# Quantum creation of traversable wormholes ex nihilo in Gauss–Bonnet-dilaton gravity

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## Abstract

We investigate a nucleation of a Euclidean wormhole and its analytic continuation to Lorentzian signatures in Gauss–Bonnet-dilaton gravity, where this model can be embedded by the type-II superstring theory. We show that there exists a Euclidean wormhole solution in this model by choosing a suitable shape of the dilaton potential. After the analytic continuation, this explains a quantum creation of a time-like traversable wormhole. Finally, we discuss relations to the information loss problem and the current literature.

## 1 Introduction

The investigation of wormholes is a very hypothetical but interesting research topic. One may study wormholes in the Lorentzian signatures; if the wormhole throat is time-like and traversable, then it should violate the averaged null energy condition by a certain way, e.g., introducing an exotic matter or quantum effects [1]. One may also study wormholes in the Euclidean signatures [2]; then this topic can have applications for quantum gravity [3] and quantum cosmology [4, 5, 6].

- (1)
Can we introduce the (at least, effective) exotic matter within the well-known safe field theory without the menace of phantom [8]?

- (2)
Why should the two separated spaces be joined at the throat, i.e., is there a mechanism to bring two spaces to one place?

However, recently, especially by string theorists, it was suggested that wormholes will do an important role within the string theory. For example, in order to overcome the inconsistency [9, 10, 11, 12, 13] of black hole complementarity [14] and resolve the firewall paradox [15, 16, 17], the Einstein–Rosen/Einstein–Podolsky–Rosen (ER=EPR) conjecture has been suggested [18], where this is related to a space-like wormhole with the Einstein–Rosen bridge. Later, it was known that semi-classical effects on the Einstein–Rosen bridge can change the bridge to be traversable [19]. Moreover, recently, it was reported that the existence of wormholes can be allowed within the string theory as we turn on suitable semi-classical quantum effects [20, 21, 22].

Therefore, it is interesting to study whether there can be other time-like wormhole solutions within the string theory or not. In this paper, we will focus on the Gauss–Bonnet-dilaton gravity, where this can be embedded by the string theory [23]. Previously, it was reported that there exists Lorentzian wormhole solutions within the theory [24]. This is quite impressive since now we can say that the string theory seems to allow time-like wormholes even at the classical level. Also, energy conditions will be effectively violated, but we may trust the model since the string theory can be regarded as a candidate of the UV-completion of gravity.

On the other hand, for all models of wormholes, their dynamical constructions were not known yet. In this paper, we will argue that Euclidean wormholes can be analytically continued to a time-like Lorentzian wormhole. We will find a Euclidean wormhole solution in the Gauss–Bonnet-dilaton gravity, and hence, the solution will be allowed by the string theory context. Therefore, based on our model, we can solve two previous difficult questions: (1) we used a string-inspired model which can be regarded as a UV-completion and (2) we explain a dynamical creation of a wormhole \(\textit{ex nihilo}\) by quantum mechanical processes.

This paper is constructed as follows. In Sect. 2, we describe the details of the Gauss–Bonnet-dilaton gravity model. In Sect. 3, we describe the details of the solution as well as its properties. In Sect. 4, we give some comments on the applications of wormholes and discuss possible future works.

## 2 Model

*R*is the Ricci scalar, \(\kappa ^{2} = 8\pi \), \(\phi \) is the dilaton field with potential \(V(\phi )\),

*c*is a constant, \(\lambda \) is the coupling constant proportional to the \(\alpha '\) parameter, and

*c*, and \(\phi _{0}\) without loss of generality.

*O*(4)-symmetric metric ansatz:

### 2.1 Equations of motion

*a*, \({\dot{a}}\), \(\phi \), and \({\dot{\phi }}\).

### 2.2 Solving techniques

Usually, one gives the initial condition at a certain point and solve equations. One important point is to give regular boundary conditions for the end of the solution if the solution is compact. In order to make this boundary value problem simpler, we solve *V* as a function of \(\tau \) rather than directly solve \(\phi \). This means that we first fix the solution \(\phi \) and solve equations for *a* and *V*.

*V*instead of \(\phi \), one can arbitrarily choose the shape of the field in principle and we use

*V*becomes \({\dot{V}} = {\dot{\phi }} V'\), where

### 2.3 Conditions for wormholes

### 2.4 Initial conditions

## 3 Solutions and properties

By choosing suitable initial conditions and model parameters, one can obtain a Euclidean wormhole with regular boundary conditions at compact boundaries \(a = 0\). We demonstrate an explicit example.

### 3.1 Quantum creation of time-like wormholes

*T*surfaces are space-like) and \(\tau \) is the space-like parameter (constant \(\tau \) surfaces are time-like). Note that both ends of \(a = 0\) correspond null surfaces. Beyond the null surfaces, one can further analytically continue by choosing \(\tau = it\), \(T = i\pi /2 +\chi \), and \(\alpha (t) = -ia(it)\):

### 3.2 Parameter dependences

In this subsection, we discuss more on the technical details of the solutions. Since there are many parameters that we can handle, there may be no straightforward way to finely tune the initial conditions in order to satisfy the boundary conditions for both ends. However, by changing several parameters around the solution that we have obtained, we can have some technical intuitions.

As an example, in Figs. 5, 6, and 7, we demonstrated three examples by varying \(\Delta \), where Fig. 5 is the same as Figs. 2 and 3. As we increase \(\Delta \), the slope of \(\phi \) becomes more and more gentle. Therefore, the contributions from the kinetic energy decreases and *V* should be modified according to the back-reactions.

By comparing \(V(\tau )\) of Figs. 5, 6, and 7, we can notice that the scales of *V* increases drastically. In Fig. 6, *V* crosses from positive to negative values with relatively steeper way, and hence, there appears a cusp-like point due to the sharp dynamics nearby \(\tau \sim 1.7\).

One another tendency is that as \(\Delta \) increases, the values of *a* at the local minimum (\({\dot{a}} = 0\) and \(\ddot{a} > 0\)) and the second local maximum (\({\dot{a}} = 0\) and \(\ddot{a} < 0\)) become relatively lower than that of the first local maximum (\(a \sim 0.35\)). This means that as \(\Delta \) increases, *a* of the local minimum and the second local maximum continuously decreases and eventually there exists a limit such that \({\dot{a}} \rightarrow 0\) as \(a \rightarrow 0\) (Fig. 7)^{1}.

In this paper, we could not report all possible variations of parameters, but these examples show that there are plenty of solutions that will have interesting physical implications.

### 3.3 Energy conditions

### 3.4 Probabilities

Of course, there is a subtlety to choose the background solution. However, it is fair to say that our solution strongly supports that there can be a well-defined tunneling process from a pure de Sitter space to a time-like wormhole space within the framework of string-inspired Gauss–Bonnet-dilaton gravity.

## 4 Discussion

In this paper, we investigated a nucleation of a Euclidean wormhole and its analytic continuation to Lorentzian signatures in Gauss–Bonnet-dilaton gravity. This model can be embedded by the type-II superstring theory. We show that there exists a Euclidean wormhole solution in this model by choosing a suitable shape of the dilaton potential. After the analytic continuation, this explains a quantum creation of a time-like wormhole ex nihilo. This work is new since we could deal the following two topics at the same time: (1) we embedded the model in the context of the string theory^{2} and (2) we explained a quantum mechanical creation process of a wormhole.

- (1)
This solution opens a possibility that a time-like wormhole can exist in the de Sitter space, e.g., in our universe. The causal structure near the throat will give specific signals in terms of gravitational waves [45]. Also, this has an implication in the context of the open inflation scenario [46].

- (2)
In the orange colored region in Fig. 4, there appears a hyperbolic time-like wormhole, where the other two time-like boundaries of the orange colored region correspond to the boundary of increasing areal radius. On this background, one may test some ideas of holography [47, 48].

## Footnotes

## Notes

### Acknowledgements

We would like to thank Daeho Ro for fruitful discussions. GT is pleased to appreciate Asian Pacific Center for Theoretical Physics (APCTP) for its hospitality during completion of this work. DY was supported by the Korea Ministry of Education, Science and Technology, Gyeongsangbuk-Do and Pohang City for Independent Junior Research Groups at the Asia Pacific Center for Theoretical Physics and the National Research Foundation of Korea (Grant No.: 2018R1D1A1B07049126). GT was supported by Institute for Basic Science (IBS) under the project code, IBS-R018-D1.

## References

- 1.M.S. Morris, K.S. Thorne, Am. J. Phys.
**56**, 395 (1988)CrossRefADSGoogle Scholar - 2.S.W. Hawking, Phys. Rev. D
**37**, 904 (1988)MathSciNetCrossRefADSGoogle Scholar - 3.S.R. Coleman, Nucl. Phys. B
**310**, 643 (1988)MathSciNetCrossRefADSGoogle Scholar - 4.P. Chen, Y.C. Hu, D. Yeom, JCAP
**1707**(7), 001 (2017). arXiv:1611.08468 [gr-qc] - 5.S. Kang, D. Yeom, Phys. Rev. D
**97**(12), 124031 (2018). arXiv:1703.07746 [gr-qc]MathSciNetCrossRefADSGoogle Scholar - 6.P. Chen, D. Yeom, arXiv:1706.07784 [gr-qc]
- 7.M. Visser,
*Lorentzian Wormholes: From Einstein to Hawking*(AIP Press, College Park, 1995)Google Scholar - 8.J.M. Cline, S. Jeon, G.D. Moore, Phys. Rev. D
**70**, 043543 (2004). arXiv:hep-ph/0311312 CrossRefADSGoogle Scholar - 9.D. Yeom, H. Zoe, Phys. Rev. D
**78**, 104008 (2008). arXiv:0802.1625 [gr-qc]MathSciNetCrossRefADSGoogle Scholar - 10.S.E. Hong, D. Hwang, E.D. Stewart, D. Yeom, Class. Quantum Gravity
**27**, 045014 (2010). arXiv:0808.1709 [gr-qc]CrossRefADSGoogle Scholar - 11.D. Yeom, H. Zoe, Int. J. Mod. Phys. A
**26**, 3287 (2011). arXiv:0907.0677 [hep-th]CrossRefADSGoogle Scholar - 12.A. Almheiri, D. Marolf, J. Polchinski, J. Sully, JHEP
**1302**, 062 (2013). arXiv:1207.3123 [hep-th]CrossRefADSGoogle Scholar - 13.A. Almheiri, D. Marolf, J. Polchinski, D. Stanford, J. Sully, JHEP
**1309**, 018 (2013). arXiv:1304.6483 [hep-th]CrossRefADSGoogle Scholar - 14.L. Susskind, L. Thorlacius, J. Uglum, Phys. Rev. D
**48**, 3743 (1993). arXiv:hep-th/9306069 MathSciNetCrossRefADSGoogle Scholar - 15.D. Hwang, B.-H. Lee, D. Yeom, JCAP
**1301**, 005 (2013). arXiv:1210.6733 [gr-qc]CrossRefADSGoogle Scholar - 16.
- 17.P. Chen, Y.C. Ong, D.N. Page, M. Sasaki, D. Yeom, Phys. Rev. Lett.
**116**(16), 161304 (2016). arXiv:1511.05695 [hep-th]CrossRefADSGoogle Scholar - 18.J. Maldacena, L. Susskind, Fortsch. Phys.
**61**, 781 (2013). arXiv:1306.0533 [hep-th]CrossRefADSGoogle Scholar - 19.P. Chen, C.H. Wu, D. Yeom, JCAP
**1706**(06), 040 (2017). arXiv:1608.08695 [hep-th]CrossRefADSGoogle Scholar - 20.P. Gao, D.L. Jafferis, A. Wall, JHEP
**1712**, 151 (2017). arXiv:1608.05687 [hep-th]CrossRefADSGoogle Scholar - 21.J. Maldacena, X.L. Qi, arXiv:1804.00491 [hep-th]
- 22.J. Maldacena, A. Milekhin, F. Popov, arXiv:1807.04726 [hep-th]
- 23.R.R. Metsaev, A.A. Tseytlin, Nucl. Phys. B
**293**, 385 (1987)CrossRefADSGoogle Scholar - 24.P. Kanti, B. Kleihaus, J. Kunz, Phys. Rev. Lett.
**107**, 271101 (2011). arXiv:1108.3003 [gr-qc]CrossRefGoogle Scholar - 25.P. Kanti, N.E. Mavromatos, J. Rizos, K. Tamvakis, E. Winstanley, Phys. Rev. D
**54**, 5049 (1996). arXiv:hep-th/9511071 MathSciNetCrossRefADSGoogle Scholar - 26.J.B. Hartle, S.W. Hawking, Phys. Rev. D
**28**, 2960 (1983)MathSciNetCrossRefADSGoogle Scholar - 27.J.B. Hartle, S.W. Hawking, T. Hertog, Phys. Rev. Lett.
**100**, 201301 (2008). arXiv:0711.4630 [hep-th]MathSciNetCrossRefADSGoogle Scholar - 28.J.B. Hartle, S.W. Hawking, T. Hertog, Phys. Rev. D
**77**, 123537 (2008). arXiv:0803.1663 [hep-th]MathSciNetCrossRefADSGoogle Scholar - 29.D. Hwang, H. Sahlmann, D. Yeom, Class. Quantum Gravity
**29**, 095005 (2012). arXiv:1107.4653 [gr-qc]CrossRefADSGoogle Scholar - 30.D. Hwang, B.-H. Lee, H. Sahlmann, D. Yeom, Class. Quantum Gravity
**29**, 175001 (2012). arXiv:1203.0112 [gr-qc]CrossRefADSGoogle Scholar - 31.D. Hwang, S.A. Kim, B.-H. Lee, H. Sahlmann, D. Yeom, Class. Quantum Gravity
**30**, 165016 (2013). arXiv:1207.0359 [gr-qc]CrossRefADSGoogle Scholar - 32.S. Koh, B.H. Lee, W. Lee, G. Tumurtushaa, Phys. Rev. D
**90**(6), 063527 (2014). arXiv:1404.6096 [gr-qc]CrossRefADSGoogle Scholar - 33.S. Koh, B.H. Lee, G. Tumurtushaa, Phys. Rev. D
**95**(12), 123509 (2017). arXiv:1610.04360 [gr-qc]CrossRefADSGoogle Scholar - 34.S. Koh, B.H. Lee, G. Tumurtushaa, arXiv:1807.04424 [astro-ph.CO]
- 35.B.H. Lee, W. Lee, D. Ro, Phys. Lett. B
**762**, 535 (2016). arXiv:1607.01125 [hep-th]MathSciNetCrossRefADSGoogle Scholar - 36.S. Kanno, M. Sasaki, J. Soda, Class. Quantum Gravity
**29**, 075010 (2012). arXiv:1201.2272 [hep-th]CrossRefADSGoogle Scholar - 37.S. Kanno, M. Sasaki, J. Soda, Prog. Theor. Phys.
**128**, 213 (2012). arXiv:1203.0612 [hep-th]CrossRefADSGoogle Scholar - 38.S.W. Hawking, N. Turok, Phys. Lett. B
**425**, 25 (1998). arXiv:hep-th/9802030 MathSciNetCrossRefADSGoogle Scholar - 39.S. Brahma, D. Yeom, Phys. Rev. D
**98**(8), 083537 (2018). arXiv:1808.01744 [gr-qc]CrossRefADSGoogle Scholar - 40.
- 41.A. Raychaudhuri, Phys. Rev.
**98**, 1123 (1955)MathSciNetCrossRefADSGoogle Scholar - 42.E. Poisson,
*A Relativist’s Toolkit: The Mathematics of Black-hole Mechanics*(Cambridge University Press, Cambridge, 2004)CrossRefGoogle Scholar - 43.R.G. Cai, B. Hu, S. Koh, Phys. Lett. B
**671**, 181 (2009). arXiv:0806.2508 [hep-th]CrossRefADSGoogle Scholar - 44.M. Bouhmadi-Lopez, C.Y. Chen, P. Chen, D. Yeom, JCAP
**1810**(10), 056 (2018). arXiv:1809.06579 [gr-qc]CrossRefADSGoogle Scholar - 45.S.H. Volkel, K.D. Kokkotas, Class. Quantum Gravity
**35**(10), 105018 (2018). arXiv:1802.08525 [gr-qc]CrossRefADSGoogle Scholar - 46.K. Yamamoto, M. Sasaki, T. Tanaka, Astrophys. J.
**455**, 412 (1995). arXiv:astro-ph/9501109 CrossRefADSGoogle Scholar - 47.J.M. Maldacena, Adv. Theor. Math. Phys.
**2**, 231 (1998)MathSciNetCrossRefADSGoogle Scholar - 48.
- 49.S.W. Hawking, Phys. Rev. D
**14**, 2460 (1976)MathSciNetCrossRefADSGoogle Scholar - 50.P. Chen, Y.C. Ong, D. Yeom, Phys. Rep.
**603**, 1 (2015). arXiv:1412.8366 [gr-qc]MathSciNetCrossRefADSGoogle Scholar - 51.S.W. Hawking, Phys. Rev. D
**72**, 084013 (2005). arXiv:hep-th/0507171 MathSciNetCrossRefADSGoogle Scholar - 52.
- 53.B.H. Lee, W. Lee, D. Yeom, Phys. Rev. D
**92**(2), 024027 (2015). arXiv:1502.07471 [hep-th]CrossRefADSGoogle Scholar - 54.P. Chen, G. Domènech, M. Sasaki, D. Yeom, JCAP
**1604**(04), 013 (2016). arXiv:1512.00565 [hep-th]CrossRefADSGoogle Scholar - 55.P. Chen, G. Domènech, M. Sasaki, D. Yeom, JHEP
**1707**, 134 (2017). arXiv:1704.04020 [gr-qc]CrossRefADSGoogle Scholar - 56.P. Chen, M. Sasaki, D. Yeom, arXiv:1806.03766 [hep-th]
- 57.N. Arkani-Hamed, J. Orgera, J. Polchinski, JHEP
**0712**, 018 (2007). arXiv:0705.2768 [hep-th]CrossRefADSGoogle Scholar - 58.S.W. Hawking, Phys. Lett. B
**195**, 337 (1987)MathSciNetCrossRefADSGoogle Scholar

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