document: to provide physical meaning to solutions of einstein equations , is an endeavour whose relevance deserves to be emphasized @xcite . 
   this is particularly true in the case of the levi - civita ( lc ) spacetime @xcite which after many years and a long list of works dedicated to its discussion still presents serious challenges to its interpretation ( @xcite , @xcite-@xcite , and references therein ) . 
   this metric has two essential constants , usually denoted by @xmath3 and @xmath0 . 
   one of them , @xmath4 has to do with the topology of spacetime and , more specifically , refers to the deficit angle . 
   it may accordingly be related to the gravitational analog of aharonov bohmm effect @xcite , @xcite . 
   it is however @xmath0 , the parameter which presents the most serious obstacles to its interpretation .    indeed , for small @xmath0 @xmath5 lc describes the spacetime generated by an infinite line mass , with mass @xmath0 per unit coordinate length . 
   when @xmath6 the spacetime is flat @xcite . 
   however , circular timelike geodesics exits only for @xmath7 becoming null when @xmath8 and being spacelike for @xmath9    furthermore , as the value of @xmath0 increases from @xmath10 to @xmath11 the corresponding kretschmann scalar diminishes monotonically , vanishing at @xmath12 , and implying thereby that the space is flat also when @xmath13    still worse , if @xmath14 the spacetime admits an extra killing vector which corresponds to plane symmetry @xcite ( also present of course in the @xmath12 case ) . 
   thus , the obvious question is : what does lc represents for values of @xmath15 outside the range ( 0,@xmath10 ) ? 
   the absence of circular test particle orbits for @xmath16 , and the fact that most of the known material sources for lc , @xcite , @xcite , @xcite , @xcite require @xmath17 , led to think that lc describes the field of a cylinder only if @xmath0 ranges within the ( 0,@xmath10 ) interval . 
   however , interior solutions matching to lc exist,@xcite , @xcite,@xcite , @xcite with @xmath16 .    furthermore , the absence of circular test particle orbits for @xmath16 may simply be interpreted , as due to the fact that the centrifugal force required to balance the gravitational attraction implies velocities of the test particle larger than 1 ( speed of light ) @xcite . 
   this last argument in turn , was objected in the past on the basis that kretschmann scalar decreases as @xmath0 increases from @xmath10 to @xmath11 , suggesting thereby that the gravitational field becomes weaker @xcite , @xcite . however , as it has been recently emphasized @xcite , @xcite , kretschmann scalar may not be a good measure of the strength of the gravitational field . instead , those authors suggest that the acceleration of the test particle represents more suitably the intensity of the field . 
   parenthetically , this acceleration increases with @xmath0 in the interval ( @xmath10,@xmath11 ) . 
   on the basis of the arguments above and from the study of a specific interior solution matched to lc @xcite , bonnor @xcite proposes to interpret lc as the spacetime generated by a cylinder whose radius increases with @xmath0 , and tends to infinity as @xmath0 approaches @xmath11 . 
   this last fact suggests that when @xmath1 , the cylinder becomes a plane . 
   this interpretation of the @xmath1 case was already put forward by gautreau and hoffman in @xcite ( observe that theirs @xmath0 is twice ours ) , though based on different considerations . 
   however , in our opinion , the question is not yet solved .    indeed , the interior solution analyzed in @xcite is not valid when @xmath12 . 
   therefore the vanishing of the normal curvatures of the coordinate lines on the bounding surface when @xmath18 , suggests but does not prove that the exterior solution with @xmath1 has a plane source . 
   the lc spacetime has no horizons . according to our present knowledge of the formation of black holes 
   , this seems to indicate that there is an upper limit to the mass per unit length of the line sources , and this limit has to be below the critical linear mass , above which horizons are expected to be formed @xcite . 
   the anisotropic fluid @xcite with @xmath19 matched to lc , produces an effective mass per unit length that has maximum at @xmath20which might explain the inexistence of horizons . furthermore , this fact might support too the previous acceleration representation of the field intensity . 
   it agrees with the result that the tangential speed @xmath21 of a test particle @xcite in a circular geodesics increases with @xmath0 , attaining @xmath22 for @xmath23 the source studied in @xcite remains cylindrical for @xmath20 producing a cosmic string with finite radius . 
   however , the effective mass density by increasing up to @xmath24 and then decreasing for bigger values of @xmath0 , raises a disturbing situation of a cylindrical distribution mass not curving spacetime exactly at its maximum value . 
   on the other hand , there exists a puzzling asymmetry between the negative and the positive mass case , for the plane source . 
   the point is that , as mentioned before , the @xmath14 case posseses plane symmetry and furthermore test particles are repelled by the singularity . therefore lc with @xmath14 , has been interpreted as the gravitational field produced by an infinite sheet of negative mass density @xcite ( though there are discrepancies on this point @xcite ) . 
   however in this case ( @xmath14 ) the space is not flat , unlike the @xmath1 case . 
   in other words , if we accept both interpretations , i.e. @xmath1 @xmath25 represents the field produced by an infinite plane with positive ( negative ) mass density , then we have to cope with the strange fact that the negative mass plane curves the spacetime , whereas the positive mass plane does not . 
   this asymmetry is , intuitively , difficult to understand .    in favor of the plane interpretation for the @xmath1 case , point the arguments presented in @xcite , although as already mentioned , they are not conclusive .    furthermore , even if we admit the arguments based on the principle of equivalence , leading to the plane interpretation of the @xmath1 case , there is a problem with the localization of the source itself ( the plane ) . 
   indeed , it seems reasonable to assume , according to the equivalence principle , that the physical components of curvature tensor of an homogeneous static field , vanish everywhere , except on the source ( the plane ) , where they should be singular . 
   however , when @xmath1 the space is flat everywhere ( everywhere meaning the region covered by the patch of coordinates under consideration ) , and therefore a pertinent question is : where is the source ?    in the @xmath14 case , the plane interpretation is supported by the plane symmetry of the spacetime , although objections to this interpretation have been raised , on the basis that the proper distance between neighbouring paths of test particles changes with time @xcite . 
   however , see a comment on this point , below eq.(10 ) .    also , in this case , the physical components of the curvature tensor , and the cartan scalars , are singular at @xmath26 , revealing the existence of a source , however they do not vanish ( except in the limit @xmath27 ) and therefore the pertinent question here is : why does the arguments based on the equivalence principle , mentioned above , do not apply , if @xmath28 corresponds to a plane ? 
   so , unless additional arguments are presented , we are inclined to think that either of the interpretations ( or both ) are wrong .    in order to delve deeper into these questions , and with the purpose of bringing forward new arguments 
   , we propose here to analyze some gedanken experiments with a gyroscope circumventing the axis of symmetry . 
   the obtained expression for the total precession per revolution , @xmath29 , will depend on @xmath0 . then , analyzying the behaviour of @xmath29 as function of different physical variables we shall be able to provide additional elements for the interpretation of lc .    in relation to this , we shall consider also the c - metric ( see@xcite-@xcite and references therein ) , which , as it is well known , describes , in the limit of vanishing mass parameter , the flat space as seen by an accelerated observer ( as the @xmath1 case ) .    as it will be seen below 
   , the discussion presented here does not lead to conclusive answers to the raised issues , but provides hints reinforcing some already given interpretations and , in some cases , creating doubts abouts formerly accepted points of view . 
   in particular it appears that the interpretation of the coordinate @xmath2 as an angle coordinate seems to be untenable in some cases . a fact already brought out in @xcite .    at any rate , 
   it is our hope that the results and arguments here presented , will stimulate further discussions on this interesting problem . 
   the paper is organized as follows . in the next section 
   we describe the lc spacetime and the c - metric . in section 3 
   we give the expression for the total precession per revolution of a gyroscope circumventing the axis of symmetry and display figures indicating its dependence upon different variables . 
   finally , results are discussed in the last section . 
   we shall first describe the lc line element , together with the notation and conventions used here . 
   next we shall briefly describe the c - metric . 
   the lc metric can be written as @xcite , @xcite @xmath30 where @xmath3 and @xmath0 are constants . 
   the coordinates are numbered @xmath31 and their range are @xmath32 with the hypersurface @xmath33 and @xmath34 being identified .    as stressed in @xcite 
   neither @xmath3 nor @xmath0 can be removed by coordinate transformations , and therefore they have to be considered as essential parameters of the lc metric .    as mentioned before , @xmath3 has to do with the topology of spacetime , giving rise to an angular deficit @xmath35 equal to @xcite @xmath36    also , as commented in the introduction , the spacetime becomes flat if @xmath37 is @xmath38 or @xmath11 . 
   in the first case , @xmath39 the line element ( 2 ) , adopts the usual form of the minkoswski interval in cylindrical coordinates ( except for the presence of @xmath3 )    in the second case , @xmath1 , the line element becomes @xmath40 this last expression corresponding to the flat spacetime described by an uniformly accelerated observer with a topological defect associated to @xmath3 . 
   indeed , putting @xmath41 for simplificity , the transformation @xmath42 casts ( 6 ) into @xmath43    then , the components of the four - acceleration of a particle at rest in the frame of ( 6 ) ( @xmath44 , @xmath45 @xmath46 ) as measured by an observer at rest in the minkowski frame of ( 8) are    @xmath47    and therefore @xmath48    thereby indicating that such a particle is accelerated , with proper acceleration @xmath49 . 
   it is perhaps worth noticing that due to ( 4 ) and ( 7 ) , the range of the minkowski coordinate @xmath50 is rather unusual . 
   also observe that bodies located at different points , undergo different accelerations . 
   this implies in turn that two bodies undergoing the same proper acceleration do not maintain the same proper distance ( see p.176 in @xcite for details ) . 
   this metric was discovered by levi - civita @xcite , and rediscovered since then by many authors ( see a detailed account in @xcite ) . 
   it may be written in the form @xmath51 with @xmath52 where @xmath53 and @xmath54 are the two constant parameters of the solution . 
   introducing retarded coordinates @xmath55 and @xmath56 , defined by @xmath57 @xmath58 the metric takes the form @xmath59 with @xmath60    if @xmath61 , and @xmath62 , the c - metric becomes schwarzschild . 
   but , if @xmath63 and @xmath64 then ( 15 ) may be written , with @xmath65 @xmath66 as @xmath67 @xmath68 which can be casted into the minkowski line element by @xmath69 @xmath70 @xmath71 @xmath72    now , for a particle at rest in the ( @xmath73 ) frame ( @xmath74 , @xmath75 @xmath46 ) the components of the four - acceleration as measured by an observer at rest in the ( @xmath76 , @xmath77 , @xmath50 , @xmath78 ) frame , are @xmath79 @xmath80    then , the absolute value of the four acceleration vector for such particle is @xmath81 indicating that the locus @xmath82 is accelerated with constant proper acceleration @xmath54 . 
   observe that in this case the ( @xmath73 ) coordinates are only restricted by @xmath83 
   let us consider a gyroscope circumventing the symmetry axis along a circular path ( not a geodesic ) , with angular velocity @xmath84 
   . then it can be shown that the total precession per revolution is given by ( see @xcite for details ) @xmath85 with @xmath86 . 
   the tangential velocity of particles along circular trayectories ( not necessarily geodesics ) on the plane ortogonal to the symmetry axis , is given by the modulos of the four - vector ( see @xcite , @xcite,@xcite ) @xmath87 ^{-1}v^\mu ,   \label{25}\ ] ] with @xmath88    then , for a particle in lc spacetime @xmath89 in terms of @xmath21 , the expression for @xmath90 becomes @xmath91 . 
   \label{28}\ ] ]      next , due to the similarity of interpretation , mentioned before , between the @xmath1 case and the c - metric with @xmath63 , we shall also calculate the total precession per revolution of a gyroscope circumventing the axis of symmetry , in the space - time of the c - metric .    using the rindler - perlick method @xcite , and writing the c - metric in the form @xcite 
   @xmath92 @xmath93 with @xmath94 @xmath95 and @xmath96 one obtains , @xmath97 @xmath98 with @xmath99    if @xmath100 we recover the usual thomas precession in a minkowski spacetime . 
   if @xmath63 and @xmath64 on the @xmath101 plane , @xmath102 ^{-1/2}\right\ }   \label{33}\ ] ] which is the thomas precession modified by the acceleration factor @xmath54 ; while if @xmath62 , @xmath61 , we recover the usual fokker - de sitter expression for precession of a gyroscope in the schwarzschild metric @xcite , @xmath103    in the general case @xmath62 , @xmath104 ( on the @xmath101 plane ) , we have from ( 31 ) that , either @xmath105 or @xmath106 . in the first case ( @xmath105 ) we obtain @xmath107 ^{-1/2}\right\ }   \label{35}\ ] ] whereas in the case @xmath106 , the result is @xmath108 @xmath109 ^{-1/2}\right\ }   \label{36}\ ] ]    however , this last case implies @xmath110 , for otherwise @xmath111 , what would change the signature of the metric .    finally , the tangential velocity of the gyroscope on the circular orbit calculated from ( 23 ) for the c - metric yields @xmath112    then replacing @xmath84 by @xmath21 with ( 37 ) , into ( 35 ) , we obtain ( @xmath113 ) @xmath114 where @xmath115 and @xmath116 where @xmath117 in the last case however , remember that @xmath53 must be negative .    if @xmath63 , ( 32 ) may be written ( with ( 37 ) ) as @xmath118 with @xmath119 indicating that the precession is retrograde for any @xmath120 and @xmath121    in the next section we shall discuss about the meaning of lc in the light of the information provided by ( 28 ) and ( 38 ) . 
   let us now analyze some figures obtained from ( 28 ) and ( 38 ) .      for @xmath124 ( @xmath16 ) 
   the precession is always forward ( @xmath125 ) as it obvious from ( 28 ) . 
   however for @xmath126 ( @xmath127 ) it may be retrograde ( @xmath128 ) depending on @xmath129 and @xmath21 , as indicated in figure(2 ) , figure(3 ) . 
   thus the cases @xmath130 ( @xmath1 ) and @xmath131 ( @xmath14 ) induce very different behaviours on gyroscopes . 
   this fact , together with the assymmetry mentioned in the introduction , reinforces our doubts about the simultaneous interpretation of both cases ( @xmath132 ) as due to infinite sheet of either positive or negative mass density . 
   next , let us consider the c - metric in the @xmath63 case . 
   figure(4 ) shows the behaviour of the gyroscope as function of the acceleration . 
   observe that the precession is retrograde , in contrast with the @xmath130 case , for which @xmath133 is always positive . 
   this behaviour is the opposite for lc and @xmath134 equal to @xmath135 ( see fig.(5 ) ) , and reinforces further the difficulty of interpreting @xmath2 ( in lc with @xmath130 ) as the usual azhimutal angle . 
   still worse , in this later case , @xmath90 always exceed @xmath136 indicating that the precession is forward even in the rotating frame .    now , since both cases ( c - metric with @xmath63 and lc with @xmath130 ) represent the same physical situation ( i.e. flat space described by an uniformly accelerated observer ) then we have to conclude that the meaning of @xmath2 in lc with @xmath130 , is different from its usual interpretation ( as an angle ) . 
   this also becomes apparent from the definition of @xmath21 given by ( 27 ) ( the tangential velocity decreases as @xmath137 ) . 
   also observe that in the case of the c - metric with @xmath63 , we recover the thomas precession in the limit @xmath138 . 
   this however is impossible in the lc case with @xmath130 . in the same order of ideas 
   it is worth noticing that in the case @xmath131 , the meaning of @xmath2 seems to correspond ( qualitatively ) to that of an azimuthal angle .    on the other hand 
   , it is clear that in the case of a plane source we should not expect @xmath2 to behave like an angle coordinate ( see also @xcite on this point ) . 
   therefore , on the basis of all comments above , we are inclined to think ( as in @xcite ) that the @xmath139 @xmath11 case corresponds to an infinite plane . the absence of singularities in the physical components of the curvature tensor , remaining unexplained , although ( probably ) related to the restrictions on the covering , of the coordinate system . by the same arguments it should be clear that the interpretation of the @xmath131 case as due to a plane , seems to be questionable .
summary: the physical meaning of the levi - civita spacetime , for some  critical  values of the parameter @xmath0 , is discussed in the light of gedanken experiments performed with gyroscopes circumventing the axis of symmetry . 
   the fact that @xmath1 corresponds to flat space described from the point of view of an accelerated frame of reference , led us to incorporate the c - metric into discussion . 
   the interpretation of @xmath2 as an angle coordinate for any value of @xmath0 , appears to be at the origin of difficulties .