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curved spacetimes for which the ricci scalar @xmath0 is identically zero have been known in general relativity ( gr ) since the discovery of the schwarzschild solution . while , in gr , the schwarzschild is a vacuum spacetime for which both @xmath0 and @xmath1 are zero , the reissner - nordstrom geometry has @xmath2 but @xmath3 , thereby implying the presence of traceless matter . it has been further shown in @xcite that a generalisation of schwarzschild spacetime with @xmath4 everywhere finite and non - zero and @xmath2 can be obtained . more recently @xcite , this @xmath2 wormhole spacetime has been found to be a solution in a scalar - tensor theory of gravity which is also the low energy , effective , on - brane gravity theory for the warped two - brane randall - sundrum model . the @xmath2 spacetime in @xcite when viewed as a solution in gr , requires matter that violates the weak energy condition ( wec ) . however , as shown in @xcite , in the context of the scalar - tensor theory , the required matter does not violate the wec and the scalar field ( radion ) is also finite and non - zero everywhere . in our work here , we further generalise the spacetime studied in @xcite . in particular , newer @xmath2 spacetimes with @xmath4 everywhere non - zero are constructed and shown to be solutions in scalar - tensor gravity . a sub - class of such spacetimes are lorentzian wormholes with the required matter satisfying the wec .    in our analysis here , we begin with a general static , spherisymmetric line element for which the @xmath2 constraint is written as a differential equation for @xmath5 . obviously , this differential equation contains @xmath6 ( or , @xmath7 where @xmath8 ) and its derivatives in its coefficients . it can therefore be solved once we provide our choice for @xmath7 . assuming @xmath7 as one should for a lorentzian wormhole , we find that for some specific choices ( eg . the schwarzschild wormhole and the ellis bronnikov wormhole @xcite ) the differential equation for @xmath9 can indeed be solved . we obtain a wide class of geometries in this manner . thereafter , we show how these geometries fare in terms of the wec violation issue . as mentioned above , in gr we require wec violation for the matter that threads non - singular wormhole spacetimes . however , it has been shown recently that , within gr , a wormhole ( which is topologically different from the morris - thorne class ) can be supported by a negative cosmological constant ( which is not quite exotic , wec violating matter ) @xcite . further , it is known that certain wormhole solutions which have been constructed in various modified theories of gravity such as brans - dicke theory ( scalar - tensor gravity with constant coupling @xmath10 ) @xcite , @xmath11-gravity @xcite , gauss - bonnet gravity @xcite , third - order lovelock gravity @xcite , eddington - inspired born - infeld gravity @xcite , mimetic gravity @xcite , dgp gravity @xcite , do not require wec violating matter .    in our work here , we consider scalar - tensor gravity with a non - constant coupling @xmath12 and in the presence of matter stress - energy . through our specific solutions , we show that there may not be wec violation for the matter required to support a lorentzian wormhole . it must be noted though that the timelike convergence condition is still violated , as it must be , for wormholes . the essential point is that the relation between the convergence condition and the energy condition is not as it is in gr , because of which there is extra freedom to avoid any violation in a modified theory of gravity ( here , scalar - tensor gravity ) @xcite . our paper is organised as follows . in section [ sec : stg ] , we briefly recall scalar - tensor gravity both as an independent theory and in the context of braneworld gravity . we provide the details of our construction of @xmath2 spacetimes in section [ sec : solution ] . section [ sec : wec ] and [ sec : m0 ] deal with the status of the wec . in section [ sec : redshift ] , we study gravitational redshift in the wormhole spacetimes . the existence of stable and unstable circular orbits and the issue of traversability are addressed in section [ sec : traversability ] . finally , we conclude in section [ sec : conclusion ] . scalar - tensor theories are well - studied from various angles and perspectives . the first such theory was , of course , the jordan - brans - dicke theory @xcite . a scalar field @xmath13 and an extra parameter @xmath10 characterises such theories . though originally considered a constant , a modified version incorporates a @xmath13-dependent @xmath10 @xcite . in general , the einstein field equations for such theories are given as : @xmath14 where the @xmath15 is the matter energy - momentum tensor and the metric @xmath16 is written in the jordan frame where the action is not in the standard canonical ( einstein - hilbert ) form . viewed as an independent competitor of gr , a scalar tensor theory could be specified by providing a form for @xmath12 . note that @xmath13 must satisfy the equation , @xmath17 various aspects of scalar - tensor theories have been extensively studied and we will not review them here . see @xcite and references therein for detailed reviews . taking the trace of ( [ eq : field_equation1 ] ) and using ( [ eq : scalar_equation ] ) , one obtains @xmath18 note that traceless matter implies @xmath2 in gr . but , in scalar - tensor gravity , traceless matter does not imply @xmath2 in general . but , for the specific form of @xmath12 given by @xmath19 @xmath20 and hence , traceless matter implies @xmath2 . here , @xmath21 is an integration constant . as we shall see , the case @xmath22 arises in the low energy , effective on - brane quasi - scalar - tensor gravity theory developed by kanno and soda @xcite in the context of the randall - sundrum two - brane model . it is well - known that scalar - tensor theories arise as effective theories of gravity in diverse contexts . for example , the low - energy effective gravity theory which emerges out of superstring theory contains a scalar known as the dilaton and the theory is a @xmath23 brans - dicke theory @xcite . in the braneworld scenario , our four dimensional visible universe is considered as a lower dimensional hypersurface called a brane , which is embedded in a higher dimensional bulk . in the low energy limit , the field equation governing bulk gravity , leads to effective field equations for gravity on the brane . the presence of the extra dimensional bulk leaves its imprint by modifying the einstein field equations on the brane . the most popular among the on - brane effective einstein equations was obtained by shiromizu - maeda - sasaki @xcite in the context of a single - brane model and contains a non - local term ( bulk - weyl dependent @xmath24 ) . the on - brane effective einstein equations obtained by kanno and soda @xcite in the context of the two - brane randall sundrum model , however , does not contain any non - local contribution . earlier , the non - local @xmath25 was used to obtain @xmath2 spacetime solutions @xcite ( mostly wormhole solutions ) in the single - brane effective theory of shiromizu - maeda - sasaki . in the effective on - brane theory of kanno and soda , we have an effective energy momentum tensor ( other than the matter on the branes ) constructed from the scalar radion field ( which measures the inter - brane distance ) and its derivatives . the radion field plays a crucial role in obtaining both static , spherically symmetric spacetimes and cosmological solutions @xcite . it has been shown that the presence of the radion field can correctly reproduce the observed virial mass of galaxy clusters and observed galaxy rotation curves . thus , the radion field can act as a possible dark matter candidate @xcite . let us now briefly review the low energy effective on - brane quasi - scalar - tensor theory developed by kanno and soda @xcite . the action is given as , @xmath26 where @xmath27 , @xmath28 and @xmath29 are the five dimensional gravitational coupling constant , the five dimensional curvature scalar and the induced metric on the branes , respectively . @xmath30 and @xmath31 are the brane - tensions . the positive tension brane a ( planck brane ) and the negative tension brane b ( visible brane ) are respectively placed at fixed bulk locations ( @xmath32 and @xmath33 ) . the bulk line element is , @xmath34 where the bulk curvature radius is @xmath35 . assuming the brane curvature radius @xmath36 as large compared to the bulk curvature radius @xmath35 , i.e. , @xmath37 , kanno and soda used a low energy expansion scheme ( the gradient expansion method ) wherein the bulk metric and the extrinsic curvature are expanded in powers of @xmath38 . they obtained the effective field equations , @xmath39 on the visible brane . here , @xmath40 , @xmath41 , @xmath42 and @xmath43 are respectively the radion field , the matter on the planck brane , the matter on the visible brane and the metric tensor on the visible brane . the proper distance @xmath44 between the branes is defined as @xmath45 the radion field @xmath13 satisfies the following equation of motion on the visible brane : @xmath46 where both the traces @xmath47 and @xmath48 are taken with respect to the metric tensor @xmath43 and the coupling function is given by @xmath49 using the radion equation of motion in the trace of the field equation ( [ eq : field_equation ] ) , we obtain @xmath50 , where @xmath0 is the curvature scalar on the visible brane . note that this is the @xmath22 case discussed earlier . therefore , traceless matter implies @xmath2 in this theory . using the transformation @xmath51 , we can express the radion equation of motion in the form @xmath52    in the subsequent sections , we obtain our new solutions . it is important to note that the solutions could either be viewed as solutions in a scalar - tensor theory which has no link with extra dimensions or as solutions in the context of braneworld gravity . in obtaining the solutions , we take the planck brane to be devoid of matter , i.e. , @xmath53 . further , henceforth , we refer to this on - brane scalar - tensor gravity as kanno - soda theory of gravity because of the specific form of the coupling function @xmath12 given in ( [ eq : coupling2 ] ) . we begin with the static , spherically symmetric line element given as @xmath54 where @xmath9 and @xmath7 are unknown functions to be determined . we are interested in vanishing curvature scalar ( @xmath2 ) solutions . the @xmath2 constraint yields the following second order differential equation for @xmath9 : @xmath55 the above equation can also be re - written as a first order differential equation for @xmath7 . attempts have been made to find out @xmath2 spacetime solutions in the past @xcite . in most of these articles , the authors have ( a ) considered known forms of @xmath9 and obtained solutions by solving the first order differential equation for @xmath7 or ( b ) used some known @xmath2 solutions present in the literature and analysed them in the context of an effective theory of gravity . in our work , we take a different route . we specify the form of @xmath7 and solve the second order differential equation for @xmath9 . this way we hope to obtain a more general solution different from those obtained earlier . another advantage of specifying the form of the shape function @xmath7 is that we can choose to have wormhole solutions with a desired spatial shape . one can take different forms of @xmath7 . in our work here , we take @xmath56 . putting @xmath57 in the @xmath2 equation , we obtain @xmath58 after some manipulations , the above equation can be re - written in the form @xmath59=0 \nonumber\ ] ] which can be integrated to obtain @xmath60 where @xmath61 and @xmath62 are integration constants . therefore , the line element becomes @xmath63 ^ 2dt^2 \nonumber \\ & & + \frac{dr^2}{1-\frac{2m}{r}-\frac{\beta}{r^2}}+r^2d\omega^2 \label{eq : general_metric}\end{aligned}\ ] ] it is useful to note that , when @xmath64 or @xmath65 , the @xmath66 is of the general form , @xmath67 . however , when @xmath68 , @xmath69 are both non - zero , this generic form does not hold and we have @xmath70 . however , it is important to note that , in all cases , the red - shift function is finite and non - zero everywhere . we now look at special cases of our general solution . some of these special cases reproduce the @xmath2 spacetimes obtained earlier by other authors . for @xmath65 , @xmath71 and @xmath72 , the spacetime takes the form @xmath73 ^ 2dt^2+\frac{dr^2}{1-\frac{2m}{r}}+r^2d\omega^2\ ] ] this spatially schwarzschild spacetime has been obtained in @xcite . this spacetime represents either a wormhole or a naked singularity for @xmath74 . for @xmath75 and @xmath76 , the spatially schwarzschild solution reduces to the schwarzschild spacetime . taking @xmath77 and @xmath78 in this solution , we obtain the spatial schwarzschild wormhole obtained in @xcite . for @xmath79 , @xmath80 , @xmath81 and @xmath82 , the general metric takes the form @xmath83 where @xmath84 . this metric has been obtained in @xcite . this metric reduces to the extreme reissner - nordstr@xmath85 m form for @xmath86 , i.e. , @xmath87 . the above metric fails to produce spatial ellis geometry because @xmath88 implies @xmath89 . however , we can obtain spatial ellis geometry from our general solution by taking @xmath64 and @xmath90 . the line element becomes @xmath91 this new metric has not been obtained before . note that , due to the presence of the @xmath92 factor in @xmath66 , the above metric is different from the spatial ellis geometry obtained in @xcite for vanishing torsion scalar in the context of modified teleparallel gravity . for @xmath93 , the general metric reduces to spatial reissner - nordstr@xmath85 m metric . the spatial reissner - nordstr@xmath85 m metric further reduces to the reissner - nordstr@xmath85 m metric for @xmath94 and @xmath95 . our focus is on the general metric ( [ eq : general_metric ] ) . depending on the signs and values of @xmath68 , @xmath69 , @xmath61 and @xmath62 , the general spacetime may represent a black hole , a wormhole or a naked singularity . the nature of the solution depends on the presence of the zeroes of @xmath9 and @xmath96 . we will focus on solutions for which @xmath97 , @xmath98 and @xmath99 . the solution for such choices does represent a lorentzian wormhole .    for later use , let us now write down the solution in isotropic coordinates . the transformation from the schwarzschild radial coordinate @xmath100 to the isotropic coordinate @xmath0 is given as @xmath101 using the transformation between @xmath100 and @xmath0 we obtain , in isotropic coordinates , the line element , @xmath102 where @xmath103 @xmath104 and @xmath105 the constants @xmath106 and @xmath107 are given as @xmath108 and @xmath109 . it is easy to show that @xmath110 and @xmath111 are related to @xmath106 and @xmath107 as follows : @xmath112 it can be shown that , for a spacetime without a horizon @xmath113 is necessary . we will use the isotropic coordinate form of the line element while discussing the weak energy condition .    defining @xmath114 one can solve the radion field equation as shown in @xcite . obviously one ends up introducing a new constant which we denote as @xmath115 . we have @xmath116 where we have omitted an overall additive constant . it is easy to see that @xmath117 is never zero as along as @xmath118 . note that the domain of @xmath119 is from @xmath120 to @xmath121 , where @xmath122 corresponds to the wormhole throat . we will now turn towards analysing the weak energy condition inequalities and demonstrate that it can indeed be satisfied for the wormhole we have constructed . the weak energy condition comprises the inequalities @xmath123 for a diagonal energy momentum tensor with energy density @xmath124 , radial pressure @xmath125 and tangential pressures @xmath126 defined in the static observer s frame . physically , the wec means that the matter energy density is always non - negative in any frame of reference .    for our case here , we have , for the l.h.s . of the wec inequalities , @xmath127\end{aligned}\ ] ]    @xmath128\end{aligned}\ ] ]    @xmath129\ ] ]    where the last equation follows from the traceless requirement on the matter energy momentum tensor .    for arbitrary values of the parameters it is not possible to satisfy these inequalities . however , one can isolate the negativity by looking at the requirements that emerge near @xmath130 . in the limit @xmath131 , we have @xmath132x^4+\mathcal{o}(x^5)\ ] ] @xmath133x^3+\mathcal{o}(x^4)\ ] ] @xmath134 where @xmath135 . note that , at the leading order , ( i.e. , terms which are @xmath136 ) , @xmath137 and @xmath138 are opposite in sign . this violates the wec in the limit @xmath131 . therefore , we must set the coefficient of @xmath139 in @xmath137 to zero . this yields the following expression for @xmath115 . @xmath140 which reduces to the expression quoted in @xcite for @xmath141 and @xmath142 . with the above condition satisfied , it can be shown that in @xmath137 , @xmath143 term vanishes as well and @xmath144 term is always positive for any positive @xmath106 and @xmath107 . therefore , it is clear from the traceless condition that , at the @xmath143 , positivity of @xmath124 implies positivity of @xmath138 . note that , at this point , @xmath106 and @xmath107 can be chosen so that @xmath124 and @xmath145 are positive . however , we choose those values of @xmath106 and @xmath107 for which the coefficient of the @xmath143 term in @xmath124 vanishes . this gives us the following requirement on @xmath145 . @xmath146 equating the above two expressions for @xmath145 , one arrives at a relation between @xmath106 and @xmath107 which must be obeyed . @xmath147 \log\frac{q_1}{q_2 } = q_1 ^ 2-q_2 ^ 2 \label{eq : q1q2}\ ] ] thus , once we are able to choose a @xmath106 and a @xmath107 which satisfies the above relation , we can use their values to find @xmath115 . the set of values for @xmath106 , @xmath107 and @xmath115 can therefore be used to write down the line element and the scalar field profile for a lorentzian wormhole in scalar - tensor gravity which will satisfy all the wec inequalities . the central issue at this point is whether we can obtain an analytical handle on the relation between @xmath106 and @xmath107 . to get rid of the logarithm , we can assume @xmath148 . this will eventually lead us to a quartic equation in @xmath107 for which we can obtain the positive , real valued roots in terms of @xmath149 . choosing @xmath149 one can then obtain @xmath106 , @xmath107 and then @xmath115 . we will now illustrate the above statements with examples and also demonstrate our claim that the wec is indeed satisfied for the wormholes we have constructed . putting @xmath148 in ( [ eq : q1q2 ] ) , we obtain @xmath150 where , @xmath151 @xmath152 therefore , we fix @xmath106 and @xmath107 and hence @xmath111 , @xmath110 and @xmath115 by choosing a particular value of @xmath149 . it can be shown that @xmath153 for the whole range @xmath154 . therefore , the definitions of @xmath106 and @xmath107 indicate that both are positive . figure [ fig : q ] shows the parametric plot for @xmath106 and @xmath107 . figure [ fig : gamma ] shows the dependence of @xmath115 on @xmath106 . figures [ fig : ec1]-[fig : ec3 ] show the l.h.s of the wec inequalities for three sets of parameter values . it is to be noted that wec inequalities are satisfied .     as a function of @xmath106 . ]     as a function of @xmath106 . ] ( blue dashed curve ) , @xmath155 ( red dotted curve ) and @xmath156 ( black dot - dashed curve ) as a function of @xmath119 for @xmath157 which corresponds to @xmath158 , @xmath159 , @xmath160 , @xmath161 and @xmath162 . ]     ( blue dashed curve ) , @xmath155 ( red dotted curve ) and @xmath156 ( black dot - dashed curve ) as a function of @xmath119 for @xmath163 which corresponds to @xmath164 , @xmath165 , @xmath166 , @xmath167 and @xmath168 . ]     ( blue dashed curve ) , @xmath155 ( red dotted curve ) and @xmath156 ( black dot - dashed curve ) as a function of @xmath119 for @xmath169 which corresponds to @xmath170 , @xmath171 , @xmath172 , @xmath173 and @xmath174 . ] let us now take the limit @xmath175 . in this limit @xmath176 and @xmath110 diverges . but , the product @xmath177 is finite . in terms of @xmath178 , we have @xmath179 therefore , @xmath106 and @xmath107 have opposite signs . in this limit , the factor @xmath180 in the energy density and pressures must be replaced by @xmath181 . it can be shown that @xmath182 does not have zeroes in the range @xmath183 for @xmath184 , thereby representing a wormhole . but , for @xmath185 , it does have a zero and represents a naked singularity . let us first consider the case @xmath184 . like the @xmath186 case , here also , one must consider the expression ( [ eq : gamma1 ] ) . with this , it can be shown that , in the limit @xmath131 , the coefficient of the @xmath143 term in @xmath124 can not be set to zero since it is negative for all @xmath184 , thereby violating the wec . it also implies that the expression in ( [ eq : gamma2 ] ) is not valid . however , we can make use of the freedom to add an additive constant to @xmath187 and make this coefficient positive . if we do so , then we find that @xmath188 in ( [ eq : gamma1 ] ) , i.e. , @xmath115 is imaginary . on the other hand , it can be shown that if we force @xmath145 to be positive , then either @xmath124 or @xmath137 or both become negative in the limit @xmath131 . therefore , the wormholes without the mass term ` @xmath68 ' violate the wec . figure [ fig : ec4 ] shows the wec violation for @xmath64 . we have carried out a similar analysis for the naked singularity case @xmath189 where we found that the wec can be satisfied in the limit @xmath131 , but the radion field @xmath187 vanishes at a point between the singularity and spatial infinity . this gives an invalid solution . therefore , wec can not be satisfied for the naked singularity . ( blue dashed curve ) , @xmath155 ( red dotted curve ) and @xmath156 ( black dot - dashed curve ) as a function of @xmath119 for @xmath190 . ] let us now look at the gravitational redshift of a light signal propagating in the @xmath2 wormhole spacetimes discussed above . note that by virtue of having a @xmath191 which is everywhere finite and non - zero , the gravitational redshift is always finite . by definition , wormhole spacetimes are horizon - free . assume that a light signal is emitted at a frequency @xmath192 from the wormhole throat ( @xmath122 ) and it travels to spatial infinity ( @xmath130 ) where a static observer receives it at a frequency @xmath193 . therefore , the fractional change in frequency due to the gravitational redshift is given by the standard formula . @xmath194 we choose values of @xmath106 and @xmath107 which satisfy eqn . ( [ eq : q1q2 ] ) and hence , the wec . figure [ fig : redshift ] shows the fractional change in frequency as a function of @xmath106 . it is worth noting that the light signal may have a negative gravitational redshift , i.e. , a gravitational blueshift ( @xmath195 ) for certain values of @xmath106 and @xmath107 . for the three sets of @xmath106 and @xmath107 given in figs . [ fig : ec1]-[fig : ec3 ] , @xmath196 are respectively @xmath197 , @xmath198 and @xmath199 . for @xmath200 and @xmath201 , @xmath196 vanishes . therefore , the wormholes can exhibit both redshift and blueshift . it can be shown that , for those @xmath106 and @xmath107 satisfying the wec , the maximum redshift is @xmath202 . this redshift does not exceed @xmath203 for such values of @xmath106 and @xmath107 . however , we can have more than @xmath203 redshift if we choose @xmath106 and @xmath107 arbitrarily , which obviously violates the wec . thus , in a certain sense , as mentioned above , the wec does restrict the gravitational redshift to an amount below @xmath203 . it is possible to use this result ( along with other signatures ) to identify whether the cause of an observed gravitational frequency shift is indeed due to a wormhole .     with @xmath107 and @xmath106 satisfying eqn . the geodesic equations for a test particle moving in the equatorial ( @xmath204 ) plane of the spacetime [ eq : general_metric ] reduce to the following first integrals : @xmath205 where the energy @xmath206 and the angular momentum @xmath36 are constants of motion . an overdot represents differentiation with respect to the parameter @xmath207 ( affine for timelike and arbitrary for null geodesics ) . the effective potential @xmath208 is given by @xmath209 where the normalisation constant @xmath210 is @xmath211 for timelike geodesics and is @xmath120 for null geodesics . we now look for circular orbits which correspond to @xmath212 ( i.e. @xmath213 ) and @xmath214 ( i.e. extrema of the effective potential ) . minima ( @xmath215 ) and maxima ( @xmath216 ) of the effective potential correspond to the stable and unstable circular orbits , respectively . for the line element in ( [ eq : general_metric ] ) , the effective potential can be written as @xmath217f^2(z)\ ] ] where @xmath218 @xmath219 and @xmath220 , @xmath221 being the throat radius . here , we have normalised @xmath9 by dividing it by @xmath222 such that @xmath223 as @xmath224 . the effective potential is plotted in figs . [ fig : potential1 ] and [ fig : potential2 ] for the parameter values satisfying the wec . by choosing @xmath149 , we find @xmath107 from ( [ eq : q2 ] ) , @xmath106 from @xmath148 and hence @xmath111 and @xmath110 from ( [ eq : mu_eta ] ) . it should be noted that both stable and unstable circular orbits exist for the timelike case . however , for the null case , no stable circular orbit exists for the parameter values satisfying the wec . but , stable circular orbits exist if we choose the parameter values arbitrarily which , of course , violates the wec . to check this , we first note that the extremum @xmath225 of the potential are given by @xmath226 using eqs . ( [ eq : ricci_flat ] ) and ( [ eq : extremum ] ) , it can be shown that @xmath227 therefore , minima ( @xmath228 ) exist if @xmath225 are real and @xmath229 . note that we must have @xmath230 . this is possible when @xmath231 . figure [ fig : potential3 ] shows the plots for the potential for parameter values chosen arbitrarily . we have checked that the wec is violated for these parameter values . therefore , stable null circular orbits exist if the wec is violated .    . red dots represent unstable circular orbits . here , we have taken @xmath232 . ]    . the green and red dots represent respectively the stable and unstable circular orbits . here , we have taken @xmath233 and @xmath232 . ] it should be noted that , for the parameter values used in fig . [ fig : potential3 ] , @xmath106 and @xmath107 become complex conjugates of each other because the square root terms in the expressions of @xmath106 and @xmath107 become imaginary . but , the metric function @xmath182 still remains real since @xmath234 and @xmath235 are real . it can also be shown that @xmath236 is real . to verify this , let @xmath237 where @xmath238 and @xmath239 are real . therefore , @xmath240 . writing @xmath241 and @xmath242 where @xmath243 , we obtain @xmath244=\frac{2\gamma\theta(x)}{b}\ ] ] which is real . traversability of a wormhole demands that the tidal force felt by a human traveller moving radially , must be within tolerable limits . in an orthonormal basis @xmath245 attached to the traveller frame , the tidal acceleration between two parts of his / her body , separated by the deviation vector @xmath246 is given by @xcite @xmath247 where @xmath248 is the riemann tensor . at the throat , the components of the tidal acceleration are given by @xmath249 @xmath250 where @xmath251 and @xmath252 is the the radial velocity of the traveler as measured by a static observer . @xmath253 denotes the radial velocity at the throat . for the wormhole solutions , we must have @xmath254 . note that , the radial component of the tidal acceleration vanishes for @xmath65 . therefore , it can be made arbitrarily small by choosing an appropriate value of @xmath69 . we now restrict the radial component below one earth gravity , i.e. , @xmath255 . thus , for a traveller of typical size @xmath256 @xmath257 meters , we obtain @xmath258 @xmath259 where we have used the expressions @xmath260 and @xmath261 . note that @xmath68 and @xmath69 are related through @xmath111 . for a given metric parameter @xmath111 , the constraint on @xmath68 ( and hence on @xmath69 ) puts a constraint on the throat radius . the constraint on @xmath221 is given by @xmath262 for @xmath263 which corresponds to fig . [ fig : ec2 ] , we obtain @xmath264 , where @xmath265 km is the earth radius . however , we can reduce the lower limit on @xmath221 if we choose @xmath111 to be close to @xmath121 .    if the wormhole is to be traversable , not only the tidal acceleration but also the magnitude of acceleration felt by the traveller as he or she travels through it , must be within tolerable limits . in the orthonormal basis of the traveler frame , the magnitude of the acceleration is given by @xcite @xmath266 let us now consider both @xmath267 and @xmath268 to be finite at the throat . therefore , at the throat @xmath221 , we have @xmath269 for the wormhole solutions , we must have @xmath113 . therefore , for a given @xmath221 and @xmath68 which restrict the tidal acceleration below one earth gravity , the magnitude of the acceleration can be made arbitrarily small by taking @xmath110 to be very close to zero which , of course , satisfies the wec ( fig . [ fig : ec2 ] ) . in this work , we have obtained a class of static , spherically symmetric @xmath2 spacetimes which generalise certain known @xmath2 line elements . we have shown that our spacetimes can arise in a scalar - tensor theory of gravity , an example of which is the low energy , effective on - brane gravity theory developed by kanno and soda @xcite in the context of the randall - sundrum two - brane model . some special cases of the general solution reproduce the @xmath2 spacetimes obtained earlier by other authors in the context of gr and in the on - brane effective gravity theory due to shiromizu - maeda - sasaki @xcite for a single - brane scenario . a sub - class of the general spacetimes found by us represents @xmath2 lorentzian wormholes . we have studied the wec for the matter that supports such wormholes and have shown that they can satisfy wec . this is in contrast to gr , where such wormholes must violate wec . we have also shown that the mass term in the metric functions is necessary to satisfy the wec for these wormholes . note that most of the wormhole solutions obtained in @xcite are in vacuum - brans - dicke theory . however , the solutions we have obtained include both matter and a non - constant , i.e. , a @xmath13-dependent coupling @xmath12 . as a additional curiosity , we also looked at the wec for the naked singularity with the parameter @xmath64 , and have found that it is violated . apart from constructing solutions and checking the wec , we have calculated the gravitational redshift of a light signal traveling in such wormholes and have found that the wec restricts the amount of redshift to be below @xmath203 . furthermore , we obtained the effective potentials for particle motion and investigated the possibility of circular orbits in these wormhole spacetimes . it has been shown that both stable and unstable circular orbits exist for the timelike case . for the null case , only unstable circular orbits are possible if the wec is satisfied . but , stable null circular orbits exist if the wec is violated . finally , we analysed traversability in such wormholes and found that they are traversable for values of the metric parameters satisfying the wec . therefore , we now have wormholes which satisfy the wec and are traversable . in addition , they may also be viewed as on - brane spacetimes if one considers the existence of warped extra dimensions of the randall - sundrum variety , in a two - brane context . in such a context the scalar field ( radion ) is related to the inter - brane distance and we have seen that this can always be kept finite and non - zero everywhere . recently , it has been argued that the universal ringdown waveforms from a binary coalescence indicate the presence of light rings , rather than of horizons @xcite . therefore , an object ( e.g. a traversable wormhole ) with a light ring will display a similar initial ringdown stage ( waveform ) , even when its quasinormal mode spectrum is completely different from that of a black hole . however , their late time ringdown stages are different @xcite . it has further been shown that , if one does not use the thin - shell wormhole ( the one used in @xcite ) obtained by special matching near the schwarzschild radius but prefers a wormhole configuration constructed without thin - shells instead , a wormhole may ring like a black hole or differently at all times , depending on the values of the wormhole and black hole parameters @xcite . this leads to the hope that the lorentzian wormholes we have obtained here , may be relevant in such studies and may have an observational consequence in astrophysics , in future . further , it is to be noted that the @xmath2 spacetime solution we have obtained can be used in the context of any other theory of gravity ( different from the one considered here ) to study the relation between the energy conditions and wormholes . we have noted that a sub - class of our @xmath2 spacetimes represents naked singularities . therefore , as a observational test , it will be interesting to study gravitational lensing in the spacetimes we have obtained here and to see whether we can find ways to distinguish between wormholes and naked singularities @xcite through observations . r. s. acknowledges the council of scientific and industrial research , india , for providing support through a fellowship . 99 n. dadhich , s. kar , s. mukherjee , and m. visser , _ @xmath2 spacetimes and self - dual lorentzian wormholes _ , phys . d * 65 * , 064004 ( 2002 ) . s. kar , s. lahiri , and s. sengupta , _ can extra dimensional effects allow wormholes without exotic matter ? 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we obtain a large class of lorentzian wormhole spacetimes in scalar - tensor gravity , for which the matter stress energy does satisfy the weak energy condition . 
 our constructions have zero ricci scalar and an everywhere finite , non - zero scalar field profile . interpreting the scalar - tensor gravity as an effective on - brane theory resulting from a two - brane randall  sundrum model of warped extra dimensions , it is possible to link wormhole existence with that of extra dimensions . 
 we study the geometry , matter content , gravitational red - shift and circular orbits in such wormholes and argue that our examples are perhaps among those which may have some observational relevance in astrophysics in future . 
 we also study traversability and find that our wormholes are indeed traversable for values of the metric parameters satisfying the weak energy condition .