# Stable Dyonic Thin-Shell Wormholes in Low-Energy String Theory

Authors: Ali Övgün; Kimet Jusufi
Journal: Advances in High Energy Physics
 (2017)

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## Abstract

Considerable attention has been devoted to the wormhole physics in the past 30 years by exploring the possibilities of finding traversable wormholes without the need for exotic matter. In particular, the thin-shell wormhole formalism has been widely investigated by exploiting the cut-and-paste technique to merge two space-time regions and to research the stability of these wormholes developed by Visser. This method helps us to minimize the amount of the exotic matter. In this paper, we construct a four-dimensional, spherically symmetric, dyonic thin-shell wormhole with electric chargeQ, magnetic charge P, and dilaton charge Σ, in the context of Einstein-Maxwell-dilaton theory. We have applied Darmois-Israel formalism and the cut-and-paste method by joining together two identical space-time solutions. We carry out the dyonic thin-shell wormhole stability analyses by using a linear barotropic gas, Chaplygin gas, and logarithmic gas for the exotic matter. It is shown that, by choosing suitable parameter values as well as equation of state parameter, under specific conditions, we obtain a stable dyonic thin-shell wormhole solution. Finally, we argue that the stability domain of the dyonic thin-shell wormhole can be increased in terms of electric charge, magnetic charge, and dilaton charge.


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## Body

## 1. Introduction

Wormholes are exotic objects predicted by Einstein’s theory of gravity which act as a space-time tunnel by connecting two different regions of the universe. Though the idea of wormholes is not new [1, 2], the interest in wormholes was recently reborn by the seminal work of Morris and Thorne [3] who studied traversable wormholes. There are, however, several problematic issues related to the possible existence of wormholes; in particular, it was shown that the existence of wormholes requires the violation of energy conditions [4, 5]. Another major problem is related to the stability analysis of wormholes. On the other hand, Visser attempted to minimize the existence of the exotic matter by constructing infinitesimally small thin-shell wormholes [6–9]. Visser’s method is based on the cut-and-paste technique by joining together two identical space-time solutions and making use of the Darmois-Israel formalism [10] to compute the surface stress-energy tensor components. Finally, these results can be used to study the wormhole dynamics with the help of Lanczos equations.This method was applied to construct a number of thin-shell wormholes (TSW), including charged TSW [11, 12], TSW with a cosmological constant [13], TSW in dilaton gravity [14], TSW from the regular Hayward black hole [15], TSW in higher-dimensional Einstein-Maxwell theory [16, 17], rotating TSW [18, 19], quantum corrected TSW in Bohmian quantum mechanics [20], primordial wormholes induced from Grand Unified Theories (GUTs) [21, 22], canonical acoustic TSW, charged TSW with dilaton field, TSW with a Chaplygin gas, traversable wormholes in the anti-de Sitter space-time, TSW with a negative cosmological constant, wormholes in mimetic gravity, TSW from charged black string, cylindrical TSW, and many other interesting papers [23–58], while the stability analysis is investigated by different models, for example, linear perturbations [9] and specific equations of state (EoS) such as linear barotropic gas (LBG), Chaplygin gas (CG), and logarithmic gas (LogG) for the exotic matter [14, 59–62].Recently, Goulart found a four-dimensional, spherically symmetric, dyonic black hole and charged wormhole solution in the low-energy effective actions of string theory or supergravity theory [63, 64]. Furthermore, in [65], a time-dependent spherically symmetric black hole solution in the context of low-energy string theory was investigated. The solution found by Goulart is of particular interest since it can be written in terms of five independent parameters: the electric charge Q, the magnetic charge Σ, the value of the dilation of infinity ϕ0, and two integration constants, r1 and r2. Inspired by this work, we aim to use this solution and construct a four-dimensional TSW wormhole in the context of Einstein-Maxwell-dilaton (EMD) theory and then investigate the role of electric charge Q, magnetic charge P, and dilaton charge Σ on the stability domain of the wormhole.The structure of this paper is as follows. In Section2, we review briefly the dyonic black hole solutions. In Section 3, using Visser’s cut-and-paste technique, we construct a dyonic thin-shell wormhole (DTSW). In Section 4, we check the stability conditions for different types of gases such as LBG, CG, and LogG for the exotic matter. In Section 5, we comment on our results.


## 2. Dyonic Black Holes in the EMD Theory

In this part, we use the dyonic black hole solutions in the EMD theory found by Goulart [63]. Firstly, we consider the action of the EMD without a dilaton potential and without an axion:(1)S=∫d4x-gR-2∂μϕ∂μϕ-WϕFμνFμν,where the field strength is given by(2)Fμν=∂μAν-∂νAμ.Furthermore, for constant axion field, the bosonic sector of SU(4) version of N=4 supergravity theory is W(ϕ)=e-2ϕ [41]. It is noted that there are five independent parameters, that is, Q,P,ϕ0,r1, and r2. Accordingly, the space-time of the general spherically symmetric solution is given by the line element [63](3)ds2=-frdt2+1frdr2+hrdθ2+sin2θdφ2,where(4)fr=r-r1r-r2r+d0r+d1,hr=r+d0r+d1,e2ϕ=e2ϕ0r+d1r+d0,Frt=e2ϕ0Qr+d02,Fθφ=Psin⁡θ,with(5)d0=-r1+r2±r1-r22+8e2ϕ0Q22,d1=-r1+r2±r1-r22+8e-2ϕ0P22.Note that the corresponding electric and magnetic charges are Q and P, respectively. ϕ0 stands for the value of the dilaton at infinity. Furthermore, there are two integration constants, that is, r1 and r2. On the other hand, d0 and d1 are dependent constants inasmuch as they transform into each other under S-duality (i.e., Q↔P and ϕ→-ϕ). It is noted that e-2ϕ, which is the dilaton coupling, is also invariant. Here, r1 and r2 are the inner and outer horizons, respectively [63].The Hawking temperature is calculated by(6)T=14πr2-r1r2+d0r2+d1,and the entropy of the black hole is(7)S=πr2+d0r2+d1. One can also define the dilaton charge as follows:(8)Σ=14π∫dΣμ∇μϕ=d0-d12,where, depending on the values of electric/magnetic charge of black hole, it can be positive or negative. Firstly, the four parameters’ (Q,P,ϕ0,M) dyonic solution is found in [37]. Here, the key point is that there is no boundary condition on r1 and r2 to make this dyonic black hole.The Ricci scalar is calculated as follows:(9)R=d0-d12r-r1r-r22r+d03r+d13. The domain of h(r)≥0 is restricted with the causality. The singularity is found at rS=-d0 for d0>d1, or at rS=-d1 for d1>d0.One of the special cases which we use to construct a DTSW whend1=-d0 is that the dilaton charge is a constant d0 such as d0=Σ. Furthermore, we suppose that (r1+r2)=2M and (r1r2)=r02 [37]. The solution becomes(10)fr=r-r1r-r2r2-Σ2,hr=r2-Σ2e2ϕ=e2ϕ0r-Σr+Σ,Frt=e2ϕ0Qr+Σ2,Fθφ=Psin⁡θ. One can find the magnetically charged solutions of [38, 39] by using Q=0 and also the Schwarzschild solution by setting P=0.


## 3. Construction of DTSW

Let us now proceed to use the cut-and-paste technique to construct a DTSW using metric (3). Consider two spherically symmetric space-time solutions of the dyonic black hole metric in 4 dimensions and then remove from each four-dimensional manifold M(±) the regions described by [9](11)M±=r±≤a∣a>rh, where a is the radius of the throat of the DTSW with an important condition a>rh. In other words, a should be greater than the event horizon in order to avoid the formation of an event horizon. Next, paste these two manifolds at the boundary hypersurface given by Σ(±)=r(±)=a,a>rh which results with a geodesically complete manifold M=M+∪M-. According to the Darmois-Israel formalism, we can choose the coordinates on M as xα=(t,r,θ,φ), while the coordinates on the induced metric Σ are ξi=(τ,θ,φ). For the parametric equation on Σ, we can write(12)Σ:Fr,τ=r-aτ=0.Our main goal is to compare various characteristics of EMD theory and dyonic black hole, such as the surface stress-energy tensor and the basic question of stability. For this purpose, we define the dynamical induced metric onΣ that can now be written in terms of the proper time τ on the shell, where a=a(τ), as follows:(13)dsΣ2=-dτ2+aτ2dθ2+sin2θdφ2.The junction conditions onΣ imply from the Lanczos equations(14)Sij=-18πKij-δijK, in which Sij=diag(-σ,pθ,pφ) is the energy momentum tensor on the thin shell and K and [Kij] are defined as K=trace[Kii] and [Kij]=Kij+-Kij-, respectively. Furthermore, the extrinsic curvature Kij is defined by(15)Kij±=-nμ±∂2xμ∂ξi∂ξj+Γαβμ∂xα∂ξi∂xβ∂ξjΣ.We can choose the unit vectorsnμ(±), such that nμnμ=1 and normal to M(±) as follows:(16)nμ±=±gαβ∂F∂xα∂F∂xβ-1/2∂F∂xμΣ.Adopting the orthonormal basis{eτ^,eθ^,eφ^}(eτ^=eτ,eθ^=[h(a)]-1/2eθ,eφ^=[h(a)sin2θ]-1/2eφ), for metric (3), the extrinsic curvature components are found as [24](17)Kθ^θ^±=Kφ^φ^±=±h′a2hafa+a˙2,Kτ^τ^±=∓2a¨+f′a2fa+a˙2,where the prime and the dot represent the derivatives with respect to r and τ, respectively. With the definitions of [Ki^j^]≡Ki^j^+-Ki^j^- and K=tr[Ki^j^]=[Ki^i^] and the introduction of the surface stress-energy tensor Si^j^=diag(σ,pθ^,pφ^), we have the Einstein equations on the shell (also called the Lanczos equations):(18)-Ki^j^+Kgi^j^=8πSi^j^,which in our case results in a shell of radius a with energy density σ and transverse pressure p=pθ^=pφ^. Using the above results from the Lanczos equations, one can easily check that the surface density and the surface pressure are given by the following relations [24, 25]:(19)σ=-fa+a˙24πh′aha,(20)p=fa+a˙28π2a¨+f′afa+a˙2+h′aha.Note that the energy density is negative at the throat because of the flare-out condition in which the area is minimal at the throat (thenh(r) increases for r close to a and h′(a)>0), so we have exotic matter. From the last two equations, we can now write the static configuration of radius a; by setting a˙=0 and a¨=0, we get(21)σ0=-fa04πh′a0ha0,(22)p0=fa08πf′a0fa0+h′a0ha0.From (21), we see that the surface density is negative (i.e., σ0<0); as a consequence of this, the WEC is violated. The amount of exotic matter concentrated at the wormhole is calculated by the following integral:(23)Ωσ=∫-gρ+prd3x.In the case of a TSW, we havepr=0 and ρ=σδ(r-a), where δ(r-a) is the Dirac delta function. The above integral can be easily evaluated if we first make use of the Dirac delta function(24)Ωσ=∫02π∫0π∫-∞∞σ-gδr-adrdθdφ.Substituting the value of energy density in the last equation, for the energy density located on a thin-shell surface in static configuration, we find(25)Ωσ=-2a0a0-r1a0-r2a02-Σ2.To analyze the attractive and repulsive nature of the wormhole, we can calculate the observer’s four-accelerationaμ=uν∇νuμ, where the four-velocity reads uμ=(1/f(r),0,0,0). For the radial component of the four-acceleration, we find(26)ar=Γttrdtdτ2=a02r1+r2-2a0Σ2+r1r2+Σ2r1+r22a02-Σ22.One can easily observe that the test particle obeys the equation of motion(27)d2rdτ2=-Γttrdtdτ2=-ar.We conclude from the last equation that ifar=0, we get the geodesic equation, while the wormhole is attractive when ar>0 and repulsive when ar<0.


## 4. Stability Analysis

In this section, using the formalism developed in Section3, we calculate the potential and define the stability method for DTSW. From the energy conservation, we have [24](28)ddτσA+pdAdτ=h′a2-2hah′′aa˙fa+a˙22ha,where the area of the wormhole is calculated by A=4πh(a). It is noted that the internal energy of the throat is located at the left side of (28) as a first term. Then, the second term represents the work done by the internal forces of the throat; on the other hand, there is a flux term in the right side of the equation. Furthermore, to calculate the equation of dynamics of the wormhole, we use σ(a) in (19) and find this simple equation:(29)a˙2=-Va,with potential(30)Va=fa-16π2hah′aσa2.A Taylor expansion to the second order of the potential V(a) around the static solution yields [24](31)Va=Va0+V′a0a-a0+V′′a02a-a02+Oa-a03.From (30), the first derivative of V(a) is(32)V′a=f′a-32π2σahah′a1-hah′′ah′a2σa+hah′aσ′a,and the last equation takes the form(33)V′a=f′a+16π2σahah′aσa+2pa.The second derivative of the potential is(34)V′′a=f′′a+16π2hah′aσ′a+1-hah′′ah′a2σaσa+2pa+hah′aσaσ′a+2p′a.Sinceσ′(a)+2p′(a)=σ′(a)[1+2p′(a)/σ′(a)], replacing the parameter p=ψ(σ) and ψ′=dp/dσ=p′/σ′, we have that σ′(a)+2p′(a)=σ′(a)(1+2ψ′), and using (35) again, we obtain(35)V′′a0=f′′a0-8π2σ0+2p02+2σ032-ha0h′′a0h′a02σ0+p01+2ψ′.The wormhole is stable if and only ifV′′(a0)>0.


### 4.1. Stability Analysis of DTSW via the LBG

In what follows, we will use three different gas models for the exotic matter to explore the stability analysis: LBG [40], CG [61, 62], and finally LogG [15].The equation of state of LBG [9, 14, 59, 60] is given by(36)ψ=ωσ, and hence(37)ψ′σ0=ω, where ω is a constant parameter. For more useful information as regards the effects of the parameters Σ, r1, and r2, we show graphically the DTSW stability in terms of ω and a0, as depicted in Figure 1.Figure 1
Stability regions of DTSW in terms ofω and radius of the throat a0 for different values of Σ, r1, and r2.
(a)
(b)
(c)
(d)By changing the values ofΣ, r1, and r2, which encode the effects of electric Q, magnetic P, and dilaton charge ϕ0 on the DTSW stability, we see from Figure 1 that in two cases the region of stability is below the curve in the interval to the right of the asymptote, while in two other cases the stability region is simply below the curve. The region of stability is denoted by S.


### 4.2. Stability Analysis of DTSW via CG

The equation of state of CG that we considered is given by [61](38)ψ=ω1σ-1σ0+p0,and one naturally finds (39)ψ′σ0=-ωσ02.After inserting (38) into (29), we plot the stability regions of DTSW supported by CG in terms of V′′(a0) and a0 as shown in Figure 2. It is worth mentioning that in three cases the region of stability is above the curve in the interval to the right of the asymptote, while in one case stability region is below the curve in the interval to the right of the asymptote. The region of stability is denoted by S.Figure 2
Stability regions of DTSW in terms ofω as a function of the throat a0 for different values of Σ, r1, and r2.


### 4.3. Stability Analysis of DTSW via LogG

In our final example, the equation of state for LogG is selected as follows [15]:(40)ψ=ωlnσσ0+p0, which leads to(41)ψ′σ0=ωσ0. After inserting the above expression into (29), we show the stability regions of TSW supported by LogG in Figure 3. In this case, we see that the region of stability is above the curve in the interval to the right of the asymptote. The region of stability is denoted by S.Figure 3
Stability regions of DTSW in terms ofω and radius of the throat a0 for different values of Σ, r1, and r2.


## 4.1. Stability Analysis of DTSW via the LBG

In what follows, we will use three different gas models for the exotic matter to explore the stability analysis: LBG [40], CG [61, 62], and finally LogG [15].The equation of state of LBG [9, 14, 59, 60] is given by(36)ψ=ωσ, and hence(37)ψ′σ0=ω, where ω is a constant parameter. For more useful information as regards the effects of the parameters Σ, r1, and r2, we show graphically the DTSW stability in terms of ω and a0, as depicted in Figure 1.Figure 1
Stability regions of DTSW in terms ofω and radius of the throat a0 for different values of Σ, r1, and r2.
(a)
(b)
(c)
(d)By changing the values ofΣ, r1, and r2, which encode the effects of electric Q, magnetic P, and dilaton charge ϕ0 on the DTSW stability, we see from Figure 1 that in two cases the region of stability is below the curve in the interval to the right of the asymptote, while in two other cases the stability region is simply below the curve. The region of stability is denoted by S.


## 4.2. Stability Analysis of DTSW via CG

The equation of state of CG that we considered is given by [61](38)ψ=ω1σ-1σ0+p0,and one naturally finds (39)ψ′σ0=-ωσ02.After inserting (38) into (29), we plot the stability regions of DTSW supported by CG in terms of V′′(a0) and a0 as shown in Figure 2. It is worth mentioning that in three cases the region of stability is above the curve in the interval to the right of the asymptote, while in one case stability region is below the curve in the interval to the right of the asymptote. The region of stability is denoted by S.Figure 2
Stability regions of DTSW in terms ofω as a function of the throat a0 for different values of Σ, r1, and r2.


## 4.3. Stability Analysis of DTSW via LogG

In our final example, the equation of state for LogG is selected as follows [15]:(40)ψ=ωlnσσ0+p0, which leads to(41)ψ′σ0=ωσ0. After inserting the above expression into (29), we show the stability regions of TSW supported by LogG in Figure 3. In this case, we see that the region of stability is above the curve in the interval to the right of the asymptote. The region of stability is denoted by S.Figure 3
Stability regions of DTSW in terms ofω and radius of the throat a0 for different values of Σ, r1, and r2.


## 5. Conclusion

In this work, we have constructed a stable DTSW in the context of EMD theory. In particular, we explore the role of three parameters—Σ, r1, and r2—which encode the effects of electric charge Q, magnetic charge P, and dilaton charge ϕ0 on the wormhole stability. The surface stress at the wormhole throat is computed via Darmois-Israel formalism while the stability analyses are carried out by using three different models. As a first model, we consider LBG and show that the wormhole can be stable by choosing suitable values of parameters Σ, r1, and r2. In the second case, we focus on the stability analyses using a CG for the exotic matter and show this by choosing suitable values of parameters Σ, r1, and r2. Finally, we use LogG for the exotic matter and show similar results. The results show that electric charge, magnetic charge, and dilaton charge play an important role in DTSW by increasing the stability domain of the wormhole. A particularly interesting finding is that, for suitable values of a0, the stable solutions exist in the DTSW for each value of ω that is chosen. We conclude that DTSW is linearly stable for variable EoS, which supports the fact that the presence of EoS and dilaton/electric/magnetic charges induces stability in the WH geometry. In particular, we can see from Figures 1(a) and 1(b) that, by keeping r1 and r2 fixed and changing Σ, the vertical asymptote is slightly shifted to the left with the increase of Σ. This clearly indicates the increase of the stability region in the right of the asymptote. On the other hand, another model of charged TSW was constructed by Eiroa and Simeone [14] in low-energy string gravity, which supports our results. Their results show ρ and p on the shell for the null dilaton coupling parameter and it is shown that exotic matter is localized; moreover, they managed to minimize the exotic matter needed using the stronger dilaton-Maxwell coupling.



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