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if the topology of space - time is @xmath5 then the only known exact solution representing a black object is the uniform black string with horizon topology @xmath6  @xcite . though this solution exists for all values of the mass it is unstable below a critical value , @xmath7 , as shown by gregory and laflamme @xcite . horowitz and maeda @xcite argued that a uniform black string can not change its topology into a black hole in finite affine time , making the possibility of such a transition questionable . they suggested the possibility of a transition to a nonuniform black string . gubser @xcite showed the existence of nonuniform black string solutions . the non - uniform black string solution in 6 dimensions was investigated numerically @xcite for a range of the mass values above @xmath7 . nonuniform black string configurations do not exist for masses below the gregory - laflamme point , in the region where the uniform black string solution is unstable  @xcite . a natural candidate for a black object in this mass range is a black hole . unfortunately , no exact black hole solutions are known in a 5 ( or more ) dimensional space with a compactified dimension . still , on the basis of physical intuition , a black hole solution should exist for very small values of the mass . when the radius of the horizon is much smaller than the size of the compactified dimension , i.e. when @xmath8 , the black hole should be unaware of the compactification . then a myers - perry  @xcite solution should become asymtotically exact . indeed , a numerical solution , extending the myers - perry solution to larger values of the mass was found by harmark and obers  @xcite and kudoh and wiseman  @xcite .    using general arguments kol  @xcite suggested that the black hole branch and the nonuniform black string branch meet at a point when the black hole fills the compact dimension . further arguments for such a transition were presented in  @xcite . recently , we have studied a related problem , namely the existence of black holes in randall - sundrum  @xcite theories . we used  @xcite an approximation scheme based on the expansion of solutions in the ratio of the radius of the horizon of the black hole to the ads curvature . in this paper we employ a similar strategy by expanding the metric and other relevant quantities in the ratio of the two natural lengths associated with a black hole configuration , ( i ) the five dimensional schwarzschild radius , @xmath1 , associated with the mass @xmath9 and ( ii ) the compactification length , @xmath2 , defined as the proper circumference of the compact dimension in the region far away from the mass . the dimensionless ratio of these two quantities serves as an excellent expansion parameter . as the solution is an even function of @xmath1 and @xmath2 we use @xmath10 as our expansion parameter . such an expansion for add black holes has recently been proposed and harmark  @xcite and gorbonos and kol  @xcite evaluated the leading term of the expansion .    to find an unique solution we must fix boundary conditions , both at infinity and at the horizon of the black hole . we find solutions in two different regions . in the asymptotic region , @xmath11 an ` asymptotic solution ' , and in the near horizon region , @xmath12 a ` near solution ' is found . using the asymptotic solution we satisfy asymptotic boundary conditions but the boundary conditions at the horizon can not be satisfied . the near solution suffers from the opposite problem , as it can not be used to satisfy asymptotic boundary conditions . to solve those problems we combine the two solutions . we match the asymptotic solution with the near solution in the region @xmath13 where both are valid . this method was used in @xcite to find small black holes in randall - sundrum scenario , and in @xcite to find small black holes on cylinders . the work in this paper is parallel to and agrees with previous results   @xcite , but here we calculate the metric up to fourth order in @xmath14 ( second order in @xmath4 ) . the organization of this paper is as follows : in section [ sec : method ] we describe the general method of our calculations , and list the parameters that will be calculated from the metric . in sections [ sec : zeroth]-[sec : second ] we present the detailed calculations up to second order . in section [ sec : summary ] we summarize our results for the metric , up to second order , and discuss the possible scenarios for black holes of increasing mass . a reader not interested in technical details should read the next section , ( [ sec : method ] ) and then proceed directly to the summary and discussions ( [ sec : summary ] ) . as we indicated in the introduction we will investigate black hole solutions of the einstein equation in 5 dimensional space when one of the dimensions is compactified . recently , this problem has attracted the attention of several groups . harmark and obers@xcite introduced the relative tension of black holes as an order parameter , wrote down a generalized smarr formula  @xcite , investigated the phase diagram for black objects , and studied the solutions numerically . gorbonos and kol  @xcite proposed an analytic approximation scheme based on the expansion in the ratio of the radius of the horizon and the compactification length . this method is similar to the expansion method we used  @xcite to investigate black holes in the randall - sundrum @xcite scenario . we will follow a similar path here and calculate further terms of the expansion .    by necessity , the perturbation method is applied in two overlapping regions . in the asymptotic region , @xmath11 , @xmath1 is the smallest scale . therefore , an expansion in @xmath14 is actually an expansion in @xmath1 and in every order the metric is a general function of the dimensionless coordinates @xmath15 , @xmath16 we refer to this solution as the ` asymptotic solution . '    in the near horizon region , @xmath12 , @xmath17 is the smallest scale . therefore , an expansion in @xmath14 is actually an expansion in @xmath17 . in every order , the metric is a general function of the dimensionless coordinates @xmath18 , @xmath19 we refer to this solution as the ` near solution . '      in each region , when we calculate the @xmath20-th order terms , einstein s equations is solved in terms of two functions : a gauge function and a wave function , which satisfies a linear , ( inhomogeneous ) , master equation . the differential operator for the master equation in the asymptotic region is @xmath21 where @xmath22 is a coordinate in the compact dimension and @xmath23 is a radial coordinate in 3-dimensional space . in the near region the operator is @xmath24 where @xmath25 is a radial coordinate and @xmath26 is an angular coordinate in 4-dimensional space . the driving term in the master equation for the @xmath20th order contribution depends on the solution in lower orders . it determines an inhomogeneous solution in the wave function . the additional homogeneous solution introduces new parameters that should be fixed by boundary conditions .      in empty space , or for the uniform black string configuration , the cylinder is invariant under translations in the compact direction . when a point mass is introduced translation invariance is broken , but a @xmath27 reflection symmetry remains around the location of the mass in the compact direction . so , if we endow the compact direction with the coordinate @xmath28 $ ] and put the mass at @xmath29 then the solution is symmetric about @xmath30 and periodic in @xmath22 with period @xmath2 . asymptotically ( far away from the mass ) , we assume that the metric is minkowski . the leading order correction to the minkowski metric is of order @xmath31 . such a term provides , among others , the four - dimensional newtonian potential . we must impose constraints on the near solution , as well . the horizon of a small black hole has @xmath32 topology . we require that there is no black string attached to the horizon . consequently , the kretchmann scalar must be regular on the @xmath22 axis everywhere , when @xmath33 . furthermore , the surface gravity must be constant , otherwise the horizon is not regular  @xcite .      to match the asymptotic and the near solutions we must work in the intermediate region @xmath34 , where @xmath35 is a four - space - dimensional radial coordinate . in this region the functions @xmath36 can be expanded in the coordinates . owing to the @xmath37 symmetry the components depend only on @xmath35 and @xmath26 , the angle between the compact direction and the four - dimensional sub - manifold . then one can use a double expansion in @xmath38 and @xmath39 to write the metric as @xmath40 the expansion includes only even powers due to the @xmath27-symmetry as will be explained later . globally , our expansion parameter is @xmath41 . in the two regions @xmath4 has a different meaning . the asymptotic solution is constructed as an expansion in @xmath1 , such that @xmath42 , so one keeps @xmath20 constant and solves the equations for all values of @xmath43 . for example , minkowski metric corresponds to the zeroth order ( @xmath44 , @xmath45 ) approximation to our problem , with appropriate coefficients , @xmath46 . the first order , @xmath47 , @xmath45 , provides linearized gravity ( newtonian potential ) , etc . the near solution is constructed as an expansion in @xmath17 , such that @xmath48 , so at fixed @xmath43 we solve for all values of @xmath20 . here , @xmath49 , with appropriately chosen coefficients , @xmath46 , corresponds to the myers and perry five dimensional black hole ( mps ) @xcite . the matching is done in the intersecting points denoted by circles , as can be seen in fig.([gridfig ] ) . for example , first order of the asymptotic solution intersects zeroth order of the near solution at @xmath50 . we use this information to fix parameters in the asymptotic solution . terms as points in the @xmath51 grid . the horizontal ( vertical ) lines describe the asymptotic ( near ) solution in increasing order of @xmath4 . the matching is done at the intersecting points denoted by circles . ]    in the intermediate region the asymptotic solution is expanded in @xmath39 , while the near solution is expanded in @xmath38 . the table shows the sequence of calculating terms in the asymptotic and near solutions . terms connected by an arrow are identified by the double expansion . a similar table has been presented in @xcite ]    the calculation is carried out order by order . both solutions ( asymptotic and near ) must be known up to order @xmath52 before calculating order @xmath53 . within order @xmath53 , one should calculate the asymptotic solution first and then the near solution . the sequence of calculation of terms in the two solutions is depicted in fig.([tablefig ] ) . two parameters can be measured in the asymptotic region , the adm mass and the relative tension around the compact dimension . @xcite expanding the asymptotic metric around the minkowski metric we write @xmath54 , then the adm mass , @xmath55 , and the tension , @xmath56 , are defined as @xmath57dw~ ,      \label{defadmmass}\\      b&=&\int d^{4}x t^{ww}=-\frac{1}{4g_{5}}\lim_{r\rightarrow\infty}r^{2 }          \int_{-l/2}^{l/2}\left[\frac{2}{r}h_{rr}+h_{tt , r}\right]dw~.      \label{deftension}\end{aligned}\ ] ] it is useful to define the relative tension , @xmath58 . in @xcite the pair @xmath59 is used to describe a phase diagram of all possible black objects . the near solution provides the thermodynamic parameters of the horizon , i.e. entropy and temperature . the entropy is defined as the three dimensional area of the horizon divided by @xmath60 . the temperature is defined through the zeroth law of black holes thermodynamics as the surface gravity divided by @xmath61 . a smarr formula @xcite relates the entropy , temperature , mass , and relative tension as  @xcite @xmath62 in the asymptotic region zeroth order in @xmath4 means that we neglect the mass . therefore , the configuration is asymptotically flat ( since the mass is confined to a small region ) . so , the asymptotic form of the metric is minkowski on a cylinder @xmath63 where @xmath22 is the compact coordinate such that @xmath28 $ ] .    in the near region zeroth order in @xmath4 means that @xmath64 . the mass point is in five dimensional asymptotic flat infinite manifold . therefore , the near solution is given by the myers - perry  @xcite solution ( mps ) , @xmath65 where @xmath1 is the five dimensional schwarzschild radius . in order to match the metrics ( [ a0 ] , [ n0 ] ) one needs to specify a coordinate transformation between @xmath66 and @xmath67 . we choose to work with a specific coordinate transformation throughout the calculation of all orders . ( one can , however , choose to redefine the transformation in each order ) . the coordinates @xmath68 , @xmath69 , and @xmath70 are shared by the coordinate systems . furthermore , we want to preserve the periodic structure in @xmath22 so we choose the transformation    [ transformation ] @xmath71~ ,      \label{z}\\      r&=&\rho\sin\psi~,\label{r}\\      \rho&=&\sqrt{r^{2}+v^{2}}=\sqrt{r^{2}+\frac{l^{2}}{\pi^{2}}\sin^{2}\frac{\pi w}{l } }      ~,\label{rho}\\      \psi&=&\arctan\frac{r}{v}=\arctan\frac{\pi r}{l\sin(\pi w / l)}~,\label{psi}\end{aligned}\ ] ]    one can verify that the metrics ( [ a0],[n0 ] ) agree up to order @xmath72 . we show now that the expansion contains only terms with integer powers of @xmath73 . it is sufficient to establish that the solution must be an even function of @xmath1 and of @xmath2 ( at fixed @xmath74 ) . the solution is an even function of @xmath1 , because the driving term , the zeroth order near metric ( [ n0 ] ) depends only on @xmath75 . the zeroth order near metric , ( [ n0 ] ) , combined with ( [ transformation ] ) is also an even function of @xmath2 . the conditions imposed on the metric also force it to be even in @xmath2 . the mass is located at @xmath76 , ( @xmath77 ) . the configuration of a mass point on a cylinder has a @xmath27 symmetry about @xmath29 , ( @xmath78 ) , requiring that the terms of the metric depend on @xmath22 and @xmath2 through functions of the form @xmath79 . in the asymptotic region , @xmath11 , we expand the metric as @xmath80 where @xmath81 is the flat metric ( [ a0 ] ) .    the general form of the solution for @xmath82 appears in many places in the literature @xcite . we re - derive it here using the matching method . we expand the einstein equation to first order in @xmath4 and get a set of homogeneous linear equations for @xmath82 . using the coordinate transformation @xmath83 we choose the gauge @xmath84 . next we solve @xmath85 for @xmath86 , and @xmath87 for @xmath88 , where @xmath89 is the einstein tensor . the solution is given by    [ h1 ] @xmath90    where the gauge function satisfies the symmetry properties @xmath91 and @xmath92 is a constant . the function @xmath93 satisfies the equations @xmath94 the solution to eq.([htteq ] ) can be written as a fourier series @xmath95 the constants @xmath96 are determined by matching ( [ httmodes ] ) to the zeroth order near solution ( [ n0 ] ) . then it follows that up to leading order in @xmath17 the two solutions must be identical , @xmath97 . the constants @xmath96 are given by the fourier expansion of @xmath98 . [ fourier1 ] @xmath99    using ( [ fourier1 ] ) we are able to sum eq.([httmodes ] ) as @xmath100    for @xmath101 the metric is given in terms of the ` zero mode , ' independent of @xmath22 , and can be summarized as    [ h1asymp ] @xmath102~,\\      g_{rw}&=&\epsilon w_{,r}(r , w)~.\end{aligned}\ ] ]    the constant @xmath92 , which appears in the metric ( [ h1asymp ] ) is related to the tension in the compact dimension @xmath103dw          = -\frac{\alpha\epsilon l}{2g_{5}}~.      \label{tension}\ ] ] the tension @xmath56 should be zero because we deal here with linearized gravity , with no interaction between the mass and its periodic images . consequently , we must set @xmath104 . dimensional analysis also requires @xmath104 . first , we can deduce from eqs.([h1 ] ) that @xmath105 . the asymptotic solution is an expansion in the mass , so @xmath92 must be independent of the mass and proportional to @xmath2 , the only quantity of the correct dimension at our disposal . as we have discussed earlier , function ( [ htt1final ] ) is even in @xmath2 . if @xmath106 then the functions @xmath86 and @xmath88 , in ( [ h1 ] ) , would have a mixed symmetry under @xmath107 . note that this symmetry argument can and will be used in higher ( @xmath20th ) orders of the expansion to eliminate terms of the form @xmath108 from @xmath109 and @xmath110 . next , we turn to evaluate the conserved adm mass using eqs.([h1asymp ] ) @xmath111dw          = \frac{3\pi\mu^{2}}{8g_{5}}~.      \label{admmass}\ ] ] the effective four - dimensional newton constant is defined by @xmath112 . using eqs.([h1asymp ] , [ admmass ] ) we find that @xmath113 . we should emphasize that the physically measurable quantities , the relative tension , the mass , and newton s constant , are independent of the gauge function @xmath114 . as we mentioned earlier , in higher orders of the expansion the relative tension in the compact dimension is not zero . however , we require that the adm mass is completely determined by the first order , such that higher orders do not change eq.([admmass ] ) . this way , we make sure that the expansion in @xmath4 is also an expansion in @xmath55 and that thermodynamic properties of the black hole are well defined in each order . we will return to this issue in section [ sec : second ] when we discuss the second order contributions . the gauge function @xmath114 , which appears in ( [ h1 ] ) , can be partially determined by matching to the zeroth order of the near solution . however , we prefer to determine the function completely by matching to the first order of the near solution , as well .      in the near region it is convenient to use the coordinates @xmath67 just like in eq.([n0 ] ) . we use the following ansatz for the metric : @xmath115 the functions @xmath116 are expanded to first order in @xmath4 as    [ n1 ] @xmath117    the first step is to use a gauge transformation of the form @xmath118 , @xmath119 , which leaves the last term of ( [ spherical ] ) unchanged to leading order , to choose a gauge where @xmath120 this gauge choice simplifies the equations for the rest of the functions to be determined . next , one can solve the equation @xmath121 for @xmath122 . the rest of the equations can be solved in terms of a single function , @xmath123 , obeying a second order partial differential equation . the solution is    [ n1solution ] @xmath124\nonumber\\ & &      -\frac{2\rho^{2}h_{1,\rho}(\rho,\psi)}{\sin^{2}\psi\,\cos\psi }      + \frac{\rho^{3}}{\rho^{2}-\mu^{2}}f_{1,\psi}(\rho,\psi )      -\rho^{2}\tan\psi f_{1,\rho}(\rho,\psi )      ~.      \label{vn1}\end{aligned}\ ] ]    the wave function @xmath125 satisfies the differential equation @xmath126 the solution of eq.([h1equation ] ) can be written as @xmath127h_{\nu}(\psi)~ ,      \label{h1nu}\\      h_{\nu}(\psi)&=&\frac{4}{\sqrt{2\pi}}\sin^{3/2}2\psi\left [      \pi\cos\alpha_{\nu}p^{3/2}_{\nu}(\cos2\psi )      -2\sin\alpha_{\nu}q^{3/2}_{\nu}(\cos2\psi)\right]\nonumber\\      & & = ( 2\nu-1)\cos[(2\nu+3)\psi-\alpha_{\nu } ]      -(2\nu+3)\cos[(2\nu-1)\psi-\alpha_{\nu}]~ ,      \label{hpsi}\end{aligned}\ ] ] where @xmath128 and @xmath129 are associated legendre functions of the first and second kind . the functions @xmath130 and @xmath131 will be fixed below using the constraints on function @xmath132 . these constraints come from symmetries , from regularity requirements , and from boundary conditions . @xmath27 symmetry about the direction @xmath78 ( @xmath29 ) implies the conditions    [ z2brane ] @xmath133    it can be verified that condition ( [ h1,psi ] ) is automatically satisfied by ( [ h1nu ] ) .      for a black hole configuration , unlike for a black string configuration , the mass is localized at the origin . then the components of the metric , ( [ n1 ] ) , are finite at @xmath134 and arbitrary @xmath35 . this will only be true if @xmath135 it can be verified that ( [ nostring ] ) implies @xmath136 . then ( [ hpsi ] ) simplifies to @xmath137 where we have omit a possible overall sign . as we mentioned earlier , the expansion in @xmath41 and the fact that the zeroth order of the near solution ( [ n0 ] ) depends only on @xmath75 imply that the metric is even in @xmath1 . this means that the function @xmath132 should be even in @xmath138 . for large @xmath25 the legendre functions in ( [ h1nu ] ) behave as @xmath139 where @xmath140 are analytic functions . since we require that @xmath132 is even in @xmath25 , @xmath141 should be integer . the integral in ( [ h1nu ] ) is reduced to a sum over integer values of @xmath141 . @xmath142\sin^{3}\psi      + 2r\sqrt{r^{2}-1}\sum_{n=1}^{\infty }      \left[a_{n }          p^{1}_{n}(2r^{2}-1)+b_{n }          q^{1}_{n}(2r^{2}-1)\right]\nonumber\\      & & \hspace{2in}\times\left[(2n-1)\sin(2n+3)\psi      -(2n+3)\sin(2n-1)\psi\right]~.      \label{h1sum}\end{aligned}\ ] ] the case @xmath44 requires special attention , since the @xmath143 . the metric ( [ spherical ] ) is static . therefore , the surface @xmath144 is a killing horizon . the normal vector @xmath145 should be null on the horizon . this implies that on the horizon @xmath146 , thus the horizon is located at constant @xmath147 . using metric ( [ n1 ] ) we expand @xmath148 in @xmath4 as @xmath149 . the conditions @xmath150 and @xmath151 restrict the gauge function @xmath152 @xmath153 the surface gravity for metric ( [ spherical ] ) is defined as @xmath154    the surface gravity should be constant on the horizon , otherwise the horizon is singular @xcite . when we expand the metric in @xmath4 , the surface gravity must be constant in every order of the expansion . we use metric ( [ n1 ] ) and conditions ( [ f1horizon ] ) to evaluate the surface gravity in order @xmath4 @xmath155 the requirement of a constant surface gravity constrains the function @xmath132 . at @xmath156 the legendre functions in eq.([h1sum ] ) take the values @xmath157 , @xmath158 . we use the representation ( [ h1sum ] ) to evaluate the surface gravity ( [ kappah1 ] ) @xmath159 the set @xmath160 is complete on the interval @xmath161 $ ] , therefore , the solution to eq.([chi1 ] ) is @xmath162 in other words , the sum in eq.([h1sum ] ) contains legendre functions of the first kind only . the functions @xmath163 are polynomials of order @xmath164 in @xmath165 . the remaining free parameters of the near solution , @xmath166 , must be determined from matching the two gauge functions , @xmath114 in the asymptotic solution ( [ a1 ] ) and @xmath167 in the near solution ( [ n1 ] ) . we use ( [ transformation ] ) to transform the asymptotic solution to @xmath168 coordinates and then we expand it in @xmath17 . the gauge function @xmath114 is also expanded as @xmath169 we keep the explicit factor @xmath170 in ( [ wn ] ) to insure that @xmath171 . we start with zeroth order in @xmath17 . we transform the asymptotic metric using ( [ transformation ] ) and expand it to zeroth order in @xmath17 . [ asymprhopsi0 ] @xmath172~ ,      \label{vasymp0}\end{aligned}\ ] ]    where the superscript @xmath173 stands for asymptotic and the functions are defined in ansatz ( [ spherical ] ) . matching ( [ asymprhopsi0 ] ) to zeroth order of the near solution ( [ n0 ] ) determines the function @xmath174 @xmath175      the asymptotic solution is    [ asymprhopsi1 ] @xmath177~ ,      \label{aasymp1}\\      u^{a}&=&1+\frac{\pi^{2}}{l^{2}}\left[\rho^{2}\cos^{2}\psi\sin^{2}\psi      -\frac{\mu^{2}\sin\psi\cos^{2}\psi(5\sin3\psi-37\sin\psi)}{12 }      -2\mu^{2}\frac{\sin\psi}{\rho } w_{2,\psi}(\rho,\psi)\right]~ ,      \label{uasymp1}\\      v^{a}&= & -\frac{\pi^{2}}{l^{2}}\left[\rho^{3}\sin\psi\cos^{3}\psi      + \mu^{2}\rho\sin\psi\cos^{3}\psi(\sin^{2}\psi+2 )      + \mu^{2}\rho\sin\psi w_{2,\rho}(\rho,\psi )      -\mu^{2}\cos\psi w_{2,\psi}(\rho,\psi)\right ] ~.      \label{vasymp1}\end{aligned}\ ] ]    the near solution ( [ n1 ] ) should be expanded to second order in @xmath1 , which appears in @xmath41 and in the rescaled coordinate @xmath138 . the large @xmath25 behavior of the terms of @xmath132 , ( [ h1sum ] ) , is @xmath178 these expressions contribute by terms of order @xmath179 in metric ( [ spherical ] ) . we deduce that @xmath180 terms contribute by negative orders in @xmath1 and must be eliminated . therefore we impose @xmath181 if @xmath180 . for similar reasons , gauge function @xmath152 should also be a polynomial of order @xmath182 in @xmath35 . thus , functions @xmath132 and @xmath152 are @xmath183\sin^{3}\psi~,\label{h10}\\      f_{1}&=&(\mu^{-2}\rho^{2}-1)f_{2}(\psi)-\zeta_{1}~,\label{f1asymp}\end{aligned}\ ] ] where we have already imposed @xmath184 , ( [ f1horizon ] ) . the exact form of the periodic function @xmath185 will be fixed below . then the near metric , expanded to second order in @xmath1 , is    [ nearmu2 ] @xmath186~ ,      \label{anear2}\\      u^{n}&=&1-\frac{2\tan^{2}\psi}{l^{2}}\left[\rho^{2}[2a_{0}+f_{2}(\psi )      + \cot\psi f_{2}'(\psi)]-\mu^{2}[4a_{0}-6b_{0}+f_{2}(\psi)+\zeta_{2 }      + \cot\psi f_{2}'(\psi)]\right]~ ,      \label{unear2}\\      v^{n}&= & \frac{\rho^{3}\tan\psi}{l^{2 } }      \left[-4a_{0}-2f_{2}(\psi)+\cot\psi f_{2}'(\psi)\right ]     ~.      \label{vnear2}\end{aligned}\ ] ]    comparing ( [ asymprhopsi1 ] ) and ( [ nearmu2 ] ) we find that @xmath187      ~.\label{w2}\end{aligned}\ ] ] at this point the near solution still contains two free parameters , @xmath188 and @xmath189 . the @xmath27-symmetry condition , ( [ f1brane ] ) , imposes one constraint on these parameters @xmath190 a constraint , determining @xmath189 , is derived from the first law of black hole thermodynamics . it will be discussed in the next section . the zeroth law of black hole thermodynamics states that the temperature of a black hole is @xmath191 where @xmath192 is the surface gravity , which is constant on the horizon . to calculate the temperature for the near solution we use eqs.([kappa ] , [ kappah1 ] , [ bn ] ) to find that @xmath193    the first law of black hole thermodynamics is @xmath194 , where the entropy is proportional to the area of the horizon @xmath195 we calculate the entropy for the near solution and find that @xmath196 according to the first law the temperature can be found as @xmath197 . the mass appears in the entropy only through @xmath9 and @xmath41 , so if we combine the first law and the zeroth law we get @xmath198=2\pi\mu[1 + 5\zeta_{1}\epsilon]~.      \label{t2}\ ] ] this fixes the last parameter . if we combine eqs.([b0],[t2 ] ) we find that @xmath199 to summarize , the near metric , to first order in @xmath4 , is    [ near1final ] @xmath200~ ,      \label{b1final}\\      a^{n}&=&1+\frac{\pi^{2}\epsilon}{3}\left[(3\frac{\rho^{2}}{\mu^{2}}-1)\cos^{4}\psi-1      \right]~ ,      \label{a1final}\\      u^{n}&=&1+\pi^{2}\epsilon(\frac{\rho^{2}}{\mu^{2}}-1)\sin^{2}\psi\cos^{2}\psi~ ,      \label{u1final}\\      v^{n}&= & -\pi^{2}\epsilon\frac{\rho^{3}}{\mu^{2}}\sin\psi\cos^{3}\psi    ~.      \label{v1final}\end{aligned}\ ] ]    the location of the horizon is at @xmath201 . the entropy and the temperature are @xmath202 these expressions are in agreement with previous results  @xcite . the only freedom left in the first order metric is the @xmath203 terms of the gauge function in the asymptotic solution ( [ wn ] ) . the @xmath44 and @xmath204 terms are fixed in the region @xmath205 by our matching procedure , as given in eqs.([w0],[w2 ] ) . in addition , in the asymptotic region , @xmath101 , the metric should be minkowski , therefore , we require that @xmath206 . a form of the gauge function which is consistent with these conditions appears in appendix [ app : asymp2 ] . prior to completing calculations in second order we describe our general procedure for calculating higher order contributions . first we consider the asymptotic solution , which is fully determined ( up to gauge freedom ) by the lower order contributions .      in the asymptotic region we expand the metric in @xmath4 as @xmath207 assume that we know the solution up to order @xmath208 and we intend to obtain the @xmath20th order solution . einstein s equation is linear in the @xmath209 but due to lower order contributions it is inhomogeneous . the solution is similar to ( [ h1 ] ) but , in addition to the solution of the homogeneous equation includes extra terms corresponding to a particular solution ( the particular solution is denoted by @xmath210 ) , as follows    [ hn ] @xmath211    where @xmath92 is a constant , and @xmath212 is a gauge function which is periodic and antisymmetric in @xmath22 . the function @xmath213 is also periodic and symmetric in @xmath22 . it satisfies the equation @xmath214 where @xmath215 is the deriving term , which depends on lower orders , @xmath216 , @xmath217 . the homogeneous solution to eq.([httneq ] ) can be written as a fourier series . consequently , we have @xmath218 the particular solution @xmath210 is completely determined by the lower orders of the asymptotic solution ( without any use of the near solution ) . the free parameters are @xmath92 and the set @xmath219 . these can be determined by matching to the lower orders of the near solution as follows . take @xmath220 from the near solution up to order @xmath52 and expand it in @xmath1 . take the term of order @xmath221 and find its fourier series just like in eq.([fourier1 ] ) . compare that fourier series with the forier series of @xmath213 to lowest order in @xmath17 , and determine the constants @xmath219 . just like in first order , we can show that @xmath222 . solution ( [ hn ] ) should be even in @xmath2 . the functions @xmath223 and @xmath213 are even functions of @xmath2 since these functions are determined by lower orders but for dimensional reasons @xmath224 is odd . therefore , @xmath225 must vanish . next , we have to make sure that the definition of the adm mass does not change . so , we require that @xmath226dw=0~.      \label{fixm}\ ] ] in general , the tension in the compact dimension , ( [ tension ] ) , does not vanish . as a result , the effective four - dimensional newton s constant , @xmath227 , acquires a correction of order @xmath53 . this means that @xmath227 depends on the mass and the _ equivalence principle is violated . _    at this point the asymptotic solution is determined up to order @xmath53 , except for the gauge function @xmath212 , which is ( partially ) determined by matching to the near solution to orders up to @xmath53 .      near the horizon it is convenient to use the metric in form ( [ spherical ] ) and expand it in @xmath4 as follows    [ emetric ] @xmath228    we need the near solution up to order @xmath208 and the asymptotic solution up to order @xmath20 to calculate the @xmath20th order near solution . again , the @xmath20th order einstein s equation is a linear inhomogeneous equation in @xmath229 . the inhomogeneity depends on @xmath230 . we solve the equations , in a way , similar to that for the first order correction . the first step is to apply a gauge transformation of the form @xmath231 , @xmath232 to arrive at a gauge , in which @xmath233 next , one can solve the inhomogeneous equation @xmath121 for @xmath234 . the rest of the equations can be solved in terms of a single function , @xmath235 , obeying a second order partial differential equation , which is similar to ( [ h1equation ] ) , @xmath236 where @xmath237 depends on the lower order corrections . the solution of eq.([hnequation ] ) can be found by the method of separation of variables , just like it has been done when we have calculated the first order correction ( [ h1nu ] ) . the eigenfunctions of eq.([hnequation ] ) are the functions @xmath238 , which appear in eq.([hpsi ] ) . the driving term , @xmath237 , can also be expanded in the set @xmath238 . in addition , the boundary conditions at @xmath78 and at @xmath134 in @xmath20th order are the same as in first order , eqs.([h1,psi ] , [ nostring ] ) . and just like in first order the function @xmath239 should be even in @xmath35 . so , the general solution of eq.([hnequation ] ) is @xmath240\sin^{3}\psi\nonumber\\      & & + \sum_{n=1}^{\infty }      \left[2r\sqrt{r^{2}-1}\left(a_{n }          p^{1}_{n}(2r^{2}-1)+b_{n }          q^{1}_{n}(2r^{2}-1)\right)+p_{n}(r)\right]\nonumber\\      & & \hspace{2in}\times\left[(2n-1)\sin(2n+3)\psi      -(2n+3)\sin(2n-1)\psi\right]~ ,      \label{hnsum}\end{aligned}\ ] ] where @xmath241 is a particular solution of the inhomogeneous equation ( [ hnequation ] ) . the particular solutions are completely determined by the lower orders of the near solution ( without using the asymptotic solution ) . the homogeneous part should be completely fixed by matching to the asymptotic solution . the matching is done as follows : take the asymptotic solution up to order @xmath53 and transform it to the @xmath168 coordinates , using ( [ transformation ] ) . expand the solution in @xmath17 , take the term of order @xmath242 and compare it to the near solution of order @xmath53 to fix the parameters @xmath243 , and the gauge functions @xmath244 and @xmath212 . at this point the function @xmath212 is determined only up to order @xmath242 and one should fix it completely before proceeding to the next order . in the next section we apply this method to calculate the second order contributions . up to first order , the asymptotic solution is given by equations ( [ a0 ] ) , ( [ a1 ] ) , ( [ h1 ] ) , and ( [ htt1final ] ) . the second order solution is described in the appendix [ app : asymp2 ] . it contains an inhomogeneous part , @xmath245 , which is completely determined by the first order solution , and a homogeneous part , which includes the wave function @xmath246 , and the gauge function @xmath247    [ h2 ] @xmath248    the wave function @xmath249 satisfies the homogeneous part of eq.([httneq ] ) , and can be written in the form of eq.([httnmodes ] ) @xmath250 to fix the constants , @xmath219 , we follow the prescription that appears after eq.([httnmodes ] ) . we take @xmath220 from the first order near solution , ( [ b1final ] ) , and expand it in @xmath1 , up to order @xmath251 , to get @xmath252      + \mu^{4}\frac{\pi^{2}(4\cos^{4}\psi-3)}{12l^{2}\rho^{2}}~.      \label{gttn1}\ ] ] then we take @xmath220 from the second order asymptotic solution and transform it to the @xmath168 coordinates , using ( [ transformation ] ) . after expanding it in @xmath17 , up to order @xmath253 , we obtain @xmath254      + \mu^{4}\left[\frac{2\cos2\psi+1}{4\rho^{4 } }      -\frac{\pi^{2}(3\cos6\psi+124\cos^{4}\psi+21\cos2\psi+4 ) }      { 96l^{2}\rho^{2}}\right]+\epsilon^{2}h_{tt}^{(h)}~.      \label{gtta2}\ ] ] we find @xmath249 from @xmath255 and calculate its fourier coefficients @xmath256\nonumber\\      & & = \frac{\pi^{3}l\sinh\frac{2\pi r}{l}\left(-\cosh\frac{4\pi r}{l }      + 16\cosh\frac{2\pi r}{l}\cos\frac{2\pi w}{l}+5\cos\frac{4\pi w}{l}-20\right ) }      { 24r\left(\cosh\frac{2\pi r}{l}-\cos\frac{2\pi w}{l}\right)^{3}}~.      \label{htth}\end{aligned}\ ] ] the asymptotic ( @xmath101 ) form of the metric is    [ h2asymp ] @xmath257    the contribution to the adm mass vanishes indeed , @xmath258dw = \frac{\pi^{3}l^{2}}{4}-2r^{2}\left.w_{,r}^{(2)}\right|_{w =- l/2}^{w = l/2}=0      ~ ,      \label{deltam2}\ ] ] where the last equality follows from boundary condition ( [ w2l2 ] ) . the tension ( [ tension ] ) is now @xmath259dw          = -m\frac{\pi^{2}\epsilon}{6}~.      \label{tension2}\ ] ]      following section [ subsec : nearn ] , we find that the second order solution is    [ emetric2 ] @xmath260~ ,      \label{b2}\\      a_{2}&=&\frac{4h_{2,\psi}(\rho,\psi)}{\sin^{2}\psi\,\cos\psi }      + 2f_{2}(\rho,\psi)+2\rho f_{2,\rho}(\rho,\psi)-\frac{\pi^{4}\cos^{2}\psi}{864\mu^{4 } }      \left[\mu^{2}(105\rho^{2}+11\mu^{2})\cos6\psi\right.\nonumber\\      & & \left.+(-64\rho^{4}+138\mu^{2}\rho^{2}+58\mu^{4})\cos4\psi      + ( 1472\rho^{4}+1383\mu^{2}\rho^{2}+237\mu^{4})\cos2\psi      -768\rho^{4}+774\mu^{2}\rho^{2}-2\mu^{4}\right]~ ,      \label{a2}\\      u_{2}&=&-2      \frac{6\rho h_{2}(\rho,\psi )      -(2\rho^{2}-\mu^{2})h_{2,\rho}(\rho,\psi)}{\rho\sin\psi\,\cos^{2}\psi }      -2\tan^{2}\psi f_{2}(\rho,\psi)-2\tan\psi f_{2,\psi}(\rho,\psi)~ ,      \label{u2}\\      v_{2}&= & \frac{\rho(2\rho^{2}-\mu^{2})}{(\rho^{2}-\mu^{2})\sin^{2}\psi\,\cos\psi }      \left[3h_{2}(\rho\psi)-\tan\psi      h_{2,\psi}(\rho\psi)\right ]      -\frac{2\rho^{2}h_{2,\rho}(\rho\psi)}{\sin^{2}\psi\,\cos\psi}\nonumber\\      & &      + \frac{\rho^{3}}{\rho^{2}-\mu^{2}}f_{2,\psi}(\rho,\psi )      -\rho^{2}\tan\psi f_{2,\rho}(\rho,\psi )      + \frac{\pi^{4}\rho(2\rho^{2}-\mu^{2})\sin2\psi}{864\mu^{4}(\rho^{2}-\mu^{2})}\left [      \mu^{2}(15\rho^{2}+\mu^{2})\cos6\psi\right.\nonumber\\      & & \left . + ( -8\rho^{4}+29\mu^{2}\rho^{2}+4\mu^{4})\cos4\psi      + ( 184\rho^{4}+185\mu^{2}\rho^{2}+11\mu^{4})\cos2\psi      -96\rho^{4}+99\mu^{2}\rho^{2}-16\mu^{4}\right]~.      \label{v2}\end{aligned}\ ] ]    the function @xmath261 satisfies the differential equation ( [ hnequation ] ) with @xmath262 where the functions @xmath263 are given in eq.([hpsisin ] ) .    the solution of eq.([hnequation ] ) with inhomogeneity ( [ s2 ] ) can be written in form eq.([hnsum ] ) with @xmath264 . for each of the non vanishing @xmath265 one can add combinations of the homogeneous solution which includes the legendre functions as appear in eq.([hnsum ] ) . these should be chosen such that @xmath265 is regular at @xmath266 , and is a polynomial in @xmath35 of the lowest order possible ( in principle , the homogeneous solution is of order @xmath267 ) . in fact , the particular solution for eq.([hnequation ] ) with inhomogeneity ( [ s2 ] ) includes polynomials of order @xmath268 in @xmath35 , only ,    [ h2p ] @xmath269    in addition , to avoid negative powers of @xmath1 , the homogeneous part of @xmath261 and the gauge function @xmath270 should be polynomials of order @xmath268 in @xmath271 . @xmath272\sin^{3}\psi      + 8a_{1}r^{2}(r^{2}-1)(1+\cos^{2}\psi)\sin^{3}\psi~,\label{h2}\\      f_{2}(\rho,\psi)&\equiv & ( r^{2}-1)^{2}f_{4}(\psi )      + ( r^{2}-1)f_{2}(\psi)+f_{0}(\psi)~.\label{f2}\end{aligned}\ ] ] we now turn to imposing boundary conditions . the horizon is located at a constant @xmath147 . we set @xmath273 and solve @xmath150 for @xmath274 . we find @xmath275~.\label{f0}\ ] ] the surface gravity is constant ( due to the fact that the legendre functions of the second kind are not included in ( [ h2 ] ) ) , @xmath276~.\label{kappa2}\ ] ]      we match the near solution , ( [ emetric2 ] ) , to the asymptotic solution , ( [ asymp2 ] ) , expanded to order @xmath277 . comparing the expressions for @xmath220 , we find that @xmath278~,\label{2f4}\\      f_{2}(\psi)&=&\frac{\pi^{4}}{17280}\left[15\cos8\psi+188\cos6\psi      + 1248\cos4\psi+3156\cos2\psi+2993\right]~.\label{2f2}\end{aligned}\ ] ] matching the other components , as well , we find the constants @xmath279 the gauge function , @xmath280 , is @xmath281~.      \label{w2lexpand}\end{aligned}\ ] ]      the metric ( [ emetric2 ] ) should be symmetric about the plane @xmath78 . as a result we find that @xmath282      we calculate the entropy of the black hole @xmath283~.      \label{entropy2}\ ] ] we compare the zeroth law @xmath284 and the first law @xmath197 to find that @xmath285    to summarize , the location of the horizon , the entropy , and the temperature are @xmath286~.      \label{rhoh2}\\      s&=&\frac{\pi^{2}\mu^{3}}{2g_{5}}\left[1+\frac{\pi^{2}\epsilon}{8 }      + \frac{\pi^{4}\epsilon^{2}}{384}\right]~.      \label{s2final}\\      t&=&\frac{1}{2\pi\mu}\left[1-\frac{5\pi^{2}\epsilon}{24 }      + \frac{43\pi^{4}\epsilon^{2}}{1152}\right]~.      \label{t2final}\end{aligned}\ ] ] the near metric is given by    [ near4final ] @xmath287      + \frac{\pi^{4}\epsilon^{2}}{5760\mu^{2}}\left[1570\mu^{2}+17\rho^{2 }      -4(14\rho^{2}-395\mu^{2})\cos2\psi\right.\right.\nonumber\\          & & \left.\left.+(52\rho^{2}+580\mu^{2})\cos4\psi      -4(2\rho^{2}-25\mu^{2})\cos6\psi      -(\rho^{2}-2\mu^{2})\cos8\psi\right]\frac{}{}\right)~ ,      \label{b4final}\\      a^{n}&=&1+\frac{\pi^{2}\epsilon}{3\mu^{2}}\left[(3\rho^{2}-\mu^{2})\cos^{4}\psi-1      \right]+\frac{\pi^{4}\epsilon^{2}}{5760\mu^{4}}\left[1800\rho^{4}+1324\mu^{2}\rho^{2 }      -307\mu^{4 }      + 4(675\rho^{4}+422\mu^{2}\rho^{2}-325\mu^{4})\cos2\psi\right.\nonumber\\          & & \left . + 8(135\rho^{4}+88\mu^{2}\rho^{2}-55\mu^{4})\cos4\psi      + 4(45\rho^{4}+26\mu^{2}\rho^{2}-15\mu^{4})\cos6\psi      + 5\mu^{2}(4\rho^{2}-\mu^{2})\cos8\psi\right]~ ,      \label{a4final}\\      u^{n}&=&1+\frac{\pi^{2}\epsilon}{\mu^{2}}(\rho^{2}-\mu^{2})\sin^{2}\psi\cos^{2}\psi      -\frac{\pi^{4}\epsilon^{2}\sin^{2}\psi}{1440\mu^{4 } }      \left[-540\rho^{4}-970\mu^{2}\rho^{2}+385\mu^{4}\right.\nonumber\\          & & \left .      -5(144\rho^{4}+229\mu^{2}\rho^{2}-385\mu^{4})\cos2\psi -5(\rho^{2}-\mu^{2})(36\rho^{2}+82\mu^{2})\cos4\psi      -55\mu^{2}(\rho^{2}-\mu^{2})\cos6\psi\right]~ ,      \label{u4final}\\      v^{n}&= & -\frac{\pi^{2}\epsilon\rho^{3}\sin\psi\cos^{3}\psi}{\mu^{2 } }      -\frac{\pi^{4}\epsilon^{2}\rho^{3}\sin2\psi}{2880\mu^{4 } }      \left[990\mu^{2}+540\rho^{2 }      + ( 720\rho^{2}+1207\mu^{2})\cos2\psi\right.\nonumber\\&+&\left.2(90\rho^{2}+89\mu^{2})\cos4\psi      + 25\mu^{2}\cos6\psi\right ]   ~.      \label{v4final}\end{aligned}\ ] ] in this paper we have calculated the metric of a small black hole in a five dimensional cylinder . the metric is found perturbatively , with an expansion parameter @xmath288 . @xmath55 is the physical adm mass and @xmath2 is the circumference of the compact dimension , both measured at infinity . we calculate the metric , up to second order in @xmath4 ( fourth order in the ratio @xmath14 ) . the asymptotic solution is given in eqs.([a1],[h1],[htt1final ] ) and in the appendix [ app : asymp2 ] . the near solution is provided in eqs . ( [ spherical],[near4final ] ) . we obtained the following expressions for the entropy and the temperature of the black hole @xmath289~ ,      \label{sfinal}\\      t&=&\frac{1}{2\pi\mu}\left[1-\frac{5\pi^{2}\epsilon}{24 }      + \frac{43\pi^{4}\epsilon^{2}}{1152}\right]~.      \label{tfinal}\end{aligned}\ ] ] these , and other results , presented below , were previously obtained in first order of @xmath4 by gorbonos and kol  @xcite and by harmark  @xcite . using the smarr formula we find the relative binding energy @xmath290    we have chosen the coordinate system such that the horizon is independent of the angle , @xmath26 , between the three dimensional space and the @xmath22 axis . as the scale of the five dimensional radial coordinate , @xmath35 , is fixed by the term of the metric , @xmath291 , the radius of the horizon is uniquely determined . the horizon is located at @xmath147 as in eq.([spherical ] ) . we find that @xmath292~.      \label{rhohfinal}\ ] ] the question of transition from black hole to black string , as a function of the mass parameter , has been discussed at length in the literature  @xcite ,  @xcite . one scenario proposed was the transition at the intersection of the black hole and nonuniform black string lines in the @xmath293 ( mass - relative tension ) phase diagram . we use the second order results for the small black hole to extrapolate to such a transition . we study three aspects of the transition ; the mass - relative tension phase diagram , comparison of entropies , and the change of topology ( using two different approaches ) . a uniform black string is described by the metric @xmath294 the relative tension is constant , @xmath295 , and the entropy is @xmath296 the entropy and relative tension of the non - uniform black string were have not been found numerically yet.@xcite . such a configuration exists for masses larger than the gregory - laflamme mass , which is , in terms of our variables , @xmath297 .    in fig.([phasediagram ] ) we draw the @xmath298 phase diagram for the uniform black string , the non - uniform black string ( just the leading order term near the gl transition ) , and the black hole branches . the black hole turns into a nonuniform black string beyond the transition point , @xmath299 . we did not include recent numerical results  @xcite , which , due to numerical difficulties , have large errors  @xcite    in fig.([entropies ] ) we draw the entropies of the black objects in units of @xmath300 .    comparing ( [ sfinal ] ) and ( [ bsentropy ] ) we find that the enropies of the uniform black string and the black hole are equal for @xmath301 . below that value the black hole has higher entropy , and above that value the black string has higher entropy . this simple check suggests that the black hole will be unstable for @xmath302 . note again that our second order expansion in @xmath4 shows good convergence . using the first order expansion only the intersection of entropies would occur at @xmath303 . a transition between a black hole and a black string requires a topology change of the horizon . this change is likely to happen when the horizon of the black hole fills the compact dimension and touch itself . at this point a small pinch of the horizon will turn it into a non - uniform black string . in the notation we use , eq.([transformation ] ) , this will happen when @xmath304 the first order formula would give @xmath305 . again , we see a fairly good convergence , making our result reasonably reliable . we will see a further indication for this point in another , independent , calculation of the critical value of @xmath4 . the apparent agreement of @xmath306 with @xmath307 ( equality of the entropies of the uniform black string and the black hole ) is somewhat problematic . we see no mathematical reason for such a agreement , though it would be interesting if these points also coincided . note that for geometric reasons the black hole branch should cease to exist above the critical value of @xmath4 so the stable state of the system is the uniform black string , which has a higher entropy than the nonuniform black string .      in terms of the coordinates we chose the horizon is located at constant @xmath35 . however , this does not mean it is spherical . compactification breaks the @xmath308 symmetry . the horizon can be an oblate ( elongated perpendicular to the compact direction ) or a prolate ( elongated parallel to the compact dimension ) ellipsoid . there is no generally accepted method for distinguishing between the oblate and the prolate configurations . some authors @xcite define the eccentricity of the black hole as @xmath309 where @xmath310 is the maximal area of a cross section of the horizon parallel to the compact dimension , and @xmath311 is the maximal area of a cross section of the horizon perpendicular to the compact dimension . @xmath312 a prolate ( oblate ) horizon has positive ( negative ) eccentricity . we use the near solution ( [ emetric2 ] ) to find that @xmath313 . we see that the horizon is prolate . another possible measure of the eccentricity is given by the intrinsic ricci scalar of the horizon , which is calculated in terms of the three metric @xmath314      ~.\label{horizonmetric}\ ] ] using the near solution , the intrinsic curvature of ( [ horizonmetric ] ) is @xmath315~.\label{intrinsicr}\ ] ] it is maximal at @xmath134 , which indicates that the horizon is indeed prolate . one might speculate that a prolate horizon will tend to grow in the periodic direction and turn into a nonuniform black string . however , the circumference of the compact direction grows as well  @xcite and the transition into a black string is not obvious . in @xcite it is suggested to measure the inter polar distance , which is the proper distance between the poles of the prolate horizon along the compact dimension . in principle , to calculate this distance we need to break the integral defining this distance into two parts . in the first part we should use the near solution and in the other we should use the asymptotic solution . in the near horizon region one should integrate along @xmath134 from @xmath147 to some arbitrary point @xmath316 . in the asymptotic region we should integrate along @xmath317 from @xmath318 to @xmath319 @xmath320 in @xcite it was found that the zeroth order approximation is @xmath321 , which means that the circumference grows enough to make room for the black hole . if the mass ( @xmath4 ) value , at which the black hole fills the compact dimension , is small enough then it is possible that the integral over the near solution is sufficient in ( [ lpoles ] ) , or , in other words , we can choose @xmath322 . then we obtain @xmath323 first of all , we see that the distance between the poles of the horizon is a decreasing function of @xmath4 . if we solve the equation @xmath324 we obtain @xmath325 , which is in a rough agreement with values we obtained for the critical value , @xmath326 , above . taking the first order term only we would obtain @xmath327 . clearly , the convergence of @xmath328 is not as good as those of @xmath326 and @xmath306 . one reason for the poorer convergence is that for very small @xmath4 ( [ lpoles2 ] ) must fail , as a significant contribution coming from the asymptotic solution was omitted . it should work , however in the region where @xmath329 . fortunately , in this region the @xmath4-expansion is still fairly reliable and the rough agreement of @xmath330 is encouraging . this work is supported in part by the u.s . department of energy grant no . de - fg02 - 84er40153 . we thank richard gass and cenalo vaz for fruitful discussions . we also thank barak kol , hideaki kudoh , evgeny sorkin , and toby wiseman for their comments , which helped us improving this work . the second order asymptotic solution is given by eqs.([h2 ] )        [ h2anh ] @xmath332-\frac{r}{8}\left[h\,h_{,r }      + 4w\,h_{,rw}\right]+(w_{,r})^{2}-r\,k_{,w }      ~,\\      \bar{h}_{ww}^{(2)}&=&\frac{r^{2}}{16}\left[(h_{,r})^{2 }      + ( h_{,w})^{2}\right]-\frac{r}{2}\left[2w_{,w}h_{,r }      + w\,h_{,rw}\right]+\frac{3}{16}h^{2}+(w_{,w})^{2}-r\,k_{,w }      ~,\\      \bar{h}_{rw}^{(2)}&=&k-\frac{r}{16}\left[h\,h_{,w}+8w_{,r}h_{,r}\right ]      + w_{,w}w_{,r}~,\end{aligned}\ ] ]    where @xmath333 , @xmath334 , and @xmath335 , @xmath336~ ,      \label{w1final}\\       k&=&\frac{\pi l^{3}}{16r^{3}}\coth\frac{2\pi r}{l }      \left[1-\frac{2\pi r}{l}\coth\frac{2\pi r}{l}\right ]      \arctan\left(\tan\frac{\pi w}{l}\coth\frac{\pi r}{l}\right )      + \frac{\pi l^{3}\sin\frac{2\pi w}{l }      \left[1-\frac{2\pi r}{l}\coth\frac{2\pi r}{l}\right ] }      { 32r^{3}\left[\cosh\frac{2\pi r}{l}-\cos\frac{2\pi w}{l}\right]}~.\end{aligned}\ ] ] the function @xmath337 was chosen such that @xmath338 . it satisfies eqs . ( [ w0],[w2 ] ) , and it has a structure similar to @xmath339 . @xmath340 is an odd function of @xmath22 , but @xmath341 . however , @xmath27-symmetry implies that @xmath342 . this means that the gauge function , @xmath280 , has nontrivial boundary conditions at @xmath343 @xmath344~.      \label{w2l2}\ ] ]    to match the asymptotic and the near solution we have to transform the asymptotic solution to the @xmath168 coordinates , using ( [ transformation ] ) , and expand to order @xmath277 . we also expand the gauge function @xmath280 as @xmath345~.      \label{w2near}\ ] ] the asymptotic metric in the @xmath168 coordinates , expanded to order @xmath277 is    [ asymp2 ] @xmath346~ ,      \label{uasymp2}\\      v^{a}&= & -\frac{\pi^{2}}{l^{2}}\left[\rho^{3}\sin\psi\cos^{3}\psi      + \mu^{2}\rho\sin\psi\cos^{3}\psi(\sin^{2}\psi+2 )      + \mu^{2}\rho\sin\psi w_{2,\rho}(\rho,\psi )      -\mu^{2}\cos\psi w_{2,\psi}(\rho,\psi)\right ] ~.      \label{vasymp2}\end{aligned}\ ] ]              t. harmark and n.a . obers , jhep 0205,032 ( 2002 ) , e - print archive : hep - th/0204047 ; t. harmark , phys.rev.d69:104015,2004 e - print archive : hep - th/0310259 t. harmark and n.a . obers , e - print archive : hep - th/0309230 h. kudoh and t. wiseman , prog.theor.phys.111:475-507,2004 e - print archive : hep - th/0310104 ; e - print archive : hep - th/0409111 ; b. kol , e - print archive : hep - th/0206220 ; b. kol and t. wiseman , e - print archive : hep - th/02304070 ; e. sorkin , b. kol , t. piran _ phys.rev _ .d 69:064032 ( 2004 ) e - print archive : hep - th/0310096 ;          d. karasik , c. sahabandu , p. suranyi , and l.c.r . wijewardhana , _ phys . rev . _ d70:064007 ( 2004 ) , e - print archive : gr - qc/0404015 t. harmark , phys.rev.d69:104015,2004 e - print archive : hep - th/0310259 ; d. gorbonos and b. kol , jhep 0406:053,200 , e - print archive : hep - th/0406002 l. smarr , j.w . york _ phys . _ d17 , 2529 ( 1978 ) i. rcz and r. m. wald , _ class . _ * 9*:2643 - 2656 ( 1992 ) .
we study black hole solutions in @xmath0 space , using an expansion to fourth order in the ratio of the radius of the horizon , @xmath1 , and the circumference of the compact dimension , @xmath2 . 
 a study of geometric and thermodynamic properties indicates that the black hole fills the space in the compact dimension at @xmath3 . at the same value of @xmath4 the entropies of the uniform black string and of the black hole are approximately equal