gr-qc] - 64.M. Headrick, T. Wiseman, Class. Quant. Grav.
**23**, 6683 (2006). arXiv:hep-th/0606086 - 65.S. Alexander, R. Brandenberger, J. Magueijo, Phys. Rev. D
**67**, 081301 (2003)ADSCrossRefMathSciNetGoogle Scholar - 66.S. Alexander, J. Magueijo, Proceedings of the XIIIrd Rencontres de Blois, pp 281, (2004). hep-th/0104093
- 67.G. Amelino-Camelia, Int. J. Mod. Phys. D
**11**, 35 (2002)ADSCrossRefzbMATHMathSciNetGoogle Scholar - 68.J.P.S. Lemos, F.S.N. Lobo, S. Quinet de Oliveira, Phys. Rev. D
**68**, 064004 (2003)ADSCrossRefGoogle Scholar - 69.F.S.N. Lobo, P. Crawford, Class. Quant. Grav.
**22**, 4869 (2005)ADSCrossRefzbMATHMathSciNetGoogle Scholar - 70.F.S.N. Lobo, Class. Quant. Grav.
**21**, 4811 (2004)ADSCrossRefzbMATHMathSciNetGoogle Scholar - 71.J.P.S. Lemos, F.S.N. Lobo, Phys. Rev. D
**69**, 104007 (2004)ADSCrossRefMathSciNetGoogle Scholar - 72.F.S.N. Lobo, Gen. Rel. Grav.
**37**, 2023 (2005)ADSCrossRefzbMATHMathSciNetGoogle Scholar - 73.J.P.S. Lemos, F.S.N. Lobo, Phys. Rev. D
**78**, 044030 (2008)ADSCrossRefMathSciNetGoogle Scholar - 74.R. Garattini, P. Nicolini, Phys. Rev. D
**83**, 064021 (2011). arXiv:1006.5418 [gr-qc] - 75.R. Garattini, EPJ Web Conf.
**58**, 01007 (2013). arXiv:1212.4311 [gr-qc] - 76.R. Garattini, Self Sustained Traversable Wormholes and Topology Change Induced by Gravity’s Rainbow, in the Proceedings of Karl Schwarzschild Meeting 2013. arXiv:1311.3146 [gr-qc] (to appear)
- 77.R. Garattini, F.S.N. Lobo, in preparationGoogle Scholar
- 78.R. Garattini, Phys. Lett. B
**666**, 189 (2008)ADSCrossRefMathSciNetGoogle Scholar - 79.R. Garattini, B. Majumder, Nucl. Phys. B
**883**, 598 (2014). arXiv:1305.3390 [gr-qc]

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