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whenever the agenda is about wormholes exotic matter ( i.e. matter violating the energy conditions ) continues to occupy a major issue in general relativity @xcite . it is a fact that einstein s equations admit wormhole solutions that require such matter for its maintenance . in quantum theory temporary violation of energy conditions is permissible but in classical physics this can hardly be justified . one way to minimize such exotic matter , even if we can not ignore it completely , is to concentrate it on a thin - shell . this seemed feasible , because general relativity admits such thin - shell solutions and by employing these shells at the throat region may provide the necessary repulsion to support the wormhole against collapse . the ultimate aim of course , is to get rid of exotic matter completely , no matter how small . in the 4-dimensional ( 4d ) general relativity with a cosmological term , however , such a dream never turned into reality . for this reason the next search should naturally cover extensions of general relativity to higher dimensions and with additional structures . one such possibility that received a great deal of attention in recent years , for a number of reasons , is the gauss - bonnet ( gb ) extension of general relativity @xcite . in the brane - world scenario our universe is modelled as a brane in a 5d bulk universe in which the higher order curvature terms , and therefore the gb gravity comes in naturally . einstein - gauss - bonnet ( egb ) gravity , with additional sources such as maxwell , yang - mills , dilaton etc . has already been investigated extensively in the literature @xcite . not to mention , all these theories admit black hole , wormhole @xcite and other physically interesting solutions . as it is the usual trend in theoretical physics , each new parameter invokes new hopes and from that token , the gb parameter @xmath2 does the same . although the case @xmath3 has been exalted much more than the case @xmath4 in egb gravity so far @xcite ( and references cited therein ) , it turns out here in the stable , normal matter thin - shell wormholes that the latter comes first time to the fore . construction and maintenance of thin - shell wormholes has been the subject of a large literature , so that we shall provide only a brief review here . instead , one class @xcite that made use of non - exotic matter for its maintenance attracted our interest and we intend to analyze its stability in this paper . this is the 5d thin - shell solution of einstein - maxwell - gauss - bonnet ( emgb ) gravity , whose radius is identified with the minimum radius of the wormhole . for this purpose we employ radial , linear perturbations to cast the motion into a potential - well problem in the background . in doing this , a reasonable assumption employed routinely , which is adopted here also , is to relate pressure and energy density by a linear expression @xcite . for special choices of parameters we obtain islands of stability for such wormholes . to this end , we make use of numerical computation and plotting since the problem involves highly intricate functions for an analytical treatment .    the paper is organized as follows . in sec . ii the 5d emgb thin - shell wormhole formalism has been reviewed briefly . we perturb the wormhole through radial linear perturbation and cast the problem into a potential - well problem in sec . iii . in sec . iv we impose constraint conditions on parameters to determine possible stable regions through numerical analysis . the paper ends with conclusion which appears in sec . the action of emgb gravity in 5d ( without cosmological constant , i.e. @xmath5 ) is@xmath6 in which @xmath7 is related to the 5d newton constant and @xmath8 is the gb parameter . beside the maxwell lagrangian the gb lagrangian @xmath9 consists of the quadratic scalar invariants in the combination @xmath10 in which @xmath11scalar curvature , @xmath12ricci tensor and @xmath13riemann tensor . variational principle of @xmath14 with respect to @xmath15 yields @xmath16where the lovelock ( @xmath17 ) and maxwell ( @xmath18 ) tensors respectively are @xmath19    @xmath20    the einstein tensor @xmath21 is to be found from our metric ansatz@xmath22 in which @xmath23 will be determined from ( 3 ) . a thin - shell wormhole is constructed in emgb theory as follows . two copies of the spacetime are chosen from which the regions@xmath24are removed . we note that @xmath25 will be identified in the sequel as the radius of the thin - shell and @xmath26 stands for the event horizon radius . ( note that our notation @xmath25 corresponds to @xmath27 in ref . other notations all agree with those in ref . @xcite ) . the boundary , time - like surface @xmath28of each @xmath29 , accordingly will be@xmath30next , these surfaces are identified on @xmath31 with a surface energy - momentum of a thin - shell such that geodesic completeness holds . following the darmois - israel formalism @xcite in terms of the original coordinates @xmath32 we define @xmath33 , with @xmath34 the proper time . the gb extension of the thin - shell em theory requires further modifications . this entails the generalized darmois - israel boundary conditions @xcite , where the surface energy - momentum tensor is expressed by @xmath35diag@xmath36 . we are interested in the thin - shell geometry whose radius is assumed a function of @xmath34 , so that the hypersurface becomes@xmath37the generalized darmois - israel conditions on @xmath38 take the form @xmath39where a bracket implies a jump across @xmath38 , and @xmath40 is the induced metric on @xmath38 with normal vector @xmath41 @xmath42 is the extrinsic curvature ( with trace @xmath43 ) , defined by @xmath44the remaining expressions are as follows . the divergence - free part of the riemann tensor @xmath45 and the tensor @xmath46 ( with trace @xmath47 ) are given by@xmath48 .\end{aligned}\]]the emgb solution that will be employed as a thin - shell solution with a normal matter @xcite is given by ( with @xmath49)@xmath50with constants , @xmath51mass and @xmath52charge . for a black hole solution the inner ( @xmath53 ) and event horizons ( @xmath54 ) are @xmath55 ^{1/2}}.\]]by employing the solution ( 14 ) we determine the surface energy - momentum on the thin - shell , which will play the major role in the perturbation . we shall address this problem in the next section . in order to study the radial perturbations of the wormhole we take the throat radius as a function of the proper time , i.e. , @xmath56 . based on the generalized birkhoff theorem , for @xmath57 the geometry will be given still by ( 6 ) . for the metric function @xmath58 given in ( 14 ) one finds the energy density and pressures as @xcite @xmath59 , \\ s_{\theta } ^{\theta } & = & s_{\phi } ^{\phi } = s_{\psi } ^{\psi } = p=\frac{1}{4\pi   } \left [ \frac{2\delta } { a}+\frac{\ell } { \delta } -\frac{4\alpha } { a^{2}}% \left ( \ell \delta -\frac{\ell } { \delta } \left ( 1+\dot{a}^{2}\right ) -2\ddot{% a}\delta \right ) \right ] , \end{aligned}\]]where @xmath60 and @xmath61 in which @xmath62we note that in our notation a dot denotes derivative with respect to the proper time @xmath34 and a prime implies differentiation with respect to the argument of the function . by a simple substitution one can show that , the conservation equation @xmath63is satisfied . the static configuration of radius @xmath64 has the following density and pressures @xmath65 , \\ p_{0 } & = & \frac{\sqrt{f\left ( a_{0}\right ) } } { 4\pi } \left [ \frac{2}{a_{0}}+% \frac{f^{\prime } \left ( a_{0}\right ) } { 2f\left ( a_{0}\right ) } -\frac{2\alpha   } { a_{0}^{2}}\frac{f^{\prime } \left ( a_{0}\right ) } { f\left ( a_{0}\right ) } % \left ( f\left ( a_{0}\right ) -1\right ) \right ] .\end{aligned}\ ] ]    in what follows we shall study small radial perturbations around the radius of equilibrium @xmath66 to this end we adapt a linear relation between @xmath67 and @xmath68 as @xcite @xmath69here since we are interested in the wormholes which are supported by normal matter , @xmath70 is the speed of sound . by virtue of eq.s ( 19 ) and ( 22 ) we find the energy density in the form @xmath71this , together with ( 16 ) lead us to the equation of motion for the radius of the throat , which reads@xmath72 = \left ( \frac{\sigma _ { 0+}p_{0}}{\beta ^{2}+1}\right ) \left ( \frac{a_{0}}{a}% \right ) ^{3\left ( \beta ^{2}+1\right ) } + \frac{\beta ^{2}\sigma _ { 0-}p_{0}}{% \beta ^{2}+1}.\]]after some manipulation this can be cast into @xmath73where @xmath74 ^{1/3}-\frac{b}{\left [ \sqrt{a^{2}+b^{3}}-a\right ] ^{1/3}}\right ) ^{2}\]]in which the functions @xmath75 and @xmath76 are @xmath77 , \\ b & = & \frac{a^{2}}{8\alpha } + \frac{1-f\left ( a\right ) } { 2}.\end{aligned}\]]we notice that @xmath78 and more tediously @xmath79 both vanish at @xmath80 the stability requirement for equilibrium reduces therefore to the determination of @xmath81 and it is needless to add that , @xmath82 is complicated enough for an immediate analytical result . for this reason we shall proceed through numerical calculation to see whether stability regions/ islands develop or not . since the hopes for obtaining thin - shell wormholes with normal matter when @xmath3 have already been dashed 5 , we shall investigate here only the case for @xmath83    in order to analyze the behavior of @xmath82 ( and its second derivative ) we introduce new parameterization as follows@xmath84accordingly , our new variables @xmath85 @xmath86 @xmath87 @xmath75 and @xmath76 take the following forms @xmath88and @xmath89 , \\ \tilde{p}_{0 } & = & \frac{\sqrt{f\left ( \tilde{a}_{0}\right ) } } { 4\pi } \left [   \frac{2}{\tilde{a}_{0}}+\frac{f^{\prime } \left ( \tilde{a}_{0}\right ) } { % 2f\left ( \tilde{a}_{0}\right ) } + \frac{2}{\tilde{a}_{0}^{2}}\frac{f^{\prime } \left ( \tilde{a}_{0}\right ) } { f\left ( \tilde{a}_{0}\right ) } \left ( f\left (   \tilde{a}_{0}\right ) -1\right ) \right ] , \end{aligned}\]]@xmath90 , \\ b & = & -\frac{\tilde{a}^{2}}{8}+\frac{1-f\left ( \tilde{a}\right ) } { 2}.\end{aligned}\]]following this parametrization our eq . ( 25 ) takes the form @xmath91where@xmath92 in the next section we explore all possible constraints on our parameters that must satisfy to materialize a stable , normal matter wormhole through the requirement @xmath93 @xmath94 ) starting from the metric function we must have    @xmath95    @xmath96 ) in the potential , the reality condition requires also that @xmath97at the location of the throat this amounts to @xmath98or after some manipulation it yields @xmath99this is equivalent to@xmath100    @xmath101 ) our last constraint condition concerns , the positivity of the energy density , which means that @xmath102this implies , from ( 31 ) that @xmath103 < 0\]]or equivalently@xmath104it is remarkable to observe now that the foregoing constraints ( @xmath105 ) on our parameters can all be expressed as a single constraint condition , namely @xmath100    we plot @xmath106 from ( 26 ) for various fixed values of mass and charge , as a projection into the plane with coordinates @xmath107 and @xmath108 in other words , we search and identify the regions for which @xmath109 , in @xmath110dimensional figures considered as a projection in the @xmath111 plane . the metric function @xmath112 and energy density @xmath113 behavior also are given in fig.s 1 - 4 . it is evident from fig.s 1 - 4 that for increasing charge the stability regions shrink to smaller domains and tends ultimately to disappear completely . for smaller @xmath114 bounds  we obtain fluctuations in @xmath115 which is smooth otherwise .    in each plot it is observed that the maximum of @xmath116 occurs at the right - below corner ( say , at @xmath117 )  which decreases to the left ( with @xmath114 ) and in the upward direction ( with @xmath107 ) . beyond certain limit ( say @xmath118 ) , the region of instability takes the start . the proper time domain of stability can be computed from ( 35 ) as @xmath119from a distant observer s point of view the timespan @xmath120 can be found by using the radial geodesics lagrangian which admits the energy integral @xmath121this gives the lifetime of each stability region determined by @xmath122once @xmath118 ( @xmath123 ) are found numerically , assuming that no zeros of @xmath124 and @xmath125 occurs for @xmath126 the lifespan of each stability island can be determined . we must admit that the mathematical complexity discouraged us to search for possible metastable region that may be triggered by employing a semi - classical treatment . our numerical analysis shows that for @xmath0 and specific ranges of mass and charge the 5d emgb thin - shell wormholes with normal matter can be made stable against linear , radial perturbations . the fact that for @xmath127 there is no such wormholes is well - known . the magnitude of @xmath8 is irrelevant to the stability analysis . this reflects the universality of wormholes in parallel with black holes , i.e. , the fact that they arise at each scale . stable regions develop for each set of finely - tuned parameters which determine the lifespan of each such region . beyond those regions instability takes the start . our study concerns entirely the exact emgb gravity solution given in ref . it is our belief that beside emgb theory in different theories also such stable , normal - matter wormholes are abound , which will be our next venture in this line of research . 1 : @xmath128 region ( @xmath129 , @xmath130 ) for various ranges of @xmath107 and @xmath108 the lower and upper limits of the parameters are evident in the figure . the metric function @xmath131 and @xmath132 , are also indicated in the smaller figures .
recently in ( phys . rev . 
 d 76 , 087502 ( 2007 ) and phys . rev . 
 d 77 , 089903(e ) ( 2008 ) ) a thin - shell wormhole has been introduced in 5-dimensional einstein - maxwell - gauss - bonnet ( emgb ) gravity which was supported by normal matter . 
 we wish to consider this solution and investigate its stability . 
 our analysis shows that for the gauss - bonnet ( gb ) parameter @xmath0 stability regions form for a narrow band of finely - tuned mass and charge . for the case @xmath1 , we iterate once more that no stable 
 , normal matter thin - shell wormhole exists .