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the important behavior of colliding gravitational waves was first shown in khan and penrose s pioneering work [ 1 ] in a plane - symmetric spacetime . the formulation for dynamics of colliding plane waves was given by szekeres [ 2 ] in an extended form with non - parallel polarization , and the solution , which is impulsive similarly to khan and penrose s one , was derived by nuktu and halil [ 3 ] . the plane - symmetric inhomogeneous cosmological models were , on the other hand , studied by gowdy [ 4 ] in a vacuum spacetime with compact invariant hypersurfaces , and models containing space - like and time - like singularities alternately were discussed by the present author [ 5 ] . studies on impulsive colliding gravitational waves were developed by chandrasekhar and his collaborators [ 6 ] who treated not only vacuum cases but also the cases with fluids and electromagnetic fields , and derived also the solutions with a time - like singularity . yurtsever [ 7 ] derived solutions which tend to the kasner solutions asymptotically , contain the cauchy horizons , and are connected with the schwarzschild and weyl vacuum solutions . the cauchy horizons in the plane - symmetric spacetime were paid attention in similarity to the cauchy horizons appearing in the neighborhood of the time - like singularity of the ressner - nordstrom black hole spacetime . around the latter cauchy horizon there is the extraordinary behavior of the spacetime such as the mass inflation [ 8 ] and infinite blue shift [ 9 ] . recently ori discussed the instability of the cauchy horizon in plane - symmetric spacetimes in connection with that in the black hole solutions [ 10 ] , and showed the possibility of a true null singularity in colliding gravitational waves [ 11 ] .    in this paper we show new exact solutions representing plane - symmetric spacetimes with true null singularities , which are deeply related to ori s local proof [ 11 ] . in  2 , we show the basic equations to be solved , for the plane - symmetric spacetime s in the case of parallel polarization , and derive their new solutions with a kind of self - similarity . it is found by analyzing one of the curvature invariants that in addition to the space - like or time - like singularity there appear true null singularities for a limited range of the model parameter . in  3 , schemes of colliding waves are shown , corresponding to the above solutions , and in  4 concluding remarks are given . in appendices a and b the tetrad components of curvature tensors and some basic formulas for hypergeometric functions are shown . the line - element of plane - symmetric spacetimes with parallel polarization is given in the rosen form @xmath0,\ ] ] where @xmath1 and @xmath2 are functions of @xmath3 and @xmath4 , and @xmath5 and @xmath6 . the vacuum einstein equations reduce to @xmath7 @xmath8 @xmath9 where @xmath10r\ ] ] with @xmath11 and so on . the derivation of the above equations are shown in appendix a , together with the components of the curvature tensor .    from eq.([eq : ba2 ] ) we have @xmath12 where @xmath13 and @xmath14 are arbitrary functions . if @xmath15 and @xmath16 , we can put @xmath17 and @xmath18 without loss of generality . in the same way , if @xmath15 and @xmath19 , we can put @xmath17 and @xmath20 . a. the case when the hypersurfaces @xmath21 are space - like    in this case we can assume @xmath15 and @xmath22 , and so @xmath23 . then eqs.([eq : ba3 ] ) and ( [ eq : ba5 ] ) lead to @xmath24 and @xmath25 ( v + u).\ ] ] here let us assume the following form of @xmath2 : @xmath26 this @xmath2 has a kind of self - similarity , because @xmath27 is a function of only @xmath28 . the parameter @xmath29 is equal to @xmath30 in ori s paper [ 11 ] . if we substitute eq.([eq : ba9 ] ) into eq.([eq : ba7 ] ) , we obtain an ordinary differential equation for @xmath31    @xmath32 { da \over dp } +   { 1 \over 4}(1 + 2\alpha)a = 0.\ ] ]    this is a hypergeometric differential equation . its independent solutions are expressed using hypergeometric functions as follows and have singular points at @xmath33 :    for the non - integer @xmath29 , we have @xmath34 and @xmath35 for the integer @xmath36 , @xmath37 and @xmath38 where @xmath39 and @xmath40 for @xmath41 and @xmath42 , respectively . the expressions of @xmath43 and @xmath40 are shown in appendix b. from eq . ( [ eq : ba8 ] ) we obtain for @xmath44 @xmath45\ ] ] and for @xmath46 @xmath47 from the two solutions ( [ eq : ba11 ] ) and ( [ eq : ba12 ] ) we have for @xmath48 @xmath49\ ] ] and @xmath50,\ ] ] respectively . if we use a formula @xmath51 it is found that the first solution ( [ eq : ba11 ] ) is a sum of the following two solutions : @xmath52 and @xmath53 and the second solution ( [ eq : ba12 ] ) is a sum of the following two solutions : @xmath54 and @xmath55 this means that the solutions are symmetric for the conversion of @xmath56 and @xmath57 .    here let us examine the singular behavior in the limit of @xmath58 . \(1 ) _ non - integer @xmath29 _    for the first solution ( [ eq : ba11 ] ) or ( [ eq : ba17 ] ) in the case of @xmath59 , we find @xmath60 from eq.([eq : ba15 ] ) and @xmath61 , which comes from the product @xmath62 in eq.(a19 ) , so that we have the invariant curvature @xmath63 . if @xmath46 , we find that @xmath64 and @xmath65 are finite , so that @xmath66 also is finite , and if @xmath67 , @xmath68 and @xmath65 is finite , so that @xmath66 is finite .    the second solution ( [ eq : ba12 ] ) or ( [ eq : ba18 ] ) is regular at @xmath69 and so @xmath64 and @xmath66 are finite in the limit of @xmath70 .    it is concluded therefore that only the first solution given by ( [ eq : ba11 ] ) or ( [ eq : ba17 ] ) for @xmath59 has the true null singularity the singular behavior in the limit of @xmath71 is the same as that in the limit of @xmath70 .    as was shown in eqs . ( [ eq : ba25 ] ) @xmath72 ( [ eq : ba28 ] ) , the first and second solutions at @xmath69 correspond to the first and second ones at @xmath73 , but this not the one - to - one correspondence . the second solution at @xmath69 corresponds to both solutions at @xmath73 , and the second solution at @xmath74 corresponds to both solutions at @xmath69 .    as for the individual tetrad components , there are divergent components even for @xmath67 . for example , @xmath75 and @xmath76 diverge in the limit of @xmath70 for @xmath77 , because they include @xmath78 . this divergence is a kind of weak singularities . \(2 ) _ integer @xmath36 _    if @xmath41 , we find that @xmath79 is finite , so that @xmath80 and @xmath81 from the first solution ( [ eq : ba13 ] ) are finite . in the second solution ( [ eq : ba14 ] ) , the term @xmath82 gives the infinity to @xmath83 and @xmath81 in the limit of @xmath70 and @xmath84 . if @xmath42 , on the other hand , @xmath79 is finite , in both solutions ( [ eq : ba13 ] ) and ( [ eq : ba14 ] ) , so that @xmath81 remains to be finite . the true null singularities appear , therefore , only for @xmath41 , if @xmath29 is integer . next let us consider the behavior of the solutions at @xmath85 . if we put @xmath86 , then eq . ( [ eq : ba10 ] ) reduces to @xmath87 { da \over dq } + { 1 \over 4 } ( 1 + 2\alpha)a = 0,\ ] ] and the following two solutions are obtained : @xmath88 and @xmath89 at @xmath90 , we have @xmath91 and @xmath92 , so that the first solution ( [ eq : ba20 ] ) is finite and the second ( [ eq : ba20a ] ) diverges as @xmath93 for @xmath94 . at @xmath95 with finite @xmath3 and @xmath4 , the first solution gives finite @xmath2 and @xmath64 , and so @xmath96 from eqs . ( [ eq : ba4 ] ) , ( [ eq : ba15 ] ) and ( [ eq : ba16 ] ) . if we define @xmath97 and @xmath98 as @xmath99 and @xmath100  ( or @xmath101 ) , we obtain @xmath102 in the neighborhood of @xmath103 . accordingly the solution reduces to the kasner solution with the kasner parameter @xmath104 in the limit of @xmath105 . the invariant curvature diverges as @xmath106 ( cf . eq.(a19 ) ) .    the second solution ( [ eq : ba20a ] ) may have a dominat contribution because of the large term @xmath107 . it gives @xmath108 and so @xmath109 , where @xmath110 . from eq . ( [ eq : ba15 ] ) we obtain @xmath111 and so @xmath112 . in this case , @xmath98 defined by @xmath100 gives the relation @xmath113 , so that we have @xmath114 where @xmath115 these exponents satisfy the kasner relation @xmath116 accordingly the solution is found to be a generalized kasner solution at @xmath85 with the kasner parameters depending on a spatial variable as @xmath117 , where @xmath118 at @xmath85 . the singularities at @xmath119 are space - like ones .    in the limit of @xmath120 and @xmath121  ( with  @xmath122 const ) , the two solutions ( [ eq : ba11 ] ) and ( [ eq : ba14 ] ) satisfy the relation @xmath123 , and so we find for @xmath124 and @xmath125 that @xmath2 vanishes , is finite and diverges , respectively . for @xmath126 in eq.([eq : ba15 ] ) vanishes and we have the kasner behavior of @xmath104 , as in the case @xmath127 . for @xmath128 and @xmath64 remain to be constant and we have the kasner behavior of @xmath104 . for @xmath129 diverges but @xmath64 diverges more strongly as @xmath130 , so that the spacetime reduces to the minkowskian spacetime with the kasner parameter @xmath131 . the above treatment is applicable to the half - plane @xmath132 . to the half - plane @xmath133 , we transform @xmath3 and @xmath4 as @xmath134 and @xmath135 . then we have @xmath136 , and the same conclusion is derived in the same way . b. the case when the hypersurfaces @xmath21 have time - like sections    now let us consider the case @xmath137 . then eqs . ( [ eq : ba3 ] ) and ( [ eq : ba5 ] ) lead to @xmath138 and @xmath139 ( v - u).\ ] ] if we assume @xmath2 in the form of @xmath140 we obtain an ordinary differential equation @xmath141 : @xmath142 { d b \over dp } +   { 1 \over 4}(1 + 2\alpha)b = 0.\ ] ] this equation reduces to eq . ( [ eq : ba10 ] ) by putting @xmath143 , so that the solutions are given by eqs . ( [ eq : ba11 ] ) @xmath72 ( [ eq : ba14 ] ) in which @xmath144 is transformed to @xmath145 . it is shown similarly to the case a that the null singularities appear at @xmath146 and @xmath147 . they are true only for non - integer @xmath59 and integer @xmath41 .    at @xmath85 , the equation for @xmath148 with respect to @xmath149 leads to @xmath150 { d b \over dq } +   { 1 \over 4}(1 + 2\alpha)b = 0.\ ] ] this equation reduces to eq . ( [ eq : ba19 ] ) by putting @xmath151 , so that the solutions are given by eqs . ( [ eq : ba20 ] ) and ( [ eq : ba20a ] ) in which @xmath152 is transformed to @xmath153 . in the limit of @xmath105 , we find from the first solution that  @xmath2 is finite and @xmath154 , and so we obtain @xmath155 in the neighborhood of @xmath103 , where @xmath156 and @xmath157  ( or @xmath158 ) . accordingly the solution tends to the kasner - type one with the kasner parameter @xmath104 , in which the time - like singularity appears in the limit of @xmath105 or @xmath159 , and the invariant curvature diverges as @xmath106 ( cf . eq.(a19 ) ) .    from the second solution we obtain similarly the generalized kasner - type solution expressed by @xmath109 and @xmath160 , where @xmath110 . in this case , a spatial variable @xmath161 defined by @xmath162 gives the relation @xmath163 , so that we have @xmath164 where @xmath165 and @xmath166 are defined by eq . ( [ eq : ba23 ] ) . accordingly the hypersurface @xmath103 gives the kasner - type time - like singularity and the kasner parameter depends on @xmath118 , which is a time variable in the case b.    in the limit of @xmath120 and @xmath167 const ) , the two solutions ( [ eq : ba11 ] ) and ( [ eq : ba14 ] ) , in which @xmath144 was replaced by @xmath145 , satisfy the relation @xmath123 , and so we find for @xmath168 and @xmath125 that @xmath2 vanishes , is finite and diverges , respectively . for @xmath126 in eq.([eq : ba15 ] ) vanishes and we have the kasner behavior of @xmath104 , as in the case @xmath127 .    in the above we considered the half - plane with @xmath169 . to the half - plane @xmath170 , we have to transform @xmath3 and @xmath4 as @xmath171 and @xmath172 , and use @xmath173 . the conclusion about singularities does not depend on this transformation . the positions of singularities are shown in figs . 1 and 2 for @xmath174 and @xmath175 , respectively .    fig.1    fig.2 now , corresponding to two cases a and b in the previous sections , let us consider a spacetime containing colliding gravitational waves , which consists of the no - wave region ( i ) , the outgoing - wave region ( ii ) , the incoming - wave region ( iii ) , and the colliding - waves region ( iv ) . for the connection of these four regions at @xmath176 and @xmath177 , we use the penrose prescription in which @xmath3 and @xmath4 in the metric are replaced by @xmath178 and @xmath179 where @xmath180 is @xmath181 and @xmath182 for @xmath183 and @xmath184 , respectively .    in the case a , the wave scheme is shown in fig . 3 , in which colliding waves are in the region iv within a triangle acd . the true space - like singularity is in the line ac , and for @xmath59 the true null singularities are in two lines be and bf , in which one of these two lines can be weak singularity due to the second solution .    in the case b , the scheme is shown in fig . similarly colliding waves are in the region iv within a triangle acd . the time - like singularity ac and null singularities bf and be are only in the region iv . in the regions ii and iii weak singularities appear in the extended part of the above lines , because @xmath81 vanishes there .    in the region iii , @xmath185 , and @xmath186 are functions of only @xmath3 , so that @xmath187 . for @xmath188 or @xmath189 , eqs . ( [ eq : ba10 ] ) , ( [ eq : ba11 ] ) , ( [ eq : ba13 ] ) , and ( [ eq : ba14 ] ) are satisfied by an arbitrary function @xmath190 , and eq . ( a11 ) gives @xmath191.\ ] ] the tetrad components of curvature tensor vanish , except for @xmath192.\ ] ] here we take the functional form @xmath193 where @xmath31 and @xmath194 are given by eqs . ( [ eq : ba11 ] ) @xmath72 ( [ eq : ba14 ] ) , corresponding to the cases a and b , respectively . then in the limit of @xmath105 ( with finite values of @xmath3 ) , we have @xmath195 and @xmath196 , and in the limit of @xmath70 , we have for @xmath44 @xmath197,\ ] ] @xmath198 \vert v   \vert^{\alpha -2}\ ] ] and we have for @xmath46 @xmath199 @xmath200 where we assumed @xmath201 . accordingly , @xmath202 diverges for @xmath203 , but since @xmath204 vanishes , these divergences give only weak singularities .    in the connection at the boundaries of @xmath176 and @xmath177 , @xmath2 is continuous , but @xmath79 and @xmath205 are not continuous , because for example @xmath206 and @xmath207 at @xmath208 and @xmath209 , respectively . accordingly , @xmath210 has the delta function @xmath211 and @xmath212 along the lines @xmath176 and @xmath177 , respectively . moreover , since @xmath213 and @xmath214 and @xmath79 and @xmath205 are not continuous at @xmath146 and @xmath147 , @xmath202 has the delta function @xmath215 and @xmath216 along the lines @xmath146 and @xmath147 . fig.3    fig.4 in the previous sections we derived exact solutions representing inhomogeneous vacuum spacetimes with space - like and null singularities or time - like and null singularities , and applied them to the problem of colliding gravitational waves . while these null singularities may be closely related to the instability of cauchy horizons , we must consider the stability of this null singularities themselves . for instance , there are questions whether they can coexist together with the electromagnetic fields and fluidal matter , and whether they can appear in colliding waves with non - parallel polarization . these problems will be discussed in separate papers . the physical meaning of the important parameter @xmath29 may be a change rate of the shear , because it is included as a power index in the expression of the kasner parameter . the above spacetimes are cosmologically interesting in that near the space - like or time - like singularity they include strong gravitational waves and at the same time show exactly the velocity - dominated [ 13 ] and anti - newtonian behavior [ 14 ] , in which the kasner parameters depend on a spatial or time variable , respectively . the author would like to thank prof . w. israel for the valuable lectures and helpful discussions at yukawa institute , kyoto university . the following tetrads are used : @xmath217 which satisfy the condition @xmath218 where the non - zero components of @xmath219 are @xmath220 and @xmath221 . the tetrad components of curvature tensor are given by @xmath222 and their components are expressed as @xmath223 @xmath224 , \ ] ] @xmath225 , \ ] ] @xmath226 , \ ] ] @xmath227 , \ ] ] @xmath228 , \ ] ] where @xmath229 etc . the components @xmath230 and @xmath231 are derived from @xmath232 and @xmath233 , respectively , by replacing @xmath234 and @xmath235 each other , as @xmath236 and @xmath237 . other components such as @xmath238 etc vanish . the tetrad components of the ricci tensor are given by @xmath239 and their non - zero components are @xmath240 @xmath241 = 0,\ ] ] @xmath242= 0,\ ] ] @xmath243= 0,\ ] ] @xmath244 = 0.\ ] ] from eqs.(a11 ) and ( a12 ) , the equations for @xmath245 and @xmath246 derived and it is shown that their differentiations with respect to @xmath247 and @xmath248 , respectively give the same expression for @xmath249 consistent with eq . ( a13 ) , when we use eqs . ( a9 ) and ( a10 ) . accordingly eqs.(a11 ) and ( a12 ) are integrable for @xmath186 , as in eq . ( [ eq : ba5 ] ) in the text . if we use the einstein equations ( a9 ) @xmath72 ( a13 ) and the conditions @xmath250 and @xmath251 , we obtain @xmath252,\ ] ] @xmath253,\ ] ] @xmath254,\ ] ] @xmath255.\ ] ] one of curvature invariants @xmath256 is proportional to @xmath257 using eqs.(a14 ) @xmath72 ( a16 ) and eq.([eq : ba4 ] ) we obtain @xmath258 with @xmath259   \big[\gamma_{,11}/(r_{,1})^2 - 2 r ( \gamma_{,1}/r_{,1})^3 \cr   & + & { 3 \over 2r}\gamma_{,1}/r_{,1 } \big ] + 2 \big[- { 1 \over 2r^2 } + 2 \big({\gamma_{,0 } \gamma_{,1 } \over r_{,0}r_{,1 } } \big)^2 \big]^2   +    \big[{1 \over 2r^2 } + 2 \big ( { \gamma_{,0 } \gamma_{,1 } \over r_{,0}r_{,1}}\big)^2 \big]^2 . \end{aligned}\ ] ] formula ( [ eq : ba24 ] ) for the hypergeometric functions and their following expressions in series are found in the texts on special functions [ 12 ] . @xmath260 @xmath261~ z^k,\end{aligned}\ ] ] @xmath262 ~z^k,\end{aligned}\ ] ]    k. khan and r. penrose , nature * 229 * , 185 ( 1971 ) . p. szekeres , j. math . * 13 * , 286 ( 1972 ) . y. nutku and m. halil , phys . lett . * 39 * , 1379 ( 1977 ) . gowdy , ann . phys . * 83 * , 203 ( 1974 ) . k. tomita , prog . theor . phys . * 59 * , 1150 ( 1978 ) . s. chandrasekhar and v. ferrari , proc . r. soc . london * a396 * , 55 ( 1984 ) ; s. chandrasekhar and b.c . xanthopoulos , ibid * a398 * , 223 ( 1985 ) , * a402 * , 37 ( 1985 ) , * a402 * , 205 ( 1985 ) , * a403 * , 189 ( 1986 ) , * a408 * , 175 ( 1986 ) . u. yurtsever , phys . rev . d*37 * , 2790 ( 1988 ) , ibid * 38 * , 1706 ( 1988 ) . e. poisson and w. israel , phys . rev . d*41 * , 1796 ( 1990 ) . r. penrose , in _ battle rencontres _ , edited by c.m . dewitt and j.a . wheeler ( benjamin , new york , 1968 ) , p222 . a. ori , phys . lett . * 67 * , 789 ( 1991 ) . see also a. ori and e.e . flanagan , phys . rev . d*53 * , r1754 ( 1996 ) . a. ori , phys . d*57 * , 4745 ( 1998 ) . h. bateman , _ higher transcendental functions , vol . 1 _ ( mcgraw - hill , new york , 1953 ) , p56 , p101 ;  m. abramowitz and i.a . stegun , _ handbook of mathematical functions with formulas , graphs and mathematical tables _ , nat . standards appl . series , u.s . government printing office , washington , d.c . * 55 * , 1 ( 1964 ) ;   c. snow , _ hypergeometric and legendre functions with applications to integral equations of potential theory _ , ibid * 19 * , 1 ( 1952 ) . d. eardley , e. liang , and r. sachs , j. math . phys . * 13 * , 99 ( 1972 ) . k. tomita , prog . . phys . * 48 * , 1503 ( 1972 ) ; k. tomita , ibid * 54 * , 730 ( 1975 ) ; phys . rev . d*48 * , 5634 ( 1993 ) .
new exact vacuum solutions with various singularities in the plane - symmetric spacetime are shown , and they are applied to the analysis of inhomogeneous cosmological models and colliding gravitational waves . 
 one of the singularities can be true null singularities , whose existence was locally clarified by ori . 
 these solutions may be interesting from the viewpoint of the variety of cosmological singularities and the instability problem of cauchy horizons inside black holes .