hep-th0204101.tex
\count100=1 
\documentstyle[12pt]{article}
\def\doit#1#2{\ifcase#1\or#2\fi}

% Underline for text or math
\catcode`@=11
\catcode`@=12

% Accents and foreign (in text):
\let\under=\b                   % bar-under (but see \un above)
\let\ced=\c                     % cedilla
\let\du=\d                      % dot-under
\let\um=\H                      % Hungarian umlaut
\let\sll=\l                     % slashed (suppressed) l (Polish)
\let\Sll=\L                     % " L
\let\slo=\o                     % slashed o (Scandinavian)
\let\Slo=\O                     % " O
\let\tie=\t                     % tie-after (semicircle 
%   connecting two letters)
\let\br=\u                      % breve

% Abbreviations for Greek letters
\def\a{\alpha} \def\b{\beta} \def\c{\chi} \def\d{\delta}
\def\e{\epsilon} \def\f{\phi} \def\g{\gamma}
\def\h{\eta} \def\i{\iota} \def\j{\psi} \def\k{\kappa}
\def\l{\lambda} \def\m{\mu} \def\n{\nu} \def\o{\omega}
\def\p{\pi} \def\q{\theta} \def\r{\rho} \def\s{\sigma}
\def\t{\tau} \def\u{\upsilon} \def\x{\xi} \def\z{\zeta}
\def\D{\Delta} \def\F{\Phi} \def\G{\Gamma} \def\J{\Psi}
\def\L{\Lambda} \def\O{\Omega} \def\P{\Pi} \def\Q{\Theta}
\def\S{\Sigma} \def\U{\Upsilon} \def\X{\Xi}

% Use the following definitions of \pmb, \bfalpha, etc, 
% for `boldfaced' Greek characters. (03/05/99) 
\def\pmb#1{\setbox0=\hbox{${#1}$}%
   \kern-.025em\copy0\kern-\wd0
   \kern-.035em\copy0\kern-\wd0
   \kern.05em\copy0\kern-\wd0
   \kern-.035em\copy0\kern-\wd0
   \kern-.025em\box0 }

\def\bfalpha{\pmb{\alpha}} \def\bfbeta{\pmb{\beta}} 
\def\bfgamma{\pmb{\gamma}}
\def\bfdelta{\pmb{\delta}} \def\bfepsilon{\pmb{\epsilon}}
\def\bfzeta{\pmb{\zeta}}  \def\bfeta{\pmb{\eta}} 
\def\bftheta{\pmb{\theta}}
\def\bfiota{\pmb{\iota}}  \def\bfkappa{\pmb{\kappa}}
\def\bflambda{\pmb{\lambda}} \def\bfmu{\pmb{\mu}}  
\def\bfnu{\pmb{\nu}}
\def\bfxi{\pmb{\xi}} \def\bfomicron{\pmb{o}}  \def\bfpi{\pmb{\pi}}
\def\bfroh{\pmb{\rho}} \def\bfsigma{\pmb{\sigma}}  
\def\bftau{\pmb{\tau}}
\def\bfupsilon{\pmb{\upsilon}} \def\bfphi{\pmb{\phi}}  
\def\bfchi{\pmb{\chi}}
\def\bfpsi{\pmb{\psi}} \def\bfomega{\pmb{\omega}}

\def\bfAlpha{\pmb{A}} \def\bfBeta{\pmb{B}} \def\bfGamma{\pmb{\Gamma}}
\def\bfDelta{\pmb{\Delta}} \def\bfEpsilon{\pmb{E}} \def\bfZeta{\pmb{Z}} 
\def\bfEta{\pmb{H}} \def\bfTheta{\pmb{\Theta}} \def\bfIota{\pmb{I}} 
\def\bfKappa{\pmb{K}} \def\bfLambda{\pmb{\Lambda}} \def\bfMu{\pmb{M}} 
\def\bfNu{\pmb{N}} \def\bfXi{\pmb{\Xi}} \def\bfOmicron{\pmb{O}} 
\def\bfPi{\pmb{\Pi}} \def\bfRoh{\pmb{P}} \def\bfSigma{\pmb{\Sigma}} 
\def\bfTau{\pmb{T}} \def\bfUpsilon{\pmb{\Upsilon}} 
\def\bfPhi{\pmb{\Phi}} 
\def\bfChi{\pmb{X}} \def\bfPsi{\pmb{\Psi}} \def\bfOmega{\pmb{\Omega}}

% Math symbols
\def\dvp{\raisebox{-.45ex}{\rlap{$=$}} 
\raisebox{-.45ex}{$\hskip .48ex { |}$}} 
\def\dvm{\raisebox{-.45ex}{\rlap{$=$}} }
\def\uvp{\raisebox{.45ex}{\rlap{$=$}} 
\raisebox{.45ex}{$\hskip .48ex { 
|}$}}   
\def\uvm{\raisebox{.45ex}{\rlap{$=$}} }
\def\DP{{\scriptsize{\dvp}}~~}\def\DM{{\scriptsize{\dvm}}~~}
\def\UP{{\scriptsize{\uvp}}~~}
\def\UM{{\scriptsize{\uvm}}~~} % 2-d vector indices
%\def\bo{{\raise.15ex\hbox{\large$\Box$}}} % D'Alembertian
\def\bo{{\raise-.46ex\hbox{\large$\Box$}}} % D'Alembertian
\def\cbo{\Sc [}                            % curly "
\def\pa{\partial}                          % curly d
\def\dell{\bigtriangledown}               % hi ho the dairy-o

\def\pr{\prod}                            % product
\def\iff{\leftrightarrow}                % <-->
\def\conj{{\hbox{\large *}}}             % complex conjugate

\def\ltap{\raisebox{-.4ex}{\rlap{$\sim$}} \raisebox{.4ex}{$<$}} %< or ~
\def\gtap{\raisebox{-.4ex}{\rlap{$\sim$}} \raisebox{.4ex}{$>$}} %> or ~
\def\TH{{\raise.2ex\hbox{$\displaystyle \bigodot$}\mskip-4.7mu %
\llap H \;}}
\def\face{{\raise.2ex\hbox{$\displaystyle \bigodot$}\mskip-2.2mu %
\llap {$\ddot
        \smile$}}}                           % happy face
\def\dg{\sp\dagger}                         % hermitian 
% conjugate
\def\ddg{\sp\ddagger}                      % double dagger
\font\tenex=cmex10 scaled 1200

% Math stuff with one argument
\def\sp#1{{}^{#1}}                 % superscript (unaligned)
\def\sb#1{{}_{#1}}                              % sub"
\def\oldsl#1{\rlap/#1}                          % poor slash
\def\sl#1{\rlap{\hbox{$\mskip 1 mu /$}}#1} % good slash for lower case

\def\Sl#1{\rlap{\hbox{$\mskip 3 mu /$}}#1}      % " upper
\def\SL#1{\rlap{\hbox{$\mskip 4.5 mu /$}}#1}    % " fat stuff (e.g., M)
\def\PMMM#1{\rlap{\hbox{$\mskip 2 mu | $}}#1}   % 
\def\PMM#1{\rlap{\hbox{$\mskip 4 mu ~ \mid $}}#1}       % 

%\def\Tilde#1{{\widetilde{#1}}\hskip 0.03in}                    
\def\Tilde#1{{\widetilde{#1}}\hskip 0.015in}     % big tilde
\def\Hat#1{\widehat{#1}}                        % big hat
\def\Bar#1{\overline{#1}}                       % big bar
\def\bra#1{\left\langle #1\right|}              % < |
\def\ket#1{\left| #1\right\rangle}              % | >
\def\VEV#1{\left\langle #1\right\rangle}        % < >
\def\abs#1{\left| #1\right|}                    % | |
\def\leftrightarrowfill{$\mathsurround=0pt \mathord\leftarrow 
 \mkern-6mu
        \cleaders\hbox{$\mkern-2mu \mathord- \mkern-2mu$}\hfill
        \mkern-6mu \mathord\rightarrow$}
\def\dvec#1{\vbox{\ialign{##\crcr
        \leftrightarrowfill\crcr\noalign{\kern-1pt\nointerlineskip}
        $\hfil\displaystyle{#1}\hfil$\crcr}}}           % <--> accent
\def\dt#1{{\buildrel {\hbox{\LARGE .}} \over {#1}}}% dot-over 
% for sp/sb
\def\dtt#1{{\buildrel \bullet \over {#1}}}              % alternate "
\def\der#1{{\pa \over \pa {#1}}}              % partial derivative
\def\fder#1{{\d \over \d {#1}}}               % functional derivative

% Math stuff with more than one argument
\def\frac#1#2{{\textstyle{#1\over\vphantom2\smash{\raise.20ex
        \hbox{$\scriptstyle{#2}$}}}}}   % fraction
\def\sfrac#1#2{{\vphantom1\smash{\lower.5ex\hbox{\small$#1$}}\over
        \vphantom1\smash{\raise.4ex\hbox{\small$#2$}}}}
       %alternate fraction
\def\bfrac#1#2{{\vphantom1\smash{\lower.5ex\hbox{$#1$}}\over
        \vphantom1\smash{\raise.3ex\hbox{$#2$}}}}       % "
\def\afrac#1#2{{\vphantom1\smash{\lower.5ex\hbox{$#1$}}\over#2}} % "
\def\partder#1#2{{\partial #1\over\partial #2}} % partial derivative 
 % of
\def\parvar#1#2{{\d #1\over \d #2}}               % variation of
\def\secder#1#2#3{{\partial^2 #1\over\partial #2 \partial #3}}  %second
 %  "
\def\on#1#2{\mathop{\null#2}\limits^{#1}}       % arbitrary accent
\def\bvec#1{\on\leftarrow{#1}}                % backward vector accent
\def\oover#1{\on\circ{#1}}                      % circle accent

% Aligned equations
\newskip\humongous \humongous=0pt plus 1000pt minus 1000pt
\def\caja{\mathsurround=0pt}
\def\eqalign#1{\,\vcenter{\openup2\jot \caja
        \ialign{\strut \hfil$\displaystyle{##}$&$
        \displaystyle{{}##}$\hfil\crcr#1\crcr}}\,}
\newif\ifdtup
\def\panorama{\global\dtuptrue \openup2\jot \caja
        \everycr{\noalign{\ifdtup \global\dtupfalse
        \vskip-\lineskiplimit \vskip\normallineskiplimit
        \else \penalty\interdisplaylinepenalty \fi}}}
\def\li#1{\panorama \tabskip=\humongous      % eqalignno
        \halign to\displaywidth{\hfil$\displaystyle{##}$
        \tabskip=0pt&$\displaystyle{{}##}$\hfil
        \tabskip=\humongous&\llap{$##$}\tabskip=0pt
        \crcr#1\crcr}}
\def\eqalignnotwo#1{\panorama \tabskip=\humongous
        \halign to\displaywidth{\hfil$\displaystyle{##}$
        \tabskip=0pt&$\displaystyle{{}##}$
        \tabskip=0pt&$\displaystyle{{}##}$\hfil
        \tabskip=\humongous&\llap{$##$}\tabskip=0pt
        \crcr#1\crcr}}

% The following is to be skipped when using harvmac.tex
\doit0{
\def\ref#1{$\sp{#1)}$}
}

% Text style parameters
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\headheight=0in                         % height of heading (LaTeX)
\headsep=0in            % separation of heading from body (LaTeX)
\textheight=9in                         % height of body (LaTeX)
%\footheight=3ex                           % height of foot (LaTeX)
\footheight=15ex                           % height of foot (LaTeX)
\footskip=4ex    % distance between bottoms of body & foot (LaTeX)
\textwidth=6in                          % width of body (LaTeX)
\hsize=6in                              % " (TeX)
\parskip=\medskipamount          % space between paragraphs (LaTeX)
\lineskip=0pt                           % minimum box separation (TeX)
\abovedisplayskip=1em plus.3em minus.5em        %space above equation 
%   (either)
\belowdisplayskip=1em plus.3em minus.5em        % " below
\abovedisplayshortskip=.5em plus.2em minus.4em  %"above 
%   when no overlap
\belowdisplayshortskip=.5em plus.2em minus.4em  %"below
\def\baselinestretch{1.2}       % magnification for line spacing 
%   (LaTeX)
\thicklines                         % thick straight lines for pictures 
%   (LaTeX)

% Title page
% \thispagestyle{empty}               % no heading or foot on title page 
%   (LaTeX)
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        \repeat
        \put(\xco,\yco){$\kbr$}}
        \put(-20,11){\tiny University of Maryland Elementary Particle
Physics University of Maryland Elementary Particle Physics % 
University of Maryland Elementary Particle Physics}
        \put(-20,-241.5){\tiny University of Maryland Elementary
Particle Physics University of Maryland Elementary Particle Physics
University of Maryland Elementary Particle Physics}
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\def\endtitle{\end{quotation}\newpage}  % end title page

% Section heading and reference stuff
\def\sect#1{\bigskip\medskip \goodbreak \noindent{\bf {#1}} %
\nobreak \medskip}
\def\refs{\sect{References} \footnotesize \frenchspacing \parskip=0pt}
\def\Item{\par\hang\textindent}
\def\Itemitem{\par\indent \hangindent2\parindent \textindent}
\def\makelabel#1{\hfil #1}
\def\topic{\par\noindent \hangafter1 \hangindent20pt}
\def\Topic{\par\noindent \hangafter1 \hangindent60pt}
\def\[{\lfloor{\hskip 0.35pt}\!\!\!\lceil}
\def\]{\rfloor{\hskip 0.35pt}\!\!\!\rceil}
\def\delsl{{{\partial\!\!\! /}}}
\def\nablasl{{{\nabla\!\!\!\!\!{\hskip 1.0pt}/}}}
\def\caldsl{{\calD\!\!\! /}}    
\def\calO{{\cal O}}
\def\asym{({\scriptstyle 1\leftrightarrow \scriptstyle 2})}
\def\Lag{{\cal L}}
\def\du#1#2{_{#1}{}^{#2}}
\def\ud#1#2{^{#1}{}_{#2}}
\def\dud#1#2#3{_{#1}{}^{#2}{}_{#3}}
\def\udu#1#2#3{^{#1}{}_{#2}{}^{#3}}
\def\calA{{\cal A}}\def\calB{{\cal B}}\def\calC{{\cal C}}
\def\calD{{\cal D}}
\def\calE{{\cal E}}\def\calF{{\cal F}}\def\calG{{\cal G}}
\def\calH{{\cal H}}
\def\calI{{\cal I}}\def\calJ{{\cal J}}\def\calK{{\cal K}}
\def\calL{{\cal L}} 
\def\calM{{\cal M}}\def\calN{{\cal N}}\def\calO{{\cal O}}
\def\calP{{\cal P}}
\def\calQ{{\cal Q}}\def\calR{{\cal R}}\def\calS{{\cal S}}
\def\calT{{\cal T}}
\def\calU{{\cal U}}\def\calV{{\cal V}}\def\calW{{\cal W}}
\def\calX{{\cal X}}
\def\calY{{\cal Y}}\def\calZ{{\cal Z}}
\def\rma{{\rm a}} \def\rmb{{\rm b}} \def\rmc{{\rm c}} 
\def\rmd{{\rm d}} 
\def\rme{{\rm e}} \def\rmf{{\rm f}} \def\rmg{{\rm g}} 
\def\Mpl{M_{\rm Pl}}
\def\mw{m\low{\rm W}}
\def\Mgut{M\low{\rm GUT}}
%\def\Re{{\rm Re}\,\,}  \def\Im{{\rm Im}\,\,} 
\def\Re{{\cal R}e\,}  \def\Im{{\cal I}m\,}
\def\tildef{{\tilde f}}
\def\calDsl{{\calD\!\!\!\! /}}
\def\plpl{{{\hskip0.03em}{}{+{\hskip -0.563em}
{\raise-0.03em\hbox{$_+$}}
{\hskip 0.05pt}}{}{\hskip0.03em}}} 
\def\mimi{{{\hskip0.03em}{}{-{\hskip -0.563em}
{\raise-0.05em\hbox{$_-$}}
{\hskip 0.05pt}}{}{\hskip0.03em}}}   
\def\E{{\cal E}}
\def\order#1#2{{\cal O}({#1}^{#2})}
\def\alp{\alpha{\hskip 0.007in}'}
\def\oalp#1{\alp^{\hskip 0.007in {#1}}}
\def\naive{{{na${\scriptstyle 1}\!{\dot{}}\!{\dot{}}\,\,$ve}}}
\def\pl#1#2#3{Phys.~Lett.~{\bf {#1}B} (19{#2}) #3}
\def\np#1#2#3{Nucl.~Phys.~{\bf B{#1}} (19{#2}) #3}
\def\prl#1#2#3{Phys.~Rev.~Lett.~{\bf #1} (19{#2}) #3}
\def\pr#1#2#3{Phys.~Rev.~{\bf D{#1}} (19{#2}) #3}
\def\cqg#1#2#3{Class.~and Quant.~Gr.~{\bf {#1}} (19{#2}) #3} 
\def\cmp#1#2#3{Comm.~Math.~Phys.~{\bf {#1}} (19{#2}) #3} 
\def\jmp#1#2#3{Jour.~Math.~Phys.~{\bf {#1}} (19{#2}) #3} 
\def\ap#1#2#3{Ann.~of Phys.~{\bf {#1}} (19{#2}) #3} 
\def\prep#1#2#3{Phys.~Rep.~{\bf {#1}C} (19{#2}) #3}
\def\jhep#1#2#3{Jour.~High Energy Phys.~{\bf {#1}} (19{#2}) #3}
\def\ptp#1#2#3{Prog.~Theor.~Phys.~{\bf {#1}} (19{#2}) #3}
\def\ijmp#1#2#3{Int.~Jour.~Mod.~Phys.~{\bf A{#1}} (19{#2}) #3}
\def\nc#1#2#3{Nuovo Cim.~{\bf {#1}} (19{#2}) #3}
\def\ibid#1#2#3{{\it ibid.}~{\bf {#1}} (19{#2}) #3}
\def\grg#1#2#3{Gen.~Rel.~Grav.~{\bf{#1}} (19{#2}) {#3} }
\def\pla#1#2#3{Phys.~Lett.~{\bf A{#1}} (19{#2}) {#3}}
\def\mpl#1#2#3{Mod.~Phys.~Lett.~{\bf A{#1}} (19{#2}) #3} 
\def\zp#1#2#3{Zeit.~f\"ur Phys.~{\bf{#1}C} (19{#2}) {#3}} 
\def\jgtp#1#2#3{Jour.~of Group Theory for Physicists, %
{\bf{#1}} (19{#2}) {#3}}
\def\rmp#1#2#3{Rev.~Mod.~Phys.~{\bf {#1}} (19{#2}) {#3}}
\def\hepth#1{hep-th/{#1}} \def\heplat#1{hep-lat/{#1}} 
\def\hepph#1{hep-ph/{#1}} 
\def\szet{{${\scriptstyle \b}$}}
\def\ula{{\underline a}} \def\ulb{{\underline b}} 
\def\ulc{{\underline c}} \def\uld{{\underline d}} 
\def\ule{{\underline e}} \def\ulf{{\underline f}} 
\def\ulg{{\underline g}} \def\ulm{{\underline m}} 
\def\ulp{{\underline p}} \def\uln{{\underline n}}
\def\ulq{{\underline q}} \def\ulr{{\underline r}}
\def\ul{\underline} 
\def\un{\underline} 
%\def\uln{{\underline n}} 
\def\uln#1{\underline{#1}}
\def\ulp{{\underline p}} \def\ulq{{\underline q}} 
\def\ulr{{\underline r}}
\def\ulA{{\un A}} \def\ulM{{\underline M}} 
\def\hatm{\hat m}\def\hatn{\hat n}\def\hatr{\hat r}\def\hats{\hat s}
\def\hatt{\hat t}
\def\<<{<\!\!<} \def\>>{>\!\!>} 
\def\Check#1{{\raise0.6pt\hbox{\Large\v{}}{\hskip -10pt}{#1}}}
\def\cdm{{\Sc D}_{--}} \def\cdp{{\Sc D}_{++}}
\def\Pisl{{\Pi\!\!\!\! /}\,}
\def\eqques{{~\,={\hskip -11.5pt}\raise -1.8pt\hbox{\large ?}
{\hskip 4.5pt}\,}}
%\def\fracmm#1#2{{{#1}\over{#2}}}
%\def\fracm#1#2{\hbox{\large{${\frac{{#1}}{{#2}}}$}}}
\def\fracm#1#2{\,\hbox{\large{${\frac{{#1}}{{#2}}}$}}\,}
\def\fracmm#1#2{\,{{#1}\over{#2}}\,}
\def\half{{\fracm12}}
\def\ha{\half}
\def\frac#1#2{{\textstyle{#1\over\vphantom2\smash{\raise -.20ex
        \hbox{$\scriptstyle{#2}$}}}}}   % fraction
%\def\dt#1{\Dot{#1}}
\def\sqrttwo{{\sqrt2}}
\def\scst{\scriptstyle}
\def\itrema{$\ddot{\scriptstyle 1}$}
\def\Bo{\bo{\hskip 0.03in}}
\def\lrad#1{ \left( A 
{\buildrel\leftrightarrow\over D}_{#1} B\right)} 
\def\derx{\partial_x} \def\dery{\partial_y} \def\dert{\partial_t} 
\def\Vec#1{{\overrightarrow{#1}}}
\def\.{.$\,$}
\def\-{{\hskip 1.5pt}\hbox{-}}
\def\kd#1#2{\d\du{#1}{#2}}

\def\footnotew#1{\footnote{\hsize=6.5in {#1}}} 

\def\low#1{\hskip0.01in{\raise -3pt\hbox{${\hskip 1.0pt}\!_{#1}$}}}
\def\ip{{=\!\!\! \mid}}
\def\gg{{\hbox{\sc g}}}
\def\tr{\,\,{\rm tr}\,} \def\Tr{\,\,{\rm Tr}\,}
\def\det{{\rm det}\,} 
\def\Norm{|\hskip -2.0pt|}
\def\Sgn{\hbox{\rm Sgn}\,}
\def\hati{{\hat i}} \def\hatj{{\hat j}} \def\hatk{{\hat k}} 
\def\hatl{{\hat l}} \def\hatm{\hat m} \def\hatn{\hat n} 
\def\hatr{\hat r}
\def\hats{\hat s} \def\hatt{\hat t}

\begin{document}

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%%%%%%%%%%%%%%%%%%%%% Worksheet %%%%%%%%%%%%%%%%%%%%%
% \end{document}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% \thispagestyle{empty}

%{\bf Preliminary Version (FOR YOUR EYES ONLY!)\hfill\today}
% \\[-0.3in] 
%{\hfill\today} \\[-0.3in] 


\doit0{\vskip -0.1in} 
{\hbox to\hsize{\hfill hep-th/0204101}}\par 
{\hbox to\hsize{\hfill CSULB--PA--02--2}}\par 
\vskip -0.1in 
\hfill 
%{(Revised Version)} 
\\ 

\begin{center} 
%\vglue .25in 

%\vskip -0.6in 
\vskip 0.01in 

{\large\bf Axisymmetric Gravitational Solutions} 
\\[0.001in] 
{\large\bf As Possible Classical Backgrounds}  
\\[0.001in] 
{\large\bf Around Closed String Mass Distributions} 
\\[.1in]  

\baselineskip 9pt 

\vskip 0.26in 

\doit0{ 
Hitoshi ~N{\small ISHINO}\footnote{E-Mail: 
nishino@nscpmail.physics.umd.edu} 
\\[.16in] 
{\it Department of Physics} \\ [.015in] 
{\it University of Maryland} \\ [.015in] 
{\it College Park, MD 20742-4111} \\[.20in]   
and \\[0.2in] 
} 

Hitoshi ~N{\small ISHINO}\footnotew{E-Mail: 
hnishino@csulb.edu} 
~and~ Subhash ~R{\small AJPOOT}\footnotew{E-Mail: 
rajpoot@csulb.edu} 
\\[.16in] 
{\it Department of Physics \& Astronomy} \\ [.015in] 
{\it California State University} \\ [.015in] 
{\it Long Beach, CA 90840} \\[0.02in]  


\vskip 1.8in 

{\bf Abstract}\\[.1in]  
\end{center} 

\baselineskip 14.5pt 

~~~By studying singularities in stationary axisymmetric   Kerr and
Tomimatsu-Sato solutions with distortion parameter $~\d =
2,~3,~\cdots$~ in general relativity, we conclude that these
singularities can be regarded as nothing other than closed string-like
circular mass distributions.  We use two different regularizations to
identify $~\d\-$function type singularities in the energy-momentum
tensor for these solutions, realizing a regulator independent
result.  This result gives supporting evidence that these
axisymmetric exact solutions may well be the classical solutions
around closed string-like mass distributions, just like Schwarzschild
solution corresponding to a point mass distribution.  In other words,
these axisymmetric exact solutions may well provide  the classical
backgrounds around closed strings.  



\vskip 0.6in

\leftline{\small PACS: ~04.20.F, 11.27, 04.70} 
\vskip -0.05in
\leftline{\small Key Words: ~Exact Solutions, General Relativity,
Singularities, Closed Strings} 
\vskip -0.05in
\vfill\eject 

\baselineskip 16pt

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\pageno=2



\leftline{\bf 1.~~Introduction} 

Since its first discovery in general relativity in 1916 
%%% 
\ref\schwarzschild{K.~Schwarzschild, Sitzungsberichte
Preuss.~Akad.~Wiss., {\bf 424} (1916).},
%%%  
it has been well understood that the Schwarzschild solution  describes
the classical gravitational background around a point mass.  If there
is a mathematically rigorous `point mass' in reality,  it is natural
to expect a basic and simple classical solution in general relativity
that describes the space-time around such a `point mass'.  
On the other hand, developments in string theory
%%% 
\ref\gsw{M.~Green, J.H.~Schwarz and E.~Witten, {\it
`Superstring Theory'}, Vols.~{\bf I} and {\bf II}, Cambridge University
Press (1987).}  
%%% 
tell us that such a `point mass' description of a
particle is unrealistic due to divergent self energy, but instead, it
must be replaced by a more `smoothed' one, like a circular mass
distribution in closed string theory \gsw.  

However, if that is really the case, and such a circular mass
distribution is a fundamental physical entity, it seems unnatural that 
no basic exact classical solution has ever been discovered as 
the fundamental classical background around such a circular 
mass distribution, as an improved version of Schwarzschild solution, 
or as more physically realistic mass distributions.  
For example, Chazy-Curzon solution 
%%% 
\ref\curzon{J.~Chazy, Bull.~Soc.~Math.~France, {\bf 52}
(1924) 17; 
H.E.J.~Curzon, Proc.~London Math.~Soc.~{\bf 23} (1924) 477.} 
%%% 
gives a simplest metric around a static axisymmetric mass
distribution.  However, its actual mass distribution is not at a finite
radius, but is confined to the origin as a certain limit.  
What is missing is the classical background solution around 
a circular closed string like mass distribution with a finite 
radius.  

The idea of studying classical solutions in general relativity
around singular mass distributions is also motivated by the 
recent developments in brane theory known as braneworld scenario 
%%% 
\ref\hw{P.~Ho\v rava and E.~Witten,
\np{460}{96}{506}; \np{475}{96}{94}.}%
%%%
\ref\arkanihamedetal{N.~Arkani-Hamed, S.~Dimopoulos and G.~Dvali,
\pl{429}{98}{263};  I.~Antoniadis, N.~Arkani-Hamed, S.~Dimopoulos and
G.~Dvali, \pl{436}{98}{257}.},    
%%% 
or Randall-Sundrum scenario 
%%% 
\ref\rs{L.~Randall and R.~Sundrum, \prl{83}{99}{3370};
\prl{83}{99}{4690}.}     
%%% 
in which the matter distributions are limited to the 
four-dimensional (4D) brane hypersurface embedded into 
5D with $~\d\-$function type singularities.  Such singularities 
on submanifolds or boundaries play an 
important role in brane theories, inducing a special effect such as 
desirable mass hierarchies crucial to phenomenology in 4D 
\rs, or embedding 10D supergravity into 11D \hw.  As a 
matter of fact, some ideas relating the singularities in the Kerr 
solution in general relativity 
%%% 
\ref\kerr{R.P.~Kerr, \prl{11}{63}{237}.} 
%%% 
to strings \gsw\ have been proposed in 
%%% 
\ref\burinskii{A.~Ya Burinskii, \pla{185} 
{94}{441}; {\it `Complex String as Source of Kerr Geometry'}, 
hep-th/9503094; Phys.~Rev.~{\bf D57} (1998) 2392;  
{\it `Structure of Spinning Particle Suggested by Gravity, 
Supergravity \& Low-Energy String Theory'}, hep-th/9910045, 
Czech.~J.~Phys.~{\bf 50S1} (2000) 201.}, 
%%% 
or a new concept of Bekenstein-Hawking black hole entropy
interpreted microscopically in terms of elementary string excitations
has been presented 
%%% 
\ref\sen{{\it See, e.g.,} A.~Sen, \mpl{10}{95}{2081}; P.H.~Frampton
and T.W.~Kephart, Mod.~Phys.~Lett.~{\bf A10} (1995)
2571; A.~Strominger and C.~Vafa, \pl{379}{96}{99};
K.~Behrndt, \np{455}{95}{188};    
J.C.~Breckenridge, D.A.~Lowe, R.C.~Myers, A.W.~Peet, A.~Strominger 
and C.~Vafa, Phys.~Lett.~{\bf B381} (1996) 423;  
C.~Callan and J.~Maldacena,  
Nucl.~Phys.~{\bf B472} (1996) 591; 
G.~Horowitz and A.~Strominger,  
Phys.~Rev.~Lett.~{\bf 77} (1996) 2368; 
J.M.~Maldacena, {\it `Black Holes in String Theory'}, 
Ph.D.~Thesis, hep-th/9607235; 
A.~Dabholkar and J.A.~Harvey, Phys.~Rev.~Lett.~{\bf 63} (1989) 478; 
A.~Dabholkar, G.W.~Gibbons, J.A.~Harvey and F.~Ruiz Ruiz, 
Nucl.~Phys.~{\bf B340} (1990) 33; 
C.G.~Callan, Jr., J.M.~Maldacena, A.W.~Peet,  
Nucl.~Phys.~{\bf B475} (1996) 645.}.   
%%% 

From these developments, it seems natural to consider the 
physical significance of the singularities in stationary axisymmetric 
exact solutions in vacuum in general relativity, in particular, their 
possible link with singular mass distributions that resemble 
closed strings.  Some of these exact solutions may well be analogous to 
Schwarzschild solution \schwarzschild\ that corresponds to a  
point mass distribution.  In our present paper, we try to show that
the series of axisymmetric solutions in vacuum, starting with the Kerr
solution \kerr\ and the Tomimatsu-Sato (TS) solutions 
%%% 
\ref\ts{A.~Tomimatu and H.~Sato, \ptp{50}{73}{95}.}% 
%%% 
\ref\yh{M.~Yamazaki and S.~Hori, \ptp{57}{77}{696};
Erratum-ibid.~{\bf 60} (1978) 1248; 
S.~Hori, Prog.~Theor.~Phys.~{\bf 59} (1978) 1870, 
Erratum-ibid.~{\bf 61} (1979) 365.}  
%%%  
with the distortion parameter $~\d\ge
2$~ seem to describe nothing other than the classical backgrounds
around circular mass distributions with coaxial radii with naked 
singularities for $~q>1$~ \ts, resembling closed strings themselves. 
To put it differently, we try to show that the naked singularities in 
TS solutions with $~\d\ge 2$~ \ts\yh\ can be interpreted as closed
string-like mass distributions.  As a methodology, we adopt 
regularization schemes similar to that in  
%%% 
\ref\nishino{H.~Nishino, \pl{359}{95}{77}.}, 
%%% 
{\it i.e.}, we use Yukawa-potential type regularizations with an 
exponential damping factor, that give the right coefficients for the 
$~\d\-$function like mass distributions.  In ref.~\nishino, 
this regularization was applied successfully to the cases of 
Schwarzschild and Kerr solutions.  In our present paper, we generalize 
this result to the more general case of TS solutions with the 
distortion parameter $~\d = 2, ~3, ~\cdots$~ \ts\yh.  We use two 
types of regulators:  The first one is equivalent to that in \nishino,
while the second one is its slight modification.  By comparing
the results of these two regulators, we will demonstrate no regulator
dependence in our results, which may be taken as the correctness of our
physical interpretation of the ring singularities as closed string-like
circular mass distributions.     



\bigskip\bigskip\medskip 

% \newpage 


\leftline{\bf 2.~~A Preliminary with Circular Charge 
Distribution for Coulomb Potential}  

We first analyze the case of circular charge  distribution for the case
of an electric field with a Coulomb potential, as a preliminary  for
the later case of TS solutions in general relativity.   Consider a
circular charge distribution at radius 
$~\r = a$~ on the equatorial $~x y\-$plane with constant line charge
density $~\s$~ in cylindrical coordinates $~(\r, \varphi, z)$.  
We need to get the Coulomb potential $~\phi(\r,\varphi, z)$~ around 
such a distribution, satisfying the Laplace equation $~\Delta_3 \phi =
0$.  Next, given such a result for $~\phi(\r, \varphi, z)$, we
want to reconstruct the original charge distribution, which should
contain $~\d(\r - a) \, \d(z)$.  The latter scenario is highly
non-trivial,  because if we blindly substitute the result for $~\phi(\r,
\varphi, z)$~  back into the Laplace equation, we get zero everywhere,
by  definition.  What is required here is the careful treatment of the 
singularity at $~\r = a$~ and $~z=0$.  In other words, we  need a good
regularization to handle the singularity.  

There are several regularizations to treat such singularities.  In this 
paper, we adopt finite mass regularizations, based on the 3D  
`massive' Laplace equation 
$$ \li{ & \D\low 3 \phi - \k^2 \phi = 0 ~~, 
&(2.1) \cr } $$ 
that replaces the original `massless' Laplace equation.  

The Coulomb potential $~\phi(\r,\varphi,z)$~ around the circular charge
distribution can be obtained by the use of a   massive Green's function
of the Yukawa-potential type $~e^{- \k r} / r$, where $~r = 
[\, (\r\cos\varphi - a \cos\vartheta)^2  + (\r\sin\varphi - a
\sin\vartheta)^2 + z^2\, ]^{1/2}~$ is the distance from the charge
between the azimuthal angles $~\vartheta \sim \vartheta + \d 
\vartheta$~ on the circle $~\r = a$~ on the $~x y\-$plane and the point
$~(\r\cos\varphi, \r\sin\varphi, z)$.  Let  us call such a small
contribution $~\d \phi$~ to the total potential $~\phi$.  The total
potential $~\phi$~ is obtained by superimposing $~\d \phi$~ 
over $~0 \le \vartheta < 2\pi$~ at $~\r = a$: 
$$ \li{ \phi(\r,\varphi,z) 
& = \sum \d \phi = \int d \phi 
% \cr 
% & 
= \int_0^{2\pi} d\vartheta\, 
      \fracmm{ \s a e^{-\k \sqrt{(\r\cos\varphi - a \cos\vartheta)^2 
     + (\r\sin\varphi - a \sin\vartheta)^2 + z^2}}} 
      { \sqrt{ (\r\cos\varphi - a \cos\vartheta)^2 
     + (\r\sin\varphi - a \sin\vartheta)^2 + z^2}} \cr 
& = \s a \int_0^{2\pi} d \vartheta \, 
      \fracmm{ e^{-\k \sqrt{\r^2 + z^2 + a^2 
     - 2 a \r\cos\vartheta}}} 
      { \sqrt{ \r^2 + z^2 + a^2 - 2 a \r\cos\vartheta}} ~~,    
&(2.2) \cr } $$ 
where the $~\varphi\-$dependence has disappeared in the final
expression, because of the periodicity of the original integrand under
$~\vartheta - \varphi \rightarrow \vartheta - \varphi + 2\pi$, also
consistent with the axial symmetry.  Due to its complicated nature, the
final result in a closed form is more involved than elliptic integrals,
and thus the $~\vartheta\-$integral here is not analytically
performed.  

Given such a Coulomb potential $~\phi(\r, z)$~ in (2.2), our next
question is whether we can `reconstruct' the original charge
distribution singular at $~\r = a,~z=0$.  As has been mentioned,
putting simply $~\k=0$~ from the outset will lead to a vanishing result
by definition.  One method to avoid this is to compute $~\int d^3 \vec x
\, \D\low 3 \phi$~ by equating it with $~\int d^3 \vec x \, \k^2 \phi$,
under the massive Laplace equation $~\D\low 3 \phi - \k^2 \phi = 0$:
$$ \li{I(\k) & \equiv \int d^3 \vec x \, \k^2 \phi 
% \cr 
% & 
= \s a \int d^3 \vec x \int_0^{2\pi} d \vartheta 
     \, \k^2\, \fracmm{ e^{-\k \sqrt{\r^2 + z^2 + a^2 
     - 2 a \r\cos\vartheta }}} 
      { \sqrt{ \r^2 + z^2 + a^2 - 2 a \r\cos\vartheta}} \cr 
& = \s a \k^2 \int_0^\infty d\r \int_{-\infty}^\infty d z 
     \int_0^{2\pi} d\varphi \int_0^{2\pi} d\vartheta\,  
     \r \, \fracmm{ e^{-\k \sqrt{\r^2 + z^2 + a^2 
     - 2 a \r\cos\vartheta }}} 
      { \sqrt{ \r^2 + z^2 + a^2 - 2 a \r\cos\vartheta}} ~~. 
&(2.3) \cr } $$
  
The question now is whether or not the integral $~I(\k)$~ in (2.3) gives
a non-vanishing finite result that corresponds to the right 
total charge $~q\equiv 2\pi a \s$, after the limit $~\k\rightarrow
0^+$.  To this  end, we first change the coordinate variables from
$~(\r,\varphi, z)$~ to $~(r, \theta, \varphi)$, such that its Jacobian
equals   the familiar value $~r$:
$$\li{ I(\k) & = 2\pi \s a \k^2 
     \int_0^{2\pi} d\vartheta\int_0^\infty d r \int_0^\pi 
     d \theta \, r\, \fracmm{ e^{-\k \sqrt{r^2 + a^2\sin\vartheta 
      }}} { \sqrt{ r^2 + a^2\sin^2\vartheta}}  \, 
     \big( r\cos\theta + a \cos\vartheta)  \cr 
& = 2\pi\s a \k^2 \int_0^{2\pi} d\vartheta 
     \int_0^\infty d r \, \fracmm{r \, e^{-\k 
     \sqrt{r^2 + a^2\sin\vartheta 
      }}} { \sqrt{ r^2 + a^2 \sin^2\vartheta}}   \, 
     \big(2 r + 2\pi a \cos\vartheta) ~~. 
&(2.4) \cr } $$ 
Here we have performed the $~\theta\-$integral.  
We have developed in
\nishino\ the following lemma about the 
$~r\-$integration  such as in (2.4) for $~m\le +1$: 
$$ \li{ & \k^2 \int_0^\infty d \eta \, \big( \eta + 1 \big)^m\,  
     e^{-\k \eta} ~~\rightarrow~~
     \cases{ 0 ~~~~ (m = 0,~-1,~-2,~\cdots) ~~, \cr 
             1 ~~~~ (m = + 1) ~~, \cr } ~~~~
     \big( \hbox{as} ~~\k\rightarrow 0^+ \big){~~.~~~~~}  
&(2.5) \cr } $$ 
To use this lemma in our case, we use a new variable $~\eta\equiv 
{\sqrt{r^2 + b^2}} - b ~~(b\equiv a \sin\vartheta)$, and expand the
integrand  around $~\eta = - 1$.  Thus the first term in (2.4)
is 
$$ \li{ 2\pi\s & \!\! a \k^2 
    \int_0^{2\pi} d \vartheta \int_0^\infty d r  
     \, \fracmm{2 r^2 e^{-\k {\sqrt{r^2 + b^2}}}}
     { {\sqrt{r^2 + b^2}}} 
   = 4\pi \s a \k^2 \int_0^{2\pi} 
     d\vartheta \int_0^\infty d\eta \, (\eta + b) 
     \, \fracmm{{\sqrt{\eta^2 + 2b \eta}}}{\eta + b} 
     e^{-\k (\eta+b)} \cr 
& = 4\pi \s a \k^2 \int_0^{2\pi} d\vartheta \, 
     e^{-\k b} \int_0^\infty d\eta \, (\eta+1) \, 
     \left[ \, 1+ \fracmm{b-1}{\eta+1} 
     + {\cal O}((\eta+1)^{-2}) \, \right] \,  e^{- \k\eta} \cr 
& = 4\pi \s a \int_0^{2\pi} d \vartheta \, 
     e^{-\k a \sin\vartheta} + {\cal O}(\k) 
     = 4\pi \s a\int_0^{2\pi}  d\vartheta\, 
      \big( 1 - \k a \sin\vartheta \big) + {\cal O}(\k)  \cr  
& \rightarrow~~ 8\pi^2 \s a~~~~(\hbox{as}~~\k\rightarrow 0^+) ~~. 
&(2.6) \cr } $$
Similarly, the second term in (2.4) can be shown to be zero after
$~\k\rightarrow 0^+$, and therefore we get the desirable result 
$$ \li{ & I(\k) ~~\rightarrow ~~ 8\pi^2 \s a 
    = 4\pi ( 2\pi a\s ) = 4\pi q ~~,  
&(2.7) \cr } $$ 
for the total charge $~q \equiv 2\pi a \s$~ on the circle.  

Since we have obtained a non-vanishing finite result for 
$~I(\k)$, it is natural to identify the integrand with 
a combination of $~\d\-$function such as $~\d(\r - a)\,\d(z)$.  
However, there are a few caveat for such an identification.  
For example, we have to make sure that the integrand before 
the $~\int d^3 \vec x\-$integration vanishes almost everywhere except
for $~(\r, z) = (a, 0)$.  We can itemize 
such criterions for the identification with the $~\d\-$functions, as
follows, by letting $~f(\r,z;\k)$~ be the integrand in 
$~I(\k) \equiv \int d^3 \vec x\, f(\r,z;\k)$: 
$$\li{ & ({\rm i}) ~~~~~ \lim_{\k\rightarrow 0^+} f(\r,z;\k)
     \Big|_{\r \neq a, \, z\neq 0} 
     = 0 ~~, 
&(2.8) \cr   
& ({\rm ii}) ~~~~~ \lim_{\k\rightarrow 0^+} \, \lim_{\r\rightarrow a} 
     f(\r,z;\k) \Big|_{z\neq 0} = 0 ~~,  
&(2.9) \cr  
& ({\rm iii}) ~~~~~ \lim_{\k\rightarrow 0^+} \, \lim_{z\rightarrow 0} 
     f(\r,z;\k) \Big|_{\r\neq 0} = 0 ~~,  
&(2.10) \cr 
& ({\rm iv}) ~~~~~ \lim_{\k\rightarrow 0^+} ~ \lim_{\r\rightarrow a,
     ~ z\rightarrow 0} f(\r,z;\k) = \infty ~~,  
&(2.11)  \cr   
&({\rm v}) ~~~~~ \lim_{\k\rightarrow 0^+} \int d^3 \vec x\, f(\r,z;\k)
     = 8 \pi^2 \s a ~~.  
&(2.12)\cr } $$  
Eq.~(2.8) is to guarantee that the integrand is almost everywhere 
zero, while (2.9), (2.10) and (2.11) imply that the integrand has 
a singularity only at points satisfying both $~\r = a$~ 
and $~z=0$, {\it i.e.}, the circle with the radius $~a$~ from the 
origin on the equatorial $~x y\-$plane.  Eq.~(2.12) has been already 
satisfied by (2.6).  Now, (2.8) is manifestly satisfied, because 
there is no other singularities in $~f(\r,z;\k)$~ other than 
the points on $~(\r,z) = (a,0)$.  For the same reason, eqs.~(2.9) and
(2.10) are also satisfied.  The remaining one (2.11) can be also
confirmed as
$$ \li { & \lim_{\r\rightarrow a, ~ z\rightarrow 0} f(\r,z;\k)
      = 4\k^2 \int_0^{\pi/4} d x \, \fracmm{e^{-\k a \sin x}}
      { 2a\sin x} = \infty~~,  
&(2.13) \cr } $$ 
for $~x\equiv \vartheta /2$.  This is due to the logarithmic divergence
at $~x \rightarrow 0$.  

Since the criterion (i) through (v) are 
satisfied, we can identify 
$$ \li{ & \lim_{\k\rightarrow 0^+} 
     \k^2 \r\int_0^{2\pi} 
     d\vartheta \, \fracmm{ e^{-\k 
     {\sqrt{ \r^2 + z^2 + a^2 - 2a \r \cos\vartheta}}}}
     {\sqrt{ \r^2 + z^2 + a^2 - 2a \r \cos\vartheta}} 
     = 4\pi\, \d(\r - a) \, \d(z) ~~.    
&(2.14) \cr } $$ 

\bigskip\bigskip\bigskip




% \newpage 


\vbox{
\leftline{\bf 3.~~Ring Singularities and Circular Mass
Distributions  for TS-Solutions} 

Once we have understood the basic case of circular charge 
distribution for Coulomb potential, it is much easier to handle 
similar singularities in TS solutions \ts\yh.  As has been 
mentioned, we will adopt the previous method in \nishino, which is 
also consistent with the criterions above.  
}

The Kerr metric \kerr\ is classified as a generalization of 
the Schwarzschild metric \schwarzschild\ due to rotation, while 
Weyl metric 
%%% 
\ref\weyl{H.~Weyl, Ann.~de Phys.~{\bf 54} (1917) 117.} 
%%% 
is another generalization due to deformation.  The generalizations due
to both rotation and deformation yield TS metrics
\ts\yh\ with the distortion parameter $~\d = 2, ~3,~\cdots$.  In
particular, the case $~\d =1$~ of the TS metric \ts\ is equivalent to
the Kerr metric \kerr, while in the cases of $~\d \ge 2$~ there are
naked ring singularities lying outside non-singular event horizons.  
These `naked' ring singularities outside event horizons serve as actual
circular mass distributions in our work.  

We start with the metric of axisymmetric TS solutions 
with the general distortion parameter $~\d = 1, ~2, ~\cdots$~ \ts: 
$$ \li{ & d s^2 = 
       - e^{2\n} (d t)^2 
     + m^2 e^{2\psi} \big( d\varphi - \Omega d t \big)^2 
     + m^2 e^{2\m_2} \big( d x \big)^2 
     + m^2 e^{2\m_3} \big( d y)^2 ~~, 
&(3.1) \cr 
& ~~~~~ \cr 
& e^{2\n} \equiv \fracmm{ p^2 (x^2-1) B} D ~~, ~~~~ 
       e^{2\psi} \equiv \fracmm{ (1-y^2) D}{\d^2 B} ~~, ~~~~ 
      \Omega \equiv - \fracmm{2\d^2 q C}{m D} ~~,  \cr 
& e^{2\m_2} \equiv \fracmm B {\d^2 p^{2\d -2} 
     (x^2 - 1) (x^2 - y^2)^{\d^2 -1} } ~~, ~~~~
     e^{2\m_3} \equiv \fracmm B {\d^2 p^{2\d -2} 
     (1-y^2) (x^2 - y^2)^{\d^2 -1} } {~~, ~~~~~} 
&(3.2) \cr } $$ 
where $~A,~B,~C$~ and $~D$~
are respectively the polynomials of $~x$~ and/or $~y$~ with the 
respective degrees $~2\d^2, ~2\d^2, ~ 2\d^2 -1$~ and $~2\d^2 + 2$.    

The ring singularities exist as the zeros of the algebraic    equation
$~B =0$ \ts\yh, lying on the $~y=0$~ equatorial plane at finite coaxial
radii $~x=x_1, ~x_2, ~\cdots,~x\low\d$~ which are the zeros of $~B
|_{y=0} =0$.  Our task is to show that the corresponding
energy-momentum tensor $~T^{\, 00}$~ have $~\d\-$function type ring
singularities at radii 
$~x=x_1, ~\cdots ,~x\low\d$, on the equatorial plane at $~y=0$.    

To this end, we look into the corresponding Einstein tensor $~G^{\, 00} 
\equiv R^{\, 00} - (1/2) g^{\, 00} R$, with an appropriate
regularization like that in \nishino.  The `regularized' total energy
or mass for such an exact solution is  given in a closed form by  
%%% 
\ref\bardeen{J.M.~Bardeen, Astrophys.~Jour.~{\bf 162}
(1970) 71.} 
%%%
$$ \li{ P^{\, 0} & = \int {\sqrt {-g}} \, T^{\, 00} \, d S_0 
%    = \int {\sqrt {-g}}  \, e\du{(0)} 0 \, e\du{(0)} 0 \, 
%      T^{\, (0) (0)}  \, d S_0
% \cr  & 
= \int {\sqrt{-g}}  \, e^{-\n} 
     e\du{(0)} 0  \, \fracm 1{8\pi} \Big( R^{(0)(0)} 
    + \frac12 R \Big) \, d S_0 \cr 
& = \fracm m 4 \int_{-1}^1 d y \int_{x_0}^\infty d x 
     \Big[ - e^{\psi - \n} 
     \big\{ e^{-\Tilde\m_2} (e^{\Tilde\m_3})_x \big\}_x   
     - e^{\psi - \n} 
     \big\{ e^{-\Tilde\m_3} (e^{\Tilde\m_2})_y \big\}_y \cr 
& ~~~~~ ~~~~~  ~~~~~ ~~~~~  ~~~~~ ~\,  - e^{-\n} 
     \big\{ e^{-\Tilde\m_3 - \Tilde\m_2} (e^\psi)_x \big\}_x   
      - e^{-\n}  
     \big\{ e^{-\Tilde\m_2 - \Tilde\m_3} (e^\psi)_y \big\}_y \cr 
&  ~~~~~ ~~~~~   ~~~~~ ~~~~~  ~~~~~ ~\,  - \frac 14 
     e^{3\psi - \tilde\m_2 + \Tilde \m_3 - 3\n } 
      \Omega_x^2     
     - \frac 14 e^{3\psi + \Tilde\m_2 - \Tilde\m_3 - 3\n} 
      \Omega_y^2 \, \Big] ~~. 
&(3.3) \cr } $$ 
The $~x_0$~ is the lower limit of the $~x\-$integration \ts.  
All subscripts of coordinates denote 
differentiations with respect to the coordinates, {\it e.g.,} 
$~ (e^\psi)_y\equiv \partial_y e^\psi \equiv \partial 
e^\psi/\partial y$~ \bardeen.    
Following ref.~\nishino, we have regularized the functions 
$~\m_2$~ and $~\m_3$~ by a Yukawa-potential type damping factor, 
denoted by $~\Tilde\m_2$~ and $~\Tilde\m_3$:
$$ \li{ & e^{\Tilde\m_2} \equiv e^{\m_2} S ~~, ~~~~ 
          e^{\Tilde\m_3} \equiv e^{\m_3} S ~~, ~~~~ 
S \equiv 1 + b\, \a\, e^{-\a \xi} ~~, 
&(3.4) \cr } $$ 
where $~b$~ is a non-zero real constant independent of $~\a>0$, to be 
fixed shortly.  Since we are interested in the case $~a > m$~ 
(equivalently $~q > 1$) to see naked singularities in TS solutions, 
we use also $~x = + i \Hat x,~p = - i \Hat p$~ appropriately  
\ts\yh% 
%%% 
\ref\krameretal{D.~Kramer, H.~Stephani, E.~Herlt and 
M.~MacCallum, {\it `Exact Solutions of Einstein's Field Equations'}, 
Cambridge University Press (1980).}.  
%%% 
The $~\xi$~ in (3.4) is defined by $~\xi \equiv \Hat x - \Hat 
x_0 \equiv -i(x - x_0)$~ with the range $~0 \le \xi < \infty$.  We use 
in this paper the exponent $~-\a\xi$~ in the regulator mimicking the 
previous Coulomb case (2.2) or as in \nishino.  However, in the next 
section, we will use a slightly modified regulator, in order to see 
possible regulator dependence.  If we take the limit 
$~\a\rightarrow 0^+$~ before the $~x$~ and $~y\-$integrations, we 
simply get zero, satisfying the basic criterion like (2.8) 
in the Coulomb case.  

Since the regulator $~S$~ in (3.4) is inserted only into $~e^{\m_2}$~ 
and $~e^{\m_3}$, the effect of the regulator in the integrand will 
arise only when the derivatives $~\partial_x$~ and/or $~\partial_y$~ 
hit the regulator $~S$~ at least once: 
$$ \li{ P^{\, 0} = & + \fracm m 4 \int_{-1}^1 d y 
      \int_{\xi_0}^\infty d \xi \, 
      \Big[ + e^{\psi -\n} \big\{ e^{-\Tilde\m_2} 
       ( e^{\Tilde\m_3} )\low\xi \big\}\low\xi 
     - e^{\psi -\n} \big\{ e^{-\Tilde\m_3} 
       ( e^{\Tilde\m_2} )_y \big\}_y \Big] \cr  
= & + \fracm m 4 \int_{-1}^1 d y \int_0^\infty d\xi \, 
      \Big[ - (e^{\psi-\n} )\low\xi \, e^{-\m_2} S^{-1} 
       (e^{\m_3} S)\low\xi  
     +  (e^{\psi-\n} )_y \, e^{-\m_3} S^{-1} 
     (e^{\m_2} S)_y \, \Big]  \cr 
& + \fracm m 4 \int_{-1}^1 d y \, 
     \Big[ \, e^{\psi-\m_2-\n} S^{-1} 
     (e^{\m_3} S)\low\xi\, \Big]_{\xi=0}^{\xi=\infty} 
   - \fracm m 4 \int_0^\infty d \xi \, 
     \Big[ \, e^{\psi-\m_3-\n} S^{-1} 
     (e^{\m_2} S)_y \, \Big]_{y=-1}^{y=1} {~~.~~~~~ ~~~ }  
&(3.5) \cr } $$ 
In the first term we have a sign flip caused by  the imaginary unit in
$~\xi \equiv \Hat x - \Hat x_0 \equiv  -i(x - x_0)$.  We have also
performed partial integrations under the $~x$~ and
$~y\-$integrations.  The last two terms on the boundaries have no
contributions when $~\a\rightarrow 0^+$, since they are at $~{\cal
O}(\a^2)$.   The second term in (3.5) is seen to be zero because 
at least one derivative $~\partial_y$~ should hit
$~S$~ which in turn is independent of $~y$, yielding a zero
contribution.  This is because unless there is a $~y\-$derivative on
$~S$, there will be a cancellation between $~S^{-1}$~ and $~S$, which 
is exactly the same as the non-regularized case.  

The only term remaining in (3.5) is  
$$ \li{ & P^{\,0} = - \fracm m 4 \int_{-1}^1 d y \int_0^\infty d\xi 
       \left[\, \fracmm1{\d p}{\sqrt{\fracmm{1-y^2}{x^2-1}}} 
      \fracmm D B\,
       \right]\!\lowlow\xi~ {\sqrt{\fracmm{x^2-1}{1-y^2}}} 
       \, S^{-1} S \low\xi ~~,   
&(3.6) \cr } $$  
which can be shown to be nonzero and finite.  To this end, we need a
special lemma for the factor 
$~e^{\psi-\n}$~ involving $~B$~ and $~D$~ for a general $~\d$:  
$$ \li{ & B = p^{2\d} x^{2\d^2} + {\cal O}(x^{2\d^2-1}) 
      ~~, ~~~~
     D = p^{2\d + 2} x^{2\d^2+2} + {\cal O}(x^{2\d^2+ 1}) ~~,  
&(3.7) \cr } $$ 
justified by the solution for a general $~\d$~
for the polynomial $~A$~ \ts:   
$$ \li{ & A \equiv |\a|^2 - |\b|^2 
     = 2 \sum_{r=1}^\d  (-1)^{r+1} \, d_{r+1} \, f_r \,  
     c\low{\,\d,r} \, F_r = \fracm\calD \D ~~, 
&(3.8\rma) \cr 
& F_r \equiv \fracm {\calD_r}{\D_r} ~~, ~~~~ 
     \calD \equiv \det\, \Big( 
     \fracmm{f_{r+r'-1}} {r+r'-1} \Big) ~~, ~~~~
   \D \equiv \det \Big(\fracmm1{r+r'-1} \Big)  ~~,  
&(3.8\rmb) \cr  
& c\low{\,\d,r} \equiv 2^{2r-1} \, \fracmm{\d (\d+r-1)!} 
      {(\d-r)! (2r)!} ~~, ~~~~ 
     d_r \equiv \fracmm{(2r-3)!!} {(2r-2)!!} ~~, ~~~~ 
     f_r \equiv p^2 (x^2-1)^r + q^2 (y^2-1)^r {~~, ~~~~~ ~~~~~}  
&(3.9) \cr } $$  
where $~\calD_r$~ (or $~\D_r$) is the co-factor of the determinant
$~\calD$~ (or $~\D$) for the $~(1,r)$~ component.  

We can prove that only the term with $~\calD_1$, among $~\d$~ terms in
the sum in (3.8), has the highest power in $~x$:  $~\calD_1 \approx
p^{2\d-2} x^{2\d^2-2} + {\cal O} (x^{2\d^2- 3})$.  Note that
$~\calD_r$~ is proportional to 
$$\li{ \calD_r & = \fracmm1{(\d-1)!} \,  
    \e^{i_1\cdots \Hat{i_r} \cdots i_\d} \, 
    \calD_{1 \, i_1} \calD_{2 \, i_2} \cdots \Hat{\calD_{r \, i_r}} 
     \cdots \calD_{\d \, i_\d} \cr 
& \approx \e^{i_1\cdots \Hat{i_r}\cdots i_\d} \, 
     {\cal O} \Big( (x^2)^{(1+2+\cdots + \Hat r + \cdots +\d) 
      + (i_1+i_2+\cdots +\Hat{i_r} + \cdots + i_\d) - (\d-1)} \Big) ~~, 
&(3.10) \cr } $$ 
where $~\calD_{ij}$~ is the $~i j\-$component of the determinant matrix 
in (3.8b), {\it e.g.,} $~\calD_{2\, i_2} \approx {\cal O}\Big(p^2 
(x^2-1)^{2+i_2 -1}  + q^2 (y^2-1)^{2 + i_2 -1}\Big) \approx {\cal O} 
\big(x^{2 + i_2 -1}\big)$, {\it etc.}, while a `hatted' factor or term 
is to be skipped in the product or sum.  Now it is clear that the 
highest power of $~x$~ arises only in the case of
$~r = 1$~  with $~i_1 =1$, so that 
$$ \li{ & \calD_1 \approx (p^2)^{\d-1} (x^2)^{2 (\d-1)(\d-2)/2 
     - \d + 1} \approx p^{2\d-2} x^{2\d^2-2} ~~.
&(3.11) \cr } $$
Therefore, only the term with $~F_1$~ in (3.8a) has the 
highest power:  
$$ \li{ & A\approx p^{2\d} x^{2\d^2}
     + {\cal O}(x^{2\d^2 -1}) ~~. 
&(3.12) \cr } $$ 
Recalling other relevant relationships \ts\yh, such as  
$$ \li{ & A \equiv |\a|^2 - |\b|^2 
     = u^2 + v^2 - m^2 - n^2 ~~, ~~~~ B \equiv 
      (u+m)^2 +(v+n)^2 ~~, \cr 
&\xi \equiv \fracm\a \b~~, ~~~~\a\equiv u+i v ~~, ~~~~
    \b \equiv m+ i n~~~~ (u,~v,~m,~n \in \IR) ~~, \cr 
& G \equiv |\b|^2 = \sum_{r=1}^\d c\low{\,\d,r} F_r ~~, 
     ~~~~ D A = p^2 (x^2-1) B^2 - 4\d^2 q^2 (1-y^2) C^2 ~~, 
&(3.13) \cr } $$
we see that 
$$\li{ & G \approx c\low{\,\d,1} F_1 \approx \d^2 p^{2\d-2} x^{2\d^2-2} 
    + {\cal O}(x^{2\d^2-3} ) \approx |\b|^2 ~~~\Longrightarrow ~~~
     \b\approx \d p^{\d-1} x^{\d^2-1} 
     \approx m {~~,~~~~~ ~~~~~}  \cr     
& A \approx u^2 - m^2 \approx p^{2\d} x^{2\d^2} ~~~ 
   \Longrightarrow ~~~ u \approx p^\d x^{\d^2} ~~, 
&(3.14) \cr 
& B\approx (u+m)^2 \approx p^{2\d} x^{2\d^2} ~~, 
     ~~~~ D \approx \fracmm{ p^2  x^2 B^2} A 
     \approx p^{2\d+2} x^{2\d^2+2} ~~, 
&(3.15) \cr } $$  
yielding (3.7).  The symbol $~\approx$~ denotes the omission of
lower power of ~$x$.  

We now use these leading terms for the evaluation of (3.6): 
$$ \li{ P^{\,0} & = \fracmm{b m \Hat p\a^2}{4\d} \int_{-1}^1 d y 
    \int_0^\infty d\xi \, {\sqrt{(\xi + \xi_0)^2 + 1}} \, 
    e^{-\a \xi} \, \big[ \, 1+{\cal O}(\a) \,\big] \cr 
& = \fracmm{b m \Hat p\a^2}{4\D} 
     \int_{-1}^1 d y \int_0^\infty d\xi \, 
      \left[\, (\xi + 1) + {\cal O} \Big( (\xi + 1)^0\Big)  
     \, \right] \, e^{-\a \xi} + {\cal O}(\a) ~~ 
\rightarrow ~~ \fracmm{b m \Hat p}{2\d} {~~, ~~~~~ ~~}
&(3.16) \cr } $$ 
as ~$\a \rightarrow 0^+$, where we have relied on the lemma
(2.5).  This gives the total mass in (3.5) 
$$ \li{ & \lim_{\a \rightarrow 0^+} P^{\, 0} = m ~~, 
&(3.17) \cr } $$ 
agreeing with the asymptotic mass 
%%% 
\ref\adm{R.~Arnowitt, S.~Deser and C.~Misner, in {\it `Gravitation'}: 
{\it `An Introduction to Current Research'}, ed.~L.~Witten (New York,
Wiley, 1962).}  
%%% 
after the identification 
$$ \li{ & b = \fracmm{2\d}{\Hat p}~~.   
&(3.18) \cr } $$

Our remaining task is to confirm the criterions for identification 
of the integrand with a desirable combination of $~\d\-$functions, like 
(2.8) - (2.12) in the Coulomb case.  Our present ones for 
the integrand $~g(x,y;\a)$~ under $~\int dy \int dx$~ in (3.3) can be
dictated as 
$$\li{ & ({\rm i}) ~~~~~ \lim_{\a\rightarrow 0^+} \, g(x,y;\a)
     \Big|_{x\neq x_i,~ y\neq 0} 
     = 0 ~~, 
&(3.19) \cr   
& ({\rm ii}) ~~~~~ \lim_{\a\rightarrow 0^+} \, \lim_{x\rightarrow x_i} 
     \, g(x,y;\a) \Big|_{y\neq 0} = 0 ~~,  
&(3.20) \cr  
& ({\rm iii}) ~~~~~ \lim_{\a\rightarrow 0^+} \, \lim_{y\rightarrow 0} 
     \, g(x,y;\a) \Big|_{x\neq x_i} = 0 ~~,  
&(3.21) \cr 
& ({\rm iv}) ~~~~~ \lim_{\a\rightarrow 0^+} ~ \lim_{x\rightarrow x_i,
     ~ y\rightarrow 0} g(x,y;\a) = \infty ~~,  
&(3.22)  \cr   
&({\rm v}) ~~~~~ \lim_{\a\rightarrow 0^+} \int d^3 \vec x ~ g(x,y;\a)
     = (\hbox{finite ~and ~non-zero}) ~~.  
&(3.23) \cr } $$ 
As before, (3.19), (3.20) and (3.21) are easily seen to be satisfied,
while (3.23) has been already confirmed in (3.16), leaving (3.22). 
The only non-trivial singularity at $~(x, y) \rightarrow (x_i,
0)~~{\scst  (i~=~1,~2,~\cdots,~\d ) }$~ in (3.22) must arise from the
first term in (3.5) out of its four terms, because only this term had
non-vanishing contribution at
$~\a\rightarrow 0^+$:  
$$ \li{ - \fracmm m 4 (e^{\psi-\n} )\low\xi & \, e^{-\m_2} \, 
     S^{-1} (e^{\m_3} S )\low\xi 
     = \fracmm m {4\d\Hat p} \bigg(
     \fracmm D {{\sqrt{x^2-1}}\, B} \bigg)\lowlow\xi \, 
     \fracmm{{\sqrt{x^2-1}} \, b \, \a^2 \, e^{-\a \xi}} 
     {1+ b\a e^{-\a \xi}} \, \biggr|_{y=0} ~~, 
&(3.24) \cr} $$ 
after the limit $~y\rightarrow 0$~ is taken.  
Now the question is about the behaviour of the combination $~D/B$~ at
$~y=0$.  To  answer this question, we have to postulate the general
polynomial  structures of $~A,~B$~ and $~C$~ at $~y=0$~ as 
$$ \li{ & A |_{y=0}  = (x-x_1) (x-x_2) \cdots (x-x\low\d) 
     \, A_{2\d^2-\d}(x) ~~, \cr 
& B |_{y=0}  = (x-x_1)^2 (x-x_2)^2 \cdots (x-x\low\d)^2 
     \, B_{2\d^2-2\d}(x) ~~, \cr 
& C |_{y=0}  = (x-x_1) (x-x_2) \cdots (x-x\low\d) 
     \, C_{2\d^2-\d-1}(x) ~~.   
&(3.25) \cr } $$ 
The $~A_{2\d^2-\d}(x) ,~B_{2\d^2-2\d}(x)$~ and $~C_{2\d^2-\d-1}(x)$~
are respectively polynomials of $~x$~ whose degrees are denoted by 
their subscripts, and they are supposed to have no zeros at $~x = x_i~
{\scst (i~=~1,~2,~\cdots,~\d)}$.    
These postulates are based on the following facts \ts.  First, the 
space-time singularities causing the Riemann tensor to blow up, 
occurs when two conditions $~u+m =0$~ and $~y=0$~ are met \ts. 
In particular, there are only $~n$~ zeros at $~x = x_i~{\scst
(i~=~1, ~2,~\cdots,~\d)}$~ for the equation $~u + m = 0$.  Second, in
terms of $~\a = u + i v,~\b = m + i n$~ as in
\ts, $~v = n =0$~ at $~y=0$.  Third, the polynomial
$~(u+m)$~ at $~y=0$~ is supposed to have simple zeros at
$~x=x_i$, and therefore $~B\approx
(u+m)^2$~ has double zeros at $~x=x_i$.  Fourth, $~A
\equiv (u+m)(u-m) + (v+n)(v-n)$~ has simple zeros at $~x=x_i$, due to 
the structure $~(u+m)(u-m)$.  Fifth, we assume the form of $~C$~  
above, based on the explicit cases of $~\d =1$~ and $~\d=2$~ \ts.  
 
Once we specify the structures (3.25), it follows from (3.13) that 
$$\li{ D |_{y=0} = \, & + p^2 (x^2-1) (x-x_1)^3 \cdots (x-x\low\d)^3
     \, \fracmm{(B_{2\d^2-2\d})^2}{A_{2\d^2 -\d}} \cr 
& - 4 \d^2 q^2 (x-x_1) \cdots (x-x\low\d) 
     \, \fracmm{(C_{2\d^2-\d-1} )^2}{A_{2\d^2 -\d}} 
     ~~. 
&(3.26) \cr } $$ 
Obviously there are simple poles in the combination
$~D/B|_{y=0} $~ at $~x=x_i$, because the second term here  
will generate simple poles after being divided by $~B$~ in (3.25).    
Therefore at least one of such poles remains after the differentiation
$~\partial\low\xi = \partial_x$~ in 
$~(D/B)\low\xi$~ in (3.24).  This establishes the singularity (3.22),
and therefore we have the satisfaction of all the criterions for 
the identification of the integrand $~g(x,y;\a)$~ with the 
combination of the $~\d\-$functions:
$$ \li{ & T^{\, 0 0} = \fracmm m{2\pi\d {\sqrt{-g}}} 
    \sum_{i=1}^\d \, \d(\Hat x-\Hat x_i) \, \d(y)~~. 
&(3.27) \cr } $$ 
The previous case for Kerr solution in \nishino\ can be now re-obtained
as the  special case of $~\d=1$.  
Recall that the coordinates $~(\Hat x,y)$~ are related to 
the familiar Weyl's canonical coordinates $~(\r, z)$~
\ts\krameretal\ as 
$$ \li{ & \r = \fracmm{m \Hat p}\d 
     {\sqrt{(\Hat x^2+1)(1-y^2)}} ~~, ~~~~ 
     z = \fracmm{m\Hat p}\d x y ~~, 
&(3.28) \cr } $$ 
so that (3.27) implies that we have the ring 
singularities at the radii $~\r=\r_i\equiv m\Hat p
{\sqrt{\Hat x_i^2+1}}/\d~~{\scst (i~=~  1,~\cdots,~\d)}$~ corresponding
to $~\Hat x = \Hat x_i$~ on the equatorial plane at
$~z =0$~ or equivalently $~y=0$, as desired.  

To turn the table around, our result can be re-interpreted in the
following way:  Suppose we are provided with the closed string-like
multiple circular mass distributions (3.27).  In total, there are
$~\d$~ closed strings with different radii at $~\r = \r_i$.  We can
then obtain the exact classical gravitational background solution around
such mass distributions, which are nothing other than the series of TS
solutions with the distortion parameter $~\d$~ \ts\yh.  In other words,
our result has the significance  that the classical gravitational
background around multi closed strings can be given by the series of TS
solutions \ts\yh.  





\bigskip\bigskip\bigskip 

\leftline{\bf 4.~~Regularization Independence} 

We have chosen so far the particular regulator (3.4), mimicking 
the case of Coulomb potential (2.3).  However, this does not have 
to be the only regulator for gravitational 
singularities.  As a matter of fact, the exponent of a regulator 
does not have to be a linear function like $~-\a\xi$, but it may be
a quadratic function, such as $~- \b \xi^2$, if its purpose is just to 
get the finite $~\xi\-$integral at $~\xi \rightarrow \infty$.   
Namely, we now consider replacing $~S$~ in (3.4) by  
$$ \li{ & \Hat S \equiv 1 + c \,\b \, e^{-\b^2 \xi^2} ~~, 
&(4.1) \cr } $$ 
with a nonzero constant $~c$, 
and we take the limit ~$0<\b\rightarrow 0^+$~ at the end of
computations.  The particular power of $~\b$~ in the exponent or that
in the outside of the exponential function have been chosen, such that 
the final result is neither trivially zero nor divergent.  
In what follows, we briefly study the potential difference between 
this and the previous result (3.16), and show that there is eventually
no regulator dependence.    

Among the four terms in (3.5), the last two terms on the boundaries
do not contribute, while the second term is also zero for reasons 
similar to the previous regulator.  The remaining first term in (3.5)
is now 
$$ \li{  & \fracmm m 4\int_{-1}^1 d y \int_0^\infty d\xi 
     \left[ \, \fracmm{{\Hat p}{\sqrt{1-y^2}}} \d  
    +{\cal O}(\xi^{-1})  \, \right] \, 
     {\sqrt{\fracmm{\Hat x^2+1}{1-y^2}}} ~  
     \fracmm{-2 c \, \b^3 \,\xi \,e^{-\b^2\xi^2}} 
       {1+ c\,\b \, e^{-\b^2\xi^2}} \, 
       \big[\, 1+ {\calO}(\b) \, \big] \cr 
& ~~~ = \fracmm{m{\Hat p} c\b^3}{2\d} 
     \int_0^\infty d\xi\, 
     \left[ \, a_2 (\xi+1)^2 +  a_1 (\xi+1) 
      + a_0 + \fracmm{a_{-1}}{\xi+1} + \fracmm{a_{-2}}{(\xi+1)^2}
      + {\cal O}\Big( (\xi+1)^{-3} \Big) \, \right] \,
      e^{-\b^2\xi^2} \cr  
& ~~~ = \fracmm{m\Hat p c}{2\d} 
      \, \Big( a_2 I_2 + a_1 I_1 
      + a_0 I_0 + a_{-1} I_{-1} + a_{-2} I_{-2} + \cdots \Big) 
      ~~. 
&(4.2) \cr } $$
Here we have expanded the integrand other then the factor
$~e^{-\b\x^2}$~ in terms of $~\xi+1$~ instead of 
$~\xi$~ itself in order to avoid the singularity at $~\xi=0$, with 
the coefficients $~a_2,~a_1, ~\cdots$.  The $~I_n$'s are
defined by 
$$ \li{ & I_n \equiv \int_0^\infty d\z \, \b^{2-n} \, 
      (\z+\b)^n \, e^{- \z^2} ~~,    
&(4.3) \cr } $$ 
with $~\z\equiv \b\xi$.  
Among these $~I_n$'s, we can show that it is only $~I_2$~ whose 
limit is non-zero when $~\b\rightarrow 0^+$, similarly to the 
lemma (2.5):   
$$\li{ & I_n ~~\rightarrow ~~ \cases{ 0 ~~~~~ ~
      (n=1,~0,~-1,~-2,~\cdots) ~~, \cr
      \fracmm{\sqrt\pi}4 ~~~~(n=+2) ~~, \cr } 
~~~~~(\hbox{as} ~~\b\rightarrow 0^+) ~~. 
&(4.4) \cr } $$ 
In fact, $~I_1 \rightarrow 0,~I_0 \rightarrow 0$~ are
easily proven, while we need the inequality $~0 < I_{-m} \le
I_{-(m-1)}$~ for $~(m=2,~3,~\cdots)$~ shown as   
$$ \li{ 0 < I_{-m} & = \b^{2+m} \int_0^\infty 
      d\z \, \fracmm{e^{-\z^2}}{(\z+\b)^m} ~~ 
     = \b^{2+(m-1)} \int_0^\infty d \z \, 
     \fracmm{e^{-\z^2}}{(\z+\b)^{m-1}} \, 
     \Big( 1- \fracm\z{\z+\b} \Big) \cr 
& \le \b^{2+(m-1)} \int_0^\infty d\z\, 
    \fracmm{e^{-\z^2}}{(\z+\b)^{m-1}} = I_{-(m-1)} ~~. 
&(4.5) \cr } $$ 
Therefore, all the $~I_{-m}$'s are bounded from above by $~I_{-1}$
which in turn goes to zero: 
$$ \li { 0 < I_{-1} & \equiv \int_0^\infty 
     d\z \, \fracmm{\b^3}{\z+\b} \, e^{-\z^2} 
     = \int_0^1 d\z \, \fracmm{\b^3}{\z+\b} \, e^{-\z^2} 
       + \int_1^\infty d\z \, \fracmm{\b^3}{\z+\b} e^{-\z^2} \cr 
& \le \int_0^1 d\z \, \fracmm{\b^3}{\z+\b}  
       + \int_1^\infty d\z \, \fracmm{\b^3} \z \, e^{-\z^2} 
%     = \b^3 \Big[ \, \ln(\z+\b) \, \Big]_0^1 
%      + \b^3 \int_1^\infty d\z\, \fracm1\z e^{-\z^2} \cr 
% & 
= \b^3\ln(\b+1) - \b^3 \ln\b + \b^3 {\cal O}(\b^0) 
   ~~ \rightarrow ~~ 0 ~~, {~~~~~ ~~~}
&(4.6) \cr } $$  
as ~$\b\rightarrow 0^+$.   
Now $~0 < I_{-m} \le I_{-(m-1)} \le \cdots \le I_{-1} \rightarrow 0$~
implies that
$~\lim_{\b\rightarrow 0^+} I_{-m} =0~~(m=1,~2,~\cdots)$.  Using
this in (4.2) yields the desirable result 
$$ \li{ & \lim_{\b\rightarrow 0^+} P^{\,0} = m~~, 
&(4.7) \cr } $$ 
if the constant $~c$~ is normalized as ~$c =
8\d/({\sqrt\pi} \Hat p)$, 
since $~a_2 =1$~ from (4.2).  

As for the criterions (3.19) - (3.23), the most important pole structure
in (3.24) out of $~\big({\sqrt{x^2-1}} D/B\big)_\xi$~  remains the 
same, even though the last factor is now replaced by $~\Hat S^{-1}\Hat 
S_\xi$.  Therefore, all criterions are satisfied  as in the previous
regularization, leading us to the same identification with the 
$~\d\-$functions as in (3.27).  

This independence of regularizations is 
the reflection of the fact that the contribution to the total integral 
comes solely from the leading terms in the energy-momentum 
tensor $~T^{\, 0 0}$~ as the integrand.  This also strongly indicates
the `physical' correctness of our interpretation of the singularity as
the closed string-like circular mass distributions.  
     


\bigskip\bigskip\bigskip

% \newpage 


\leftline{\bf 5.~~Concluding Remarks} 

In this paper, we have shown by explicit computation, with 
the appropriate regulator (3.4) that there exist ring singularities in 
the energy-momentum tensor for general TS solutions at the 
coaxial radii $~\r = \r_i~{\scst (i~=~1,~\cdots,~\d)}$~ on the 
equatorial plane at $~z=0$, that can be identified with the
$~\d\-$function like singularity (3.27).  As the guiding principle, we
followed the case of circular  charge distribution for Coulomb
potential with a massive  Yukawa-type damping factor, which
also provides the criterion to be satisfied for the identification of
the integrand with a combination of $~\d\-$functions.  Thanks to 
the generalization in ref.~\yh\ of the TS solutions in \ts\ to $~1\le \d
< \infty$, we have been able to generalize our result to an
arbitrary number of the distortion parameter $~\d$~ which  
corresponds to the number of ring singularities.  In particular, 
there are naked singularities outside the event horizons in the cases
$~\d\ge 2$~ which may well be actual closed
string-like mass distributions.    

It has been well-known that the asymptotic mass for a given exact 
axisymmetric gravitational solution can be obtained by the asymptotic
behaviour of such a solution at $~r\approx \infty$~ \adm.  
However, such a method is not explicit enough for us to identify 
the singularities in the solution with the closed string type 
mass distribution.  One of the reasons is that the asymptotic
behaviors are associated with global boundary effects, which are not
detailed enough to describe `microscopic' mass distributions.  
In other words, our methodology provides more direct
and explicit links between exact solutions and mass distributions.  
We stress that the mass distribution in energy-momentum tensor is a
`physical entity', and it is not just an `effective mass' observed only 
at infinity which is indirect or implicit.  

We have also used in this paper two distinct regularizations
(3.4) and (4.1), and have demonstrated that our result $~P^{\, 0} =
m$~ is independent of a particular choice of regulators.  This is
because the contributions to the total integral came from the `leading'
terms in the expansions, such as the highest power in (3.16), or the
structure of zeros in (3.25) and (3.26).  The analogous result in the
Coulomb case also provides a justification of such a regularization
based on a `physical' consideration.  In other words, our method is not
a reflection of a mathematical artifact, but is a
`physical entity' with the total mass $~m$.  We also mention that,
as is usually the case in physics, once physical interpretation is
correct, a regularized finite result does not depend on the choice of
regularizations.  

If superstring theory \gsw\ really describes the ultimate microscopic
scales around the Planck mass, it implies the existence of  singular
mass distributions as `physical objects'.   These mass distributions
are required to be rigorously singular, {\it i.e.,} there must be no
smearing effect on such singularities.  From
this viewpoint, it is natural that the gravitational background around
such singular mass distributions exist in a relatively simple and
fundamental form, such as generalized TS solutions for $~\d \ge 2$~
\ts\yh.  Our result in this paper has provided supporting evidence for
such philosophy.    

In the Randall-Sundrum scenario \rs, the 4D non-gravitational fields 
are supposed to exist only on branes embedded in 5D.  Accordingly, all 
the non-gravitational physical fields are represented as a 
$~\d\-$function singularity in the energy-momentum tensor \rs.  
Similarly in the braneworld scenario \hw\arkanihamedetal, 10D
supergravity can be embedded into 11D supergravity, again within
certain $~\d\-$function type singularities.  The common feature
between the braneworld scenarios and  our result is that the
$~\d\-$function singularities represent important physical entities. 
Therefore it is natural to identify the singularities in the
energy-momentum tensor in certain axisymmetric exact gravitational
solutions in vacuum with 
$~\d\-$function type closed string mass distributions.  

In the cases of $~\d \ge 2$~ of TS solutions, there 
are in total $~\d$~ ring singularities, some of which may be  
completely hidden by event horizons, in addition to `naked' ones
lying outside of non-singular event horizons \ts\yh.  
Our original motivation for studying the TS solutions with 
$~\d\ge 2$~ was that these naked singularities are more `physical' than
the hidden ones, because they are more likely to be actual closed 
string-like mass distribution.  However, in our actual 
computation (3.16) for the identification $~P^{\, 0} = m$, we did not
distinguish them.  This is due to the philosophy that it is the total
mass $~m$~ that should agree with the asymptotic mass $~m$~ \adm, which
of course includes those mass distributions {\it inside} event 
horizons.  Or more intuitively, any mass distribution, even if 
it is hidden inside of event horizon, is `visible' from 
outside the horizon.  To put it differently, we know as a simple fact
that even a point mass inside the event horizon in Schwarzschild metric
definitely attracts a test particle outside of the event horizon.  
From these viewpoints, there is nothing contradictory
in our prescription of identifying all the ring singularities.  It is
also true that the cases of $~\d\ge2$~ \ts\yh\ have more `naked' closed
string-like mass distributions which are much more `real' than those
hidden by event horizons.  

Despite the fact that we may need more supporting evidence based on
alternative regularizations, our first results provide strong evidence
of closed string like mass distribution for axially symmetric exact
solutions.  In other words, the general TS solutions can be
regarded as classical backgrounds around closed strings.  Since the
distortion parameter $~\d$~ corresponds also to the number of the ring 
singularities, {\it i.e.,} the number of rings, it is legitimate to
regard $~\d$~ as nothing other than the number of real closed
strings with $~\d\-$function type mass singularities.  
We believe that the encouraging results in this paper provide an
entirely new directions to be explored both in the context of 
superstring theory and general relativity.    



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