hep-th0107216.tex
%        10        20        30        40        50        60        70  
\count100=1 
% \input defrrr.tex 
\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
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\newif\ifdtup
\def\panorama{\global\dtuptrue \openup2\jot \caja
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        \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{$\sp{#1)}$}
}

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%\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)
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\abovedisplayskip=1em plus.3em minus.5em        %space above equation 
%   (either)
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\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
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%   (LaTeX)
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        \put(-20,11){\tiny University of Maryland Elementary Particle
Physics University of Maryland Elementary Particle Physics % 
University of Maryland Elementary Particle Physics}
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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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\doit0{\vskip -0.1in} 
{\hbox to\hsize{\hfill hep-th/0107216}}\par 
{\hbox to\hsize{\hfill CSULB--PA--01--1}}\par 
\vskip -0.1in 
\hfill 
{(Revised Version)} 
\\ 

\begin{center} 
%\vglue .25in 

%\vskip -0.6in 
\vskip 0.3in 

{\large\bf Teleparallel Complex Gravity as} 
\\[0.026in] 
{\large\bf Foundation for Noncommutative Gravity}
\\[.1in]  

\baselineskip 9pt 

\vskip 0.26in 

\doit0{
Hitoshi ~N{\smallcmr 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{\smallcmr ISHINO}\footnote{E-Mail: 
hnishino@csulb.edu} 
~and~ Subhash ~R{\smallcmr AJPOOT}\footnote{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 2.1in 

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

\baselineskip 14.5pt 

~~~We present a teleparallel complex gravity as the foundation for 
the formulation of noncommutative gravity theory.  The 
negative energy ghosts in the conventional formulation with 
$~U(1,3)$~ local Lorentz connection no longer exists, since the local
Lorentz invariance is broken down to $~U(1,3)$~ global Lorentz
symmetry.  As desired, our teleparallel complex gravity theory also
passes the key classical test of perihelion advance of Mercury.  Based on
this result, we present a lagrangian for  the noncommutative teleparallel
gravity theory.  



\vskip 0.8in

\leftline{\small PACS: ~02.40.H,M, ~02.40.G,K, ~04.20.F, ~11.10.E}
\vskip -0.05in
\leftline{\small Key Words: Non-Commutative Geometry, Complex Gravity, 
Lorentz Invariance,}
\vskip -0.05in
\leftline{\small {\hskip 0.8in} Teleparallelism}

\vfill\eject 

\baselineskip 15pt

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\pageno=1



\leftline{\bf 1.~~Introduction} 

The recent developments of open strings or D-branes lead to the 
consideration of constant background antisymmetric field $~B_{\m\n}$,
which in turn implies that the coordinates of space-time should be
noncommutative 
%%%
\ref\noncomm{A.~Connes, M.R.~Douglas and A.~Schwarz, JHEP {\bf 9802} 
(1998) 003; 
Y.K.E.~Cheung and M.~Krogh, Nucl.~Phys.~{\bf B528} (1998) 185;
C.-S.Chu and P.-M.~Ho, Nucl.~Phys.~{\bf B528} (1999) 151; 
V.~Schomerus, JHEP {\bf 9906} (1999) 030; 
F.~Ardalan, H.~Arfaei and M.M.~Sheikh-Jabbari, JHEP {\bf 9902} (1999) 016;
J.~Hoppe, Phys.~Lett.~{\bf B250} (1990) 44;
D.B.~Fairlie, P.~Fletcher and C.K.~Zachos, Phys.~Lett.~{\bf B218} 
(1989) 203; 
N.~Seiberg and E.~Witten, JHEP {\bf 9909} (1999) 032,
hep-th/9908142.}.    
%%%
It has been well-known that the noncommutative  
generalization \noncomm\ of gravity 
theory necessarily needs a complex metric 
%%%
\ref\chamseddine{A.~Chamseddine, Commun.~Math.~Phys.~{\bf 218} (2001) 283,
\hepth{0005222}; Int.~Jour.~of Mod.~Phys.~{\bf A16} (2001) 759, 
\hepth{0010268}.},   
%%% 
because the introduction of the $~\star~$ product with 
$~i\theta^{\m\n}$~ makes the metric complex.  However, once the metric 
becomes complex, we must consider new components present in the theory, 
such as the antisymmetric component $~B_{\m\n} \equiv \Im\big( 
g\low{\m\n} \big)$.  In particular, the presence of its kinetic term, or 
the antisymmetric part of the vierbein gravitational field equation 
$~\partial S / \partial e\du\m a = 0$~ of the total action 
$~S$~ should be studied in the light of a consistent gravitational 
theory.   

As the first attempt to formulate such a complex gravity as the
preliminary for noncommutative gravity, a lagrangian has been presented
in \chamseddine.  However, the problem in this formulation was that the 
new components $~B_{\m\n}$~ acquire a kinetic term, and some of its 
components become non-physical 
%%%
\ref\ddm{T.~Damour, S.~Deser and J.~McCarthy, 
Phys.~Rev.{\bf D47} (1993) 1541.}.   
%%%
These components are negative energy ghosts, and are not acceptable
at the level of classical field theory.  

The origin of such ghost components can be traced back to the
introduction  of $~U(1,3)$~ local Lorentz symmetry in the system, {\it
i.e.}, the introduction of the Lorentz connection $~\o\du{\m a} b$~ as
its gauge field.  A similar situation has been encountered in
noncommutative non-Abelian gauge field theories.  This is because the
ordinary noncommutative gauge theories require that the gauge groups to
be $~U(n)$.  To avoid this problem, certain formulation that enables the
gauge groups to be other than $~U(n)$, such as $~SO(n)$~ or  $~Sp(n)$,
has been presented 
%%% 
\ref\noncomsympl{L.~Bonora, M.~Schnabl, M.M.~Shaikh-Jabbari and
A.~Tormasiello, Nucl.~Phys.~{\bf B589} (2000) 461, \hepth{0006091}; 
I.~Bars, M.M.~Sheikh-Jabbari and M.A.~Vasiliev,
Phys.~Rev.~{\bf D64} (2001) 086004, \hepth{0103209}.}. 
%%% 
Applying similar techniques to this $~U(1,3)$~ local Lorentz 
symmetry, alternative noncommutative gravity theories were formulated
based on the noncommutative diffeomorphism $~ISO(1,3)$~ group  
%%% 
\ref\chamseddineiso{A.~Chamseddine, Phys.~Lett.~{\bf 504B} (2001) 33.}, 
%%% 
or $~CSO(3,1)$~ group for complex symmetric metric 
%%%
\ref\moffat{J.W.~Moffat, Phys.~Lett.~{\bf B491} (2000)
345, hep-th/0007181.}.  
%%%

In this Letter, we present a different approach as a remedy for the 
negative energy ghosts \ddm, by freezing the $~U(1,3)$~ local Lorentz 
symmetry into to a global one.  In ordinary gravity theory  
with real metrics alone, such a 
formulation is sometimes called `teleparallel gravity formulation'  
%%% 
\ref\hs{K.~Hayashi and T.~Nakano, Progr.~Theor.~Phys.~{\bf 38} (1967) 491;
Y.M.~Cho, Phys.~Rev.~{\bf D14} (1976) 2521, 3335 and 3341; 
F.W.~Hehl, P.~von der Heyde, D.~Kerlick and J.M.~Nester, Rev.~Mod.
Phys.~{\bf 48} (1976) 393; 
K.~Hayashi and T.~Shirafuji, Phys.~Rev.~{\bf D19} (1979) 3524.}
%%% 
in which the $~SO(1,3)$~ local Lorentz symmetry is frozen down to a 
global $~SO(1,3)$~ symmetry, and therefore there is no gauge field or 
spin connection ~$\o\du{\m m} n$~ for the Lorentz symmetry.  Our strategy
for complex gravity is similar, namely, we freeze the $~U(1,3)$~ local
symmetry into a global $~U(1,3)$~ symmetry, requiring
teleparallelism without any introduction of its gauge fields, and thus
avoid the problem of negative energy ghosts.  The importance
of teleparallel gravity in the context of noncommutative geometry
has been pointed out in 
%%% 
\ref\ls{E.~Langmann and R.J.~Szabo, Phys.~Rev.~{\bf D64} 
(2001) 104019, \hepth{0105094}.},      
%%% 
in which teleparallel gravity is shown to arise out of dimensional
reduction of noncommutative gauge theory.  However, we will rely on a
teleparallel gravity theory as the foundation  of complex gravity from
the outset, in order to resolve the problem with the negative energy
ghosts in the $~U(1,3)$~ local Lorentz covariant formulation
\chamseddine.  We stress that teleparallel gravity as the foundation of
noncommutative gravity is the legitimate starting  point, since the
constant
$~\theta^{\m\n}$~ in noncommutative gravity manifestly breaks Lorentz
symmetry.   



% \bigskip\bigskip\bigskip

\newpage


\leftline{\bf 2.~~Teleparallel Complex Gravity}  

Since we are considering the explicit breaking of local Lorentz symmetry, 
it is crucial to understand the degrees of freedom of the vierbein 
components $~e\du\m a$~ and its hermitian conjugates $~e_{\m a} \equiv 
\big( e\du\m a\big)^\dagger$.\footnotew{We follow the notation of  
ref.~\chamseddine\ in this paper, unless otherwise noted.}  In the local 
Lorentz covariant formulation \chamseddine, all together there are
originally $~4\times (4+ 4) = 32$~ components in $~e\du\m a$~ and $~e_{\m
a}$.  However, the $~U(1,3)$~ local Lorentz symmetry with 16
parameters deletes 16 components, leaving only $~16$~
components.  These 16 components are equivalent to the symmetric part
$~G_{\m\n} \equiv g\low{(\m\n)} \equiv \Re\big( g\low{\m\n} \big) $,
and  the antisymmetric part $~B_{\m\n} \equiv -i g\low{\[ \m\n\] } \equiv 
\Im\big( g\low{\m\n} \big)$~ of the metric tensor 
$~g\low{\m\n}$~ \chamseddine.  In our formulation, on the other
hand, the original $~32$~ components are not deleted by the $~U(1,3)$~
local Lorentz  symmetry, and all of them are intact.  This formulation
has the  advantage of deleting the kinetic term for the $~B\-$field.  
The price to be paid is that there are 16 additional components in the
vierbeins whose effect must be carefully investigated.  

We first give preliminaries for the formulation for teleparallel 
complex gravity.  If the $~U(1,3)$~ local Lorentz symmetry is manifest
from the outset, we must introduce the Lorentz connection for
local Lorentz covariance.  This causes the problem of negative energy
ghosts \chamseddine\ which we would like to avoid.  Therefore, it is
natural to consider the formulation in which local Lorentz symmetry is not
built-in, or at least it is not manifest from the outset.

The most fundamental relationships among geometric 
quantities, such as the vierbein and metric are
$$ \li{ & \big( \eta\low a{}^b \big) = \hbox{diag}.~(-,+,+,+)~~,  \cr 
& e\du \m a e\du a \n \equiv \d\du \m \n ~~, ~~~~ 
     e\du a \m e\du \m b \equiv \d\du a b~~, 
     ~~~~ e_{\m a} \equiv \big( e\du \m a \big)^\dagger ~~, ~~~~
      e^{a\m} \equiv \big( e\du a \m \big)^\dagger~~, \cr 
& \big( g\low{\m\n} \big)^\dagger \equiv g\low{\n\m} ~~, ~~~~
      \big(g^{\m\n} \big)^\dagger \equiv g^{\n\m} ~~, 
      ~~~~ e\equiv \det \big( e\du \m a \big) ~~, ~~~~
     \Bar e \equiv e^\dagger \equiv 
     \det \big( e_{\m a} \big) ~~, \cr 
& g\low{\m\n} \equiv e_{\m a} \eta\low b{}^a e\du\n b ~~, ~~~~
      g^{\m\n} \equiv e\du a \m \eta\du b a e^{b \n} ~~, 
      ~~~~ g\low{\m\n} g^{\n\r} = \d\du \m\r ~~, ~~~~
     g^{\m\n} g\low{\n\r} = \d\du\r\m {~~, ~~~~~ ~~}   
&(2.1) \cr } $$
where the symbol $~^\dagger$~ is for hermitian conjugations.  
As usual, the metric $~g\low{\m\n}$~ has both symmetric and antisymmetric
components.  The most basic global
$~U(1,3)$~ transformation rules are 
$$ \li{ & \d_\a e\du\m a = - \a\du b a e\du\m b~~, ~~~~
       \d_\a e\du a\m = + \a\du a b e\du b \m ~~, \cr 
& \d_\a e_{\m a} = + \eta\du a c \a\du c d \eta\du d b e_{\m b} ~~, ~~~~
       \d_\a e^{a\m} = - \eta\du c a \a\du d c \eta\du b d e^{b\m} ~~, 
  \cr 
& \big( \a\du a b \big)^\dagger 
       = - \eta\low b{}^c \a\du c d \eta\du d a ~~, 
&(2.2) \cr } $$
where $~\a\du a b$~ is the space-time independent parameters 
for our global $~U(1,3)$, complying with the notation in
\chamseddine. Accordingly, the metric itself does not transform: $~ \d_\a
g\low{\m\n} = 0 , ~ \d_\a g^{\m\n} = 0$. Relevantly, the $~U(1,3)$~
invariant product is 
$~\big(U_a \big)^\dagger \eta\du a b V_b$, because 
$~\d_\a\big[ \,\big(U_a \big)^\dagger \eta\du a b V_b\,\big] =0$. 
For simplicity, we use the {\it bars} instead of the {\it daggers}
whenever it is not confusing, such as $~\Bar e\equiv e^\dagger$~ in
(2.1).   

There are other important geometrical equations for 
later purposes. One of them is the definition of the covariant
derivative:  
$$\li{ & D_\m V^\n \equiv \partial_\m V^\n 
           + \G\du{\m\r}\n V^\r ~~, 
&(2.3) \cr } $$ 
for a complex vector $~V^\n \equiv V^a e\du a\n$.  Since the vierbein 
$~e\du a \n$~ but {\it not} its hermitian conjugate $~e^{a \n}$~ is used 
here, we use $~D_\m$~ instead of its hermitian conjugate $~\Bar D_\m$.  
The latter is used, when we take the hermitian conjugate of the whole
equation of (2.3):    
$$ \li{ & \Bar D_\m \Bar V{\hskip0.02in}^\n 
    \equiv \partial_\m \Bar V{\hskip0.02in}^\n 
       + \Bar\G\du{\m\r}\n \Bar V^\r ~~,    
&(2.4) \cr } $$ 
for $~\Bar V{}^\n \equiv \Bar V_a e^{a\n}$.  
Relevantly, the commutation relations between the $~D_\m$'s 
and the resulting Bianchi identity are 
$$ \li{ \[ D_\m, D_\n \] = - C\du{\m\n}\r D_\r~~,  ~~~~
& D_{\[\m} C\du{\n\r\]} \s + C\du{\[\m\n|}\t C\du{\t|\r\]}\s 
     \equiv 0 ~~, ~~~~ 
&(2.5\rma)  \cr 
\[ \Bar D_\m, \Bar D_\n \] 
      = - \Bar C\du{\m\n}\r \Bar D_\r~~,  ~~~~
& \Bar D_{\[\m} \Bar C\du{\n\r\]} \s 
      + \Bar C\du{\[\m\n|}\t \Bar C\du{\t|\r\]}\s 
     \equiv 0 ~~, ~~~~     
&(2.5\rmb) \cr } $$  
where $~C\du{\m\n}a$~ are anholonomy coefficients:
$$\li{ & C\du{\m\n} \r \equiv C\du{\m\n} a e\du a\r ~~, ~~~~
         C\du{\m\n} a 
        \equiv \partial_\m e\du\n a - \partial_\n e\du\m a ~~, \cr
& \Bar C\du{\m\n} \r \equiv \big( C\du{\m\n} \r \big)^\dagger 
       = \Bar C_{\m\n\, a} e^{a\r} ~~, ~~~~
        \Bar C_{\m\n\, a}  
       \equiv \big( C\du{\m\n} a \big)^\dagger 
      = \partial_\m e_{\n a} - \partial_\n e_{\m a}  ~~,  \cr 
& \Bar C\ud{\m\n}\r \equiv \big( C\ud{\m\n}\r \big)^\dagger 
     = \big( g^{\s\m} g^{\t\n} g_{\r\l} C\du{\s\t}\l \big)^\dagger 
     = g^{\m\s} g^{\n\t} g_{\l\r} \Bar C\du{\s\t}\l ~~, 
    ~~~~ \hbox{\it etc}. 
&(2.6) \cr } $$ 
The $~D_\m$'s in (2.5a) uses only $~\G\du{\m\n}\r$~ but none of  its
hermitian conjugates $~\Bar\G\du{\m\n}\r$, because $~C\du{\m\n}\r$~ is
composed only of $~e\du\m a$~ and $~e\du a\m$~ but none of their hermitian
conjugates $~e_{\m a}$~ and $~e^{a\m}$, as seen from  (2.6).  If  we had
manifest local Lorentz covariance, there would be an additional term in
(2.5b) proportional to the Lorentz curvature tensor.  This term is now
absent, due to the lack of manifest Lorentz covariance in our
teleparallel gravity.  

Another important equation comes from the vierbein postulate that leads
to the expression of $~\G\du{\m\n}\r$~ in terms of vierbein: 
$$ \li{ & D_\m e\du \n a 
       = \partial_\m e\du \n a - \G\du{\m\n}\r e\du\r a = 0
       ~~~\Longrightarrow ~~~
      \G\du{\m\n}\r = e\du a \r \partial_\m e\du\n a ~~, ~~~~
      \Bar \G\du{\m\n}\r = e^{a \r} \partial_\m e_{\n a} {~~.~~~~~ } 
&(2.7) \cr } $$ 
The reason for using $~D_\m$'s instead of $~\Bar D_\m$~ in the first
equation here has been already stated, and shows how important it is to
distinguish $~\G\du{\m\n}\r$~ from its hermitian conjugate
$~\Bar\G\du{\m\n}\r$.  



\bigskip\bigskip\bigskip 

% \newpage

\vbox{
\leftline{\bf 3.~~Teleparallel Complex Gravity -- Lagrangian and 
Field Equations} 

Once the transformation properties of basic quantities are in place, 
we are ready to present a lagrangian which yields the usual
Einstein-Hilbert  action, with no manifest $~U(1,3)$~ local Lorentz 
covariance.  
}

It is worth noting that we have to be cautious about
the significance of local Lorentz symmetry.  In the
ordinary case with real metric with the familiar $~SO(1,3)$~ local 
Lorentz symmetry, even if we start with quantities 
such as the anholonomy coefficients $~C_{\m\n}{}^a$, and write down the
lagrangian in terms of its quadratic products, the resulting theory may be
still locally Lorentz invariant.  The reason is that the local Lorentz
symmetry is realized as a `hidden' symmetry at the lagrangian level. 
For example, it is well-known \hs\ that a certain combination of
quadratic products of $~C_{\m\n}{}^a \equiv \partial_\m e\du\n a - 
\partial_\n e\du\m a $~ yields an action identically equal to a
Einstein-Hilbert action  up to a total divergence: 
$$ \li{ S_{\rm E H} \equiv \, & \int d^4 x\, e  
     \Big[ - \frac 18 C_{\m\n a} C^{\m\n a} 
		+ \frac 1 4 C_{\m\n\r} C^{\n\r\m} 
		+ \frac 12 \big( C_{\m \n}{}^\n \big)^2 \, \Big] 
  \equiv \int d^4 x \, 
		\Big( - \frac 14 e R \, \Big) {~~. ~~~~~}  
&(3.1) \cr } $$ 
Here the vierbeins are the usual real one $~\big( e\du\m a
\big)^\dagger = e\du\m a$, before considering any complex gravity, and
$~R$~ is the usual scalar curvature in terms
of the Riemann-Christoffel connection: $~{\scst\left\{  {
{\r}\atop{\m\n}} \right\}}$.  In other words, even though each term in 
(3.1) is not locally Lorentz invariant, the combination of the 
quadratic products of $~C_{\m\n}{}^a$~ with the appropriate relative
coefficients makes the whole expression locally Lorentz invariant.  

We now consider complex gravity.  Our construction heavily relies 
on the usage of the anholonomy coefficients and its hermitian conjugates
defined in (2.6).   Now our candidate lagrangian is the analog of (3.1), 
and is given by  
$$ \li{ |e| ^{-1} \Lag_0 \equiv &  a_1 g^{\m\r} g^{\n\s} 
      \eta\du a b C\du{\m\n} a \Bar C_{\r\s b}  
      + a_2 g^{\m\s} e^{b \n} e\du a \r C\du{\m\n} a 
     \Bar C_{\r\s b} 
      + a_3 g^{\m\r} e^{b\s} e\du a \n C\du{\m\n} a \Bar C_{\r\s b} 
      {~~~~~~~} 
&(3.2\rma) \cr 
=  & a_1 g^{\m\n} g^{\r\s} g\low{\l\o} C\du{\m\r}\o 
     \Bar C\du{\n\s}\l 
       + a _2 g^{\m\s} C\du{\m\n} \r \Bar C\du{\r\s} \n  
      + a_3 g^{\m\n} C_\m \Bar C_\n      
&(3.2\rmb) \cr 
= & a_1 C\ud{\n\s}\l \Bar C{}\du{\n\s}\l 
     + a_2 C\udu\s\n\r \Bar C{}\du{\r\s} \n 
    + a_3 C^\n \Bar C_\n ~~. 
&(3.2\rmc) \cr } $$
Here $~|e|^2 \equiv e \, \Bar e$, while $~a_1,~a_2$~ and
$~a_3$~ are real constants.  In particular, the case of 
$$ \li{ & a_3 = + 2 a_2 = - 4 a_1 \equiv - 4 a = + \frac 1{32} ~~ 
&(3.3) \cr } $$
is the direct analog of (3.1) in the conventional teleparallel gravity 
with the real vierbein \hs.  The ~$C_\m$~ and
$~\Bar C_\m$~ in (3.2b) are defined by 
$$ \li{ & C_\m \equiv C\du{\m\n}\n ~~, ~~~~
        \Bar C_\m \equiv \big( C\du{\m\n}\n\big)^\dagger 
     = \Bar C\du{\m\n}\n {~~, ~~~~~ ~~} 
&(3.4) \cr } $$
while $~C\udu\s\n\r$~ and $~C^\n$~ in (3.2c) are 
defined by 
$$ \li{ & C\udu\s\n\r \equiv g^{\t\s} C\du{\t\n}\r ~~, ~~~~
    C^\n \equiv g^{\r\n} C_\r ~~.   
&(3.5) \cr } $$ 
In terms of these anholonomy coefficients, each term in (3.2c) is
manifestly invariant under the global Lorentz transformation (2.2), 
because   
$$ \li{ & \d_\a C\du{\m\n}\r = 0 ~~, ~~~~
     \d_\a \Bar C\du{\m\n}\r = 0~~. 
&(3.6) \cr } $$ 
This also explains the reason why we need $~\eta\du a b$~ 
in the first term in (3.2a), while it is not needed in the second term. 
To put it differently, in terms of the anholonomy coefficients  with
curved indices, it is more straightforward to construct globally Lorentz
invariant terms for a lagrangian like (3.2c).  
For readers' convenience, we give the proof of the hermiticity of the 
second term in (3.2c):    
$$ \li{ \big( C\udu\s\n\r \Bar C\du{\r\s} \n\big)^\dagger
     = & \big( g^{\t\s} C\du{\t\n}\r \big)^\dagger\,  
        C\du{\r\s}\n 
     = g^{\s\t} \Bar C\du{\t\n}\r \, C\du{\r\s} \n \cr 
= & - g^{\s\t} C\du{\s\r} \n \Bar C\du{\t\n}\r  
     = + C\udu\t\r\n  \, \Bar C\du{\n\t}\r 
     = C\udu\s\n\r \Bar C\du{\r\s} \n ~~, 
&(3.7) \cr } $$ 
despite its `non-symmetric' appearance.  

For explicitness, we give our action after the use of (3.3):
$$ \li{ S_0 \equiv & \int d^4 x\, e  
     \Big[ - \frac18  C\ud{\n\s}\l \Bar C{}\du{\n\s}\l 
     + \frac 14 C\udu\s\n\r \Bar C{}\du{\r\s} \n 
     + \frac 12 C^\n \Bar C_\n \, \Big]  
  \equiv \int d^4 x \,\Lag_0 {~~. ~~~~~}  
&(3.8) \cr } $$
Some readers may wonder how the same relative coefficients as in (3.1)
can yield a theory now without the kinetic term for
$~B_{\m\n}$.  This is because the identity in (3.1) is
valid only for a real metric, but we now have the complex metric.  As
will be seen, our proposed lagrangian (3.8) has no kinetic term
for $~B_{\m\n}$~ at the quadratic order, so it is distinct from the
lagrangian in \chamseddine.  

The gravitational field equation from $~\Lag_0$~ for the vierbein
$~e\du\m a$~ is 
$$ \li{ F^{\m\n} = 
& - 2 a g^{\r\m} D_\s \Bar C{\hskip0.02in}\ud{\s\n}\r
      + 2 a g^{\r\m} D_\s \Bar C{\hskip0.02in}\du\r{\s\n}  
      - 2 a g^{\r\m} D_\s \Bar C{\hskip0.02in}\du\r{\n\s}
     + 4 a g^{\n\m} D_\r \Bar C{\hskip0.02in}^\r 
     - 4 a g^{\r\m} D_\r \Bar C{\hskip0.02in}^\n \cr 
& - 2 a \big( C_\r - i I_\r \big) \Bar C{\hskip0.02in}^{\r\n\m}  
      - 4 a g^{\s\m} \big( C_\r - i I_\r \big) 
      \Bar C{\hskip0.02in}\du\s{\[ \n \r \]} \cr   
& + 4 a g^{\n\m} \big( C_\r - i I_\r
     \big) \Bar C{\hskip0.02in}^\r 
     - 4 a \big( C^\m - i I^\m \big) \Bar C{\hskip0.02in}^\n \cr  
& -2 a C\ud{\m\r}\s \Bar C{\hskip0.02in}\udu\n\r\s 
     + a C^{\r\s\n} \Bar C{\hskip0.02in}\du{\r\s}\m 
					+ \frac12 a g^{\n\m}C\ud{\r\s}\t \Bar C{\hskip0.02in}\du{\r\s}\t \cr 
& - 2 a C\udu\m\r\s \Bar C{\hskip0.02in}\udu\n\s\r 
      + a g^{\n\m} C\udu\r\s\t \Bar C{\hskip0.02in}\du{\r\t}\s 
      + 4 a C^\m \Bar C{\hskip0.02in}^\n - 2 a g^{\n\m} C^\r 
     \Bar C{\hskip0.02in}_\r \eqdot 0  
     {~~, ~~~~~ ~~~~~} 
&(3.9) \cr } $$
where  
$$ \li{ & F^{\m\n} \equiv g^{\r\m} F\du\r\n 
      \equiv g^{\r\m} e\du \r a F\du a \n ~~, ~~~~
       \calF \du a \m \equiv |e|\, F\du a \m 
       \equiv \fracm{\d\Lag_0}{\d e\du\m a} ~~.    
&(3.10) \cr } $$ 
Here we have already used the condition (3.3) up to $~a \equiv -1/8$.  
As usual in complex field theories, the Euler derivatives with respect to 
$~e\du\m a$~ and its hermitian conjugate $~e_{\m a}$~ are treated as 
independent.   The symbol $~\eqdot$~ denotes a field equation,
distinguished from  an identity.  The
symbol $~I_\m$~ represents the imaginary part of $~\G_\m \equiv
\G\du{\m\n}\n$: $I\du{\m\n}\r \equiv \Im \big( \G\du{\m\n}\r
\big),~ I_\m \equiv I\du{\m\n}\n$.  

Our field equation (3.9) can be confirmed based on the general invariance
of our action (3.8) under 
$$ \li{ & \d_\xi e\du\m a = \xi^\n \partial_\n e\du \m a 
    + \big( \partial_\m\xi^\n \big) \, e\du\n a ~~, ~~~~
      \d_\xi e\du a \m = \xi^\n \partial_\n e\du a \m  
    - \big( \partial_\n\xi^\m \big) \, e\du a\n ~~, 
&(3.11) \cr } $$ 
leading to the Noether identity\footnotew{As is well-known 
in conventional gravity theory 
%%%
\ref\mtw{{\it See, e.g.,} C.W.~Misner, K.S.~Thorne and J.A.~Wheeler, {\it
`Gravitation'}~ (W.H.~Freeman and Company, NY, 1972); 
S.~Weinberg, {\it `Gravitation and Cosmology'} (John Wiley \& Sons, 
NY, 1971); L.D.~Landau and E.M.~Lifshitz, {\it `The Classical Theory of
Fields'}, 4th English ed., Pergamon Press (1975).}            
%%% 
this identity is equivalent to a
combination of Bianchi identity (2.5).} 
$$ \li{ & \Big[ \, D_\m F\du\n\m 
     - \big( C_\m - i I_\m \big) F\du\n\m 
     - C\du{\n\r}\t F\du\t\r \, \Big] + \hbox{h.c.} \equiv 0 ~~, 
&(3.12) \cr } $$ 
with the {\it real} parameter $~\xi^\m$.   Here `+ h.c.' implies the
addition of the hermitian conjugate of the preceding brackets.  The
covariant derivative $~D_\m$~ in (3.12) contains only 
$~\G\du{\m\n}\r$, while its hermitian conjugate $~\Bar D_\m$~  contains
$~\Bar\G\du{\m\n}\r$.  Note that eq.~(3.12) is an {\it identity} based on
the general invariance of our action, but {\it not} field equation.  

Let us now consider the degrees of freedom of our field variables.  There
are originally $~32$~ degrees of freedom for $~e\du\m a$~ and $~e_{\m
a}$, and $~32$~ independent components in $~{\cal F}^{\m\n}$.    
However, eventually we have $~10$~ for $~G_{\m\n}$~ and $~6$~ for
$~B_{\m\n}$~ as the physical components.  The gap between 32 and 16 should
be understood as the redundancy of 16 components in $~{\cal F}^{\m\n}$, by
the use of 
$$ \li{ & \fracm{\d e\du\m a} {\d G_{\r\s}}       
     = + \frac 12 \d\du\m{(\r|} e^{b | \s)} \eta\du b a~~, ~~~~
       \fracm{\d e\du a\m} {\d G_{\r\s}} 
       = + \frac 12 e\du a{(\r|} g^{\m|\s)} ~~, \cr 
& \fracm{\d e\du\m a} {\d B_{\r\s}}       
     = - \frac i 2  \d\du\m{\[ \r|} e^{b | \s \]} \eta\du b a~~, ~~~~
       \fracm{\d e\du a\m} {\d B_{\r\s}} 
       = + \frac i 2 e\du a{\[ \r|} g^{\m|\s\] } ~~, 
&(3.13) \cr } $$ 
leading to   
$$ \li{ & \fracm{\d\Lag_0}{\d G_{\m\n}} 
     \equiv + \Re \big({\cal F}^{(\m\n)} \big) ~~, ~~~~
    \fracm{\d\Lag_0}{\d B_{\m\n}} 
     \equiv - \Im \big({\cal F}^{\[\m\n\]} \big) ~~.  
&(3.14) \cr } $$ 
In other words, the components $~\Re \big( {\cal F}^{\[\m\n\]} \big)
\eqdot 0$~ and $~\Im \big( {\cal F}^{(\m\n)} \big) \eqdot 0$~ yield   
extra constraints on the components in the
vierbeins $~e\du\m a$~ and $~e_{\m a}$~ different from the directions 
of $~G_{\m\n}$~ and $~B_{\m\n}$.      

We next analyze our lagrangian (3.8) in terms of 
linearized gravity: 
$$\li{  & e\du\m a = \big( \eta\du\m a + h\du\m a + a\du\m a \big) 
      + i \big( B\du\m a + b\du\m a \big)  \equiv \eta\du\m a 
      + H\du\m a ~~, \cr 
& e_{\m a} = \big( \d_{\m a} + h_{\m a} + a_{\m a} \big) 
      - i \big( B_{\m a} + b_{\m a} \big)  \equiv \d_{\m a} 
      + \Bar H_{\m a} ~~, \cr
& g\low{\m\n} = \d_{\m\n} + h_{\m\n} + i B_{\m\n} ~~, ~~~~
      g^{\m\n} = \d^{\m\n} - h^{\m\n} - i B^{\m\n} 
     + {\cal O} (\varphi^2) ~~,  \cr  
& h_{\m a} = + h_{a \m} ~~, ~~~~ a_{\m a} = - a_{a \m} ~~, ~~~~
     B_{\m a} = - B_{a \m} ~~, ~~~~ b_{\m a} = + b_{a \m} ~~.   
&(3.15) \cr } $$ 
The fields $~h_{\m\n}, ~a_{\m\n},~B_{\m a}, ~b_{\m a}$~
are real, and $~{\cal O}(\varphi^2)$~ denotes any quadratic terms
in fields.  The $~\d_{\m\n}$~ or 
$~\d^{\m\n}$~ are the usual Minkowsky metric with the signature 
$~(-,+,+,+)$, avoiding $~\eta^{\m\n}$~ which
is confusing with $~\eta\du a b$.  The quadratic-order terms in (3.8) are 
now computed as 
$$ \li{ \Lag_0 \, |_{\rm quadratic} 
= & + \big(\! + 2a_1 -a_2 \big) \big(\partial_\m h_{\r\s} \big)^2  
     + \big( \! - 2a_1 + a_2 + a_3 \big) 
        \big( \partial_\m h^{\m\n} \big)^2  \cr 
& ~~~ - 2a_3 \big(\partial_\m h \du\r\r\big)
      \big( \partial_\n h^{\m\n} \big) 
     + a_3 \big(\partial_\m h\du \r\r \big) ^2 \cr  
& + \big( \! + 4 a_1 -2a_2 + 2a_3 \big) \, 
      \big[ \, \big(\partial_\r h\du\m\r \big)  
    \big( \partial_\s a^{\m\s} \big) 
     + \big( \partial_\r B\du\m\r \big) 
     \big( \partial_\s b^{\m\s} \big) \, \big]  \cr 
& + \big( \! + 2a_1 + a_2 \big) \big( \partial_\m a_{\r\s} 
     + \partial_\m B_{\r\s} \big)^2 
     + \big( \! - 2a_1 - 3a_2 + a_3 \big) 
     \big(\partial_\n a^{\m\n} +  \partial_\n B^{\m\n} \big)^2 \cr
& + \big( \! +2a_1 - a_2 \big) \big( \partial_\m b_{\r\s} \big)^2 
      + \big( \! -2a_1 +a_2 + a_3 \big) 
      \big(\partial_\n b^{\m\n} \big)^2  \cr     
& ~~~ - 2a_3 \big( \partial_\m b\du\r\r \big) 
      \big( \partial_\n b^{\m\n} \big) 
     + a_3 \big( \partial_\m b\du\r\r \big)^2 ~~.     
&(3.16) \cr } $$ 
Since all the fields are real, we can use the usual  Minkowsky
metric $~\d_{\m\n}$~ or $~\d^{\m\n}$~ for contractions.  
The desirable aspects of our lagrangian are summarized as:  (i) The
kinetic terms for the antisymmetric fields $~a_{\m\n}$~ and $~B_{\m\n}$,
as well as all the $~h$-$a$~ and $~B$-$b$~ mixture terms disappear upon
the condition (3.3);  (ii) There is no mixture terms between  the two
sets  
$~\big( h_{\m\n},~a_{\m\n}\big) $~ and $~\big( B_{\m\n},~ b_{\m\n}\big)$,
such as $~a B\-$terms, due to the hermiticity of the
lagrangian;  (iii)  The kinetic terms for 
$~h_{\m\n}$~ coincide exactly with the quadratic terms of the
Einstein-Hilbert action $~S_{\rm E H}$;  (iv) At the quadratic level,
there is no negative energy ghost, as $~S_{\rm E H}$~ does
not;  (v)  Interestingly, the field 
$~b_{\m\n}$~ also acquires its kinetic term, exactly with the same
coefficients as that of $~h_{\m\n}$, so that there are two sorts of spin
2 fields.  We will come back to this point shortly.   

We next analyze field equation (3.9) covariant to all orders.  To this
end, we need an additional constraint 
$$ \li{& \Im \big( \G\du{\m\n}\r\big) \eqdot 0~~.  
&(3.17) \cr } $$ 
Note that (3.17) is weaker than a direct condition $~\Im 
(g_{\m\n}) = 0$.  However, at least by perturbation, we can show
that the field equation (3.9) is satisfied with no
bad effect on the propagation of the physical graviton
$~h_{\m\n}$.  

We first note that the constraint (3.17)
is covariant under general coordinate transformations (3.11): 
$$ \li{ & \d_\xi \G\du{\m\n}\r = \xi^\s\partial_\s\G\du{\m\n}\r 
      + \big(\partial_\m \xi^\s \big) \G\du{\s\n}\r 
      + \big(\partial_\n \xi^\s \big) \G\du{\m\s}\r
      - \big(\partial_\s \xi^\r \big) \G\du{\m\n}\s 
      + \partial_\m\partial_\n \xi^\r ~~.    
&(3.18) \cr } $$ 
With its hermitian conjugate subtracted, the last purely
real term $~\partial_\m\partial_\n\xi^\r$~ vanishes in the combination of
$~\d_\xi \big[ \Im \big(\G\du{\m\n}\r \big)\,\big] $,
establishing its covariance.  Now consider (3.17) in terms of the
linearized gravity (3.15):   
$$\li{ & \Im\big(\G\du{\m\n}\r\big) 
    = \big( \partial_\m B\du \n\r + \partial_\m b\du \n\r \big)  
    + {\cal O}(\varphi^2) = 0  ~~.   
&(3.19) \cr } $$ 
This condition is equivalent to two conditions 
$$\li{ & \Im\big(\G_{\m\[\n\r\]} \big) 
    = \partial_\m B_{\n\r} + {\cal O}(\varphi^2) = 0 ~~, ~~~~
    \Im\big(\G_{\m (\n\r )} \big) 
    = \partial_\m b_{\n\r} + {\cal O}(\varphi^2) = 0 ~~.  
&(3.20) \cr } $$ 
In other words, the components $~B_{\m\n}$~ and $~b_{\m\n}$~ are frozen
with no space-time dependence.  In fact, as long as Lorentz covariance is
respected\footnotew{Lorentz covariance is respected, if we exclude
solutions such as $~B_{\m\n} = \hbox{const.}$~ or $~b_{\m\n} =
\hbox{const}$.   For example, the particular case $~B_{\m\n} =
\hbox{const.}$~ in noncommutative gravity is not included in this context
for an obvious reason.}, the only solutions are
$~B_{\m\n} = {\cal O}(\varphi^2),~ b_{\m\n} = {\cal O}(\varphi^2)$, so 
that $\Im \big( e\du\m a\big) = {\cal O}(\varphi^2), ~\Im \big( C\du{\m\n}
a \big) = {\cal O}(\varphi^2)$.  Therefore,  we can conclude that at least
perturbatively under the  constraint
$~\Im\big(\G\du{\m\n}\r\big) =0$, all the linear  terms in (3.9) are real
up to cubic order terms: 
$~\Im \big({\cal F}^{\m\n}\big) = {\cal O}(\varphi^3)$, and hence our
field eq.~(3.9) is equivalent to the general relativity up to  the cubic
terms, with no interference of extra components with the physical
components $~h_{\m\n}$.  This nice feature is the result of our constraint 
(3.17), restricting the possible solutions for extra fields always one
order higher: $~B_{\m\n} = {\cal O}(\varphi^2),~ b_{\m\n} = {\cal
O}(\varphi^2)$.  To put it differently, these extra fields do no harm on
the propagation of the physical $~h_{\m\n}\-$field, up to cubic-order
terms in the field equation, and quartic-order terms in the lagrangian.  

Analyzing the covariant field equation (3.9), we have seen that the
additional spin 2 component 
$~b_{\m\n}$~ can be eliminated\footnotew{This additional spin 2 field 
may well have consistent interactions in our lagrangian (3.8), but we
take rather a conservative viewpoint in this paper, leaving such a
possibility for future studies.} by the constraint $~\Im 
\big(\G\du{\m\n}\r\big) \eqdot 0$~ imposed `by hand', perturbatively up to
the quartic terms in the lagrangian.  Our next question is how to
automatically implement such a constraint at the lagrangian level.  To
this end, we rely on the method in 
%%%  
\ref\siegel{W.~Siegel, \np{238}{84}{307}.},   
%%% 
using some multiplier field in a quadratic constraint
lagrangian:   
$$ \li{ &\Lag_{\rm \L} \equiv \frac 12 \, |e| \, \L\ud{\m\n\s\t}{\r\l}
    \, \Im \big( \G\du{\m\n}\r \big) \, 
     \Im \big( \G\du{\s\t}\l \big) ~~. 
&(3.21) \cr } $$ 
The advantage of such constraint lagrangian \siegel\ is not to affect the
field equations of other non-constrained  fields, due to the quadratic
nature of this term.\footnotew{Even though there might be some subtlety
with this constraint lagrangian at quantum level with  path-integral, we
do not go that far in this paper.}  In fact, the field
equation for $~\L\ud{\m\n\s\t}{\r\l}$~ yields 
$$ \li{& \Im\big(\G\du{\m\n}\r\big) \, 
     \Im\big(\G\du{\s\t}\l\big) \eqdot 0~
&(3.22) \cr } $$ 
for an arbitrary indices $~\du{\m\n\s\t}{\r\l}$, if there is no
index-symmetry for $~\L\ud{\m\n\s\t}{\r\l}$.  We can
prove that (3.22) implies the constraint 
$~\Im\big(\G\du{\m\n}\r\big) \eqdot 0$~ by `reduction to absurdity'.  
Since this proof is straightforward, we will skip it in this
Letter.   

After such a treatment, $~h\low{\m\n}$~ in $~G_{\m\n} =
\d\low{\m\n} + h\low{\m\n} = \Re\big( g_{\m\n} \big) = g\low{(\m\n)}$~ is
the only physical field with its kinetic terms in our formulation.  In
particular, the additional spin 2 component $~b_{\m\n}$~ has been deleted
by our constraint lagrangian (3.21).  Recall that this is due to the
analysis from (3.19) and (3.20) leading to $~B_{\m\n} = {\cal
O}(\varphi^2), ~b_{\m\n} = {\cal O}(\varphi^2)$.   

We have another justification of our theory, based on 
the degrees of freedom.  The extra components $~a_{\m\n}$, 
$~B_{\m\n}$~ and $~b_{\m\n}$~ have respectively $~6,~6$~ and
$~10$~ degrees of freedom.  The field equations for
$~a_{\m\n},~ B_{\m\n}$~ and $~b_{\m\n}$~ are\footnotew{In `real'
teleparallel gravity with the lagrangian (3.1), the equation
$~\calF^{\[\m\n\]} \eqdot 0$~ is identically zero, due to the {\it
hidden} local Lorentz invariance of the action (3.1).  However, our
action (3.8) has {\it no} such hidden local Lorentz invariance, as can be
easily confirmed by the
$~U(1,3)$~ local Lorentz analog of (3.6) with an additional derivative
term $~\partial_\m\a\du a b$.  Therefore, the second field equation in
(3.23), which is equivalent to the $~B_{\m\n}$-field equation {\it via}
(3.6), is {\it not} identically zero.}  
$$\li{ & \Re\big( \calF^{\[ \m \n\]} \big) \eqdot 0~~, ~~~~ 
       \Im\big(\calF^{\[\m\n\]} \big) \eqdot 0~~, ~~~~ 
      \Im \big( \calF^{(\m\n)}\big) \eqdot 0~~. 
&(3.23) \cr } $$  
These independent components of the whole field equation 
$~\calF^{\m\n} \eqdot 0$~ give respectively 
$~6 + 6 + 10$~ equations to these unphysical auxiliary fields.  On the
other hand, we have seen from (3.20) that the constraint $~\Im
\big( \G\du{\m\n}\r\big) \eqdot 0$~ is equivalent to freezing $~6 + 10 $~
components $~B_{\m\n}$~ and $~b_{\m\n}$.   Even though the condition 
$~\Im\big(\G\du{\m\n}\r \big) = 0$~ is weaker than $~\Im
\big( g_{\m\n} \big) = 0$~ due to possible cross-terms in 
$~\G\du{\m\n}\r$, we have seen in (3.20) perturbatively that the
extra components $~B_{\m\n},~b_{\m\n}$~ do no harm on the propagation 
of $~h_{\m\n}$~ in the field equation $~\Re\big( \calF^{( \m \n)} \big)
\eqdot 0$.   As for the possible bad effect by the remaining extra
component $~a_{\m\n}$, this has been already clarified based on the past
teleparallel formulations \hs.  Namely, once 
$~B_{\m\n}$~ and $~b_{\m\n}$~ become irrelevant, $~\Re\big(
\calF^{\[ \m \n\]} \big) \eqdot 0$~ is automatically satisfied by 
Bianchi identities \hs.   
Therefore we conclude that there is perturbatively no undesirable effect 
on the propagation of $~h_{\m\n}$~ by the extra components
$~B_{\m\n}$~ and $~b_{\m\n}$~ or $~a_{\m\n}$.  

Before summarizing this section, we analyze our field
equation (3.9) for two important cases, when some of the extra components
are absent:  

\noindent (i)  When the vierbein is real, the 
anholonomy coefficients becomes pure real $~C\du{\m\n}\r = \Bar
C\du{\m\n}\r$, and $~I_\m =0$, while the covariant derivative
$~D_\m$~ coincides with its hermitian conjugate $~\Bar D_\m$, and in
particular, $~D_\m g\low{\n\r} \equiv 0$:     
$$\li{ |e|^{-1} {\cal F}^{\m\n} 
     \Big|_{\hbox{\gothsmall Im} (g\low{\r\s})\, = \,0} 
      = & - 4a D_\r C^{\r\[ \m\n \]} + 4 a g^{\m\n} D_\r C^\r 
      - 4 a D^\m C^\n - 2 a D_\r C^{\m\n\r} \cr 
& - 4 a C_\r C^{\r(\m\n)} - 2 a C\ud{\m\r}\s C\udu\n\r\s 
     + a C\du{\r\s}\m C^{\r\s\n} \cr  
& - 2 a C\udu\m\r\s C\udu\n\s\r - 2 a C_\r C^{\m\n\r} 
     + 2 a g^{\m\n}  \big( C_\r\big)^2 
     + \frac 12 a g^{\m\n} \big( C_{\r\s\t}\big)^2  {~~.~~~~~ ~~~}  
&(3.24) \cr } $$ 
The symmetric terms $~{\cal F}^{(\m\n)} $~ can be shown to
be equivalent to the usual Einstein tensor in general relativity
in terms of anholonomy coefficients \hs.  On the
other hand, all the antisymmetric terms $~{\cal F}^{\[\m\n\]}$~ cancel
each other upon the Bianchi identity (2.5).  We can also mimic this
procedure at the lagrangian level (3.8), reducing (3.8) into the
Einstein-Hilbert action (3.1).  Note that we have {\it not} imposed the
absence of the antisymmetric part 
$~a_{\m\n}$, but it is decoupled from the whole field equation.  
There is nothing unusual about this, because upon the 
constraint $~\Im \big(e\du\m a \big) =0$, our system is 
reduced to the teleparallel `re-formulation' of general relativity 
with no kinetic term for $~a_{\m\n}$~ as in \hs, and thus it
is equivalent to the ordinary general relativity.  

\noindent (ii)  When the vierbein and metric are complex
constant, then $~\G\du{\m\n}\r\equiv 0$~ holds due to the 
postulate (2.7), leading to $~C\du{\m\n}\r\equiv 0$.  Accordingly,  
the covariant derivatives $~D_\m = \Bar D_\m$~ become the ordinary
derivative $~\partial_\m$.  Therefore, each term in (3.9) vanishes
identically, confirming that such a metric is a trivial 
solution to our field equation.  This aspect is important, since
we can now have the constant but complex metric $~g^{\m\n} 
= \d^{\m\n} + i\theta^{\m\n}$~ used in noncommutative gravity
\noncomm.  

To summarize, we have obtained the following important results:  

\Item{(1)} The analysis of the quadratic terms in our lagrangian 
(3.8) shows no negative energy ghosts.  
In terms of linearized fields, $~h_{\m\n}$~ is the ordinary graviton,
while $~a_{\m\n}$~ and $~B_{\m\n}$~ are auxiliary
with no kinetic terms.  In particular, the $~B_{\m\n}\-$field has 
no negative energy ghost, in contrast
to \chamseddine.  There is an additional spin 2 field $~b_{\m\n}$~ 
with physical kinetic terms in (3.16), which can be further deleted by a
constraint  lagrangian (3.21), as in items (4) and (5) below.    

\Item{(2)}  The field equations $~\Re \big( \calF^{\[\m\n\]}
\big) \eqdot 0,~\Im \big( \calF^{\[\m\n\]} \big) \eqdot 0$~ and 
$~ \Im \big( \calF^{ (\m\n) } \big) \eqdot 0$~ provide 
$~6 + 6 + 10 = 22$~ equations.  These are $~6+6$~ constraints for
the extra non-propagating components
$~a_{\m\n},~B_{\m\n}$~ and $~10$~ field equations for spin 2 field
$~b_{\m\n}$.   Moreover, the constraint $~\Im\big(\G\du{\m\n}\r \big)=0$~
can freeze $~B_{\m\n}$~ and $~b_{\m\n}$, as in item (4) below.  This
constraint is also covariant under general coordinate transformation.  

\Item{(3)} The constraint $~\Im\big(\G\du{\m\n}\r \big)=0$~ is
automatically realized, by adding the constraint lagrangian $~\Lag_\L$~
(3.21), as in \siegel.   

\Item{(4)} Under the constraint $~\Im\big(\G\du{\m\n}\r \big)\eqdot 0$, 
the only perturbative solutions are $~B_{\m\n} ={\calO}(\varphi^2) $~ and
$~b_{\m\n}={\calO}(\varphi^2)$.  Hence all the worrisome quadratic terms
with $~B_{\m\n},~b_{\m\n}$~ in the field equation $~\Re(\calF^{(\m\n)})
\eqdot 0$~ of $~h_{\m\n}$~ become actually cubic order (one-order
higher).  Therefore, at least perturbatively, they will do no harm on
$~h_{\m\n}$~ at the quadratic order at the field equation level, and
the cubic order at the lagrangian level.       

\Item{(5)} Our covariant field equation (3.9) is reduced to the standard 
Einstein equation in general relativity, when the 
vierbeins becomes real.   In other words, any real vierbein
solution to the Einstein equation automatically
satisfies our field equation as a special case.  This corresponds to the
symmetric real part $~\Re\big({\cal F}^{(\m\n)} \big)\eqdot 0$.  

\Item{(6)}  The constant but generally complex metric, breaking the
ordinary Lorentz covariance, is a simple solution to our field equation
(3.9).  This includes the important case of $~g^{\m\n} = \d^{\m\n} + i
\theta^{\m\n}$.   

\bigskip

Before closing this section, we mention one possibility of generalizing 
our lagrangian (3.8).  Based on teleparallel  real gravity in \hs, we can
add the kinetic term 
$$ \li{ & \Lag_4 \equiv a_4 |e |^{-1} \, g\low{\t\m}  
      \e^{\m\n\r\s} \e^{\t\l\o\psi} C_{\n\r\s} 
      \Bar C_{\l\o\psi} ~~, ~~~~       
&(3.25) \cr } $$ 
so that the component $~a_{\m\n}$~ is now propagating with its kinetic 
term like $~\big(\partial_{\[\m} a_{\n\r\]} \big)^2$~ with no negative
energy states, with an appropriate sign for $~a_4$.   
As long as we have {\it no} term like (3.25), all the terms in our
lagrangian (3.8), apart from those produced by the imaginary part of
$~e\du\m a$, coincide with the Einstein-Hilbert lagrangian, even if there
are antisymmetric components $~a_{\m\n}$~ in  $~e\du\m a$~ 
\hs.  This is due to a `hidden' local Lorentz invariance of our action 
when $~\Im \big(e\du\m a\big) =0$.  This feature has been
well-known in teleparallelism formulation \hs.    


 
\bigskip\bigskip\bigskip

% \newpage



\leftline{\bf 4.~~Key Classical Test of Perihelion Advance}  

We have seen that our teleparallel complex gravity has no negative 
energy ghosts based on the quadratic order analysis.  Moreover, 
our system reproduces general relativity at the cubic  
order in the lagrangian with no bad effect by the extra components.   
However, some readers may be wondering if our system passes the important
key classical tests with more higher-order interactions, like the
perihelion advance of Mercury.  

To study this, we start by postulating an action for a point mass:    
$$ \li{ & I_{\rm M} = \int d s \, \Big(  
     G_{\m\n} \fracm{d x^\m} {d s} \fracm{d x^\n} {d s} \Big)^{1/2} 
    ~~. 
&(4.1) \cr } $$ 
Here we have put only the real part of our metric $~G_{\m\n} \equiv \Re
\big( g_{\m\n}\big) = G_{\n\m}$.  Even though it is natural to have only
the real part $~G_{\m\n}$~ in $~I_{\rm M}$, since we do not have the
total metric $~g_{\m\n}$~ here, the general coordinate invariance $~\d_\xi
I_{\rm M} =0$~ under (3.11) is non-trivial.  However, this can be easily 
confirmed, once we realize from (3.11) that  
$$\li{& \d_\xi G_{\m\n} = \xi^\r \partial_\r G_{\m\n} 
     + \big( \partial_\m \xi^\r \big) G_{\r\n} 
     + \big( \partial_\n \xi^\r \big) G_{\m\r} ~~.   
&(4.2) \cr } $$ 
In other words, the lack of the imaginary part
$~\Im\big(g_{\m\n}\big)$~ does not upset the desirable 
transformation property of $~G_{\m\n}$.  Due to this
property (4.2), it is now clear that our action
$~I_{\rm M}$~ has the general coordinate invariance $~\d_\xi I_{\rm
M}=0$.  
        
Once we have put only the real part $~G_{\m\n}$~ into $~I_{\rm M}$, the
usual geodesic equation of motion for a point mass
follows, exactly as in general relativity:  
$$ \li{ & \fracm{d^2 x^\m} {d s^2} 
     + \Christoffel\m\r\s \fracm{d x^\r}{d s} \fracm{d x^\s}{d s} 
     \eqdot 0 ~~, 
&(4.3) \cr } $$
with $~\Christoffel\m\r\s \equiv 
(1/2) G^{\m\t} \big( \partial_\r G_{\s\t} + 
\partial_\s G_{\r\t} - \partial_\t G_{\r\s} \big)$~ involving only 
$~G_{\m\n}$~ and its inverse $~G^{\m\n}$.  

Our next question is whether the conventional Schwarzschild metric 
$$ \li{ & d s^2 = - \Big( 1- \fracm{2m} r \Big) \,  
    + \fracmm1{ 1- \fracm{2m} r} \, d r^2 
    +  r^2 \, \big( d \theta^2 + \sin^2 \theta \, d\varphi^2 \big)  
&(4.4) \cr } $$
satisfies our gravitational field equation (3.9). However, as explained
in item (5) at the end of the last section, this Schwarzschild metric is
indeed a solution to (3.9).  In  fact, since $~F^{\m\n}\eqdot 0$~ is
reduced to the conventional Einstein tensor $~R^{\m\n} (G) - (1/2)
G_{\m\n} R \eqdot 0 $~ upon the restriction $~\Im\big(g_{\m\n}\big) =0$~
\hs, the  Schwarzschild metric (4.4) indeed satisfies our gravitational 
field equation (3.9), as the simplified case.  On the other hand,  we
have seen that a point mass  satisfies exactly the same geodesic equation
(4.2) as in general relativity in terms of $~G_{\m\n}$.  Therefore, the
computation of perihelion advance in our theory exactly agrees with that
in general relativity.  

We mention also the compatibility of our theory with 
equivalence principle.  Since the experimental tests of
`equivalence principle' so far have been always for classical macroscopic
objects, it is enough to consider only the point mass action  
$~I_{\rm M}$.  As we have seen, it is only the real Christoffel 
connection $~\Christoffel\r\m\n$~ that couples to a point mass.
If that is the case, all the extra components $~a_{\m\n},~B_{\m\n}$~ 
and $~b_{\m\n}$~ are irrelevant to the question of equivalence 
principle associated with point masses.\footnotew{As for other 
questions related to other particle theoretic fields with
different spins such as fermions, we do not discuss them here, because
they are beyond the scope of this Letter.}  

To put it differently, we have seen that our constraint $~\Im
\big(\G\du{\m\n}\r\big) \eqdot 0$~ (3.17) deletes the covariant and 
imaginary part of the affinity $~\G\du{\m\n}\r$, while the real part 
$~\Re\big( \G\du{\m\n}\r\big) $~ obeys the same transformation rule  as
general relativity.  Therefore, by choosing a geodesic coordinate  system
under (3.17), we can make also the real part $~\Re\big(
\G\du{\m\n}\r\big)$~ vanish, as in general relativity.  Eventually, the
geodesic equation (4.3) becomes a `free-fall' equation $~d^2
\Tilde x{}^\m / d s^2 \eqdot 0$, as in general relativity.  As long as
we use a classical macroscopic object as a test point mass, we can not
tell the violation of equivalence principle in our theory.  

We emphasize that our gravitational theory has indeed passed the most
important key test of perihelion advance of Mercury and equivalence
principle, at least at the classical field level.  

% \bigskip\bigskip\bigskip

\newpage


\leftline{\bf 5.~~A Lagrangian for Noncommutative Gravity}  

Once we have understood the teleparallelism formulation of complex
gravity,  it is straightforward to generalize that result to the
noncommutative gravity, with the standard $~\star~$ product:
$$ \li{ & f \star g \equiv f \, 
      \exp\big( i 
      \leftarrowoverdel_\m {} \theta^{\m\n} {} 
      \rightarrowoverdel_\n \big) \, g 
      \equiv \sum_{n=1}^\infty \fracm{i^n}{n!} \theta^{\m_1\n_1} \cdots
      \theta^{\m_n\n_n} 
      \big( \partial_{\m_1}\cdots\partial_{\m_n} f \big) ~   
      \big( \partial_{\n_1}\cdots\partial_{\n_n} g \big)  ~~. 
&(5.1) \cr } $$
The hermiticity of a noncommutative product of complex functions $~f_1,
~\cdots,~ f_n$~ is understood as 
$~ \big( f_1 \star f_2 \star \cdots \star 
     f_{n-1} \star f_n \big)^\dagger
     = f_n^\dagger \star f_{n-1}^\dagger \star \cdots \star f_2 \star
     f_1^\dagger$.  
We can replace all the products and matrix inverses by
$~\star$~ products and $~\star$~ inverses everywhere in (3.8) +
(3.21):    
$$ \li{ \Lag_\star \equiv & + {\root\star\of{e_\star}}\,{}^\dagger \star
 \Big[\, 
       - \frac18 g_\star^{\m\r} \star C\du{\m\n} a \star 
      \eta\du a b \star \Bar C_{\r\s \, b} \star g_\star^{\n\s} 
      + \frac 14 e_\star^{b \n} \star C\du{\m\n} a 
     \star g_\star^{\m\s}    
      \star \Bar C_{\r\s \, b} \star e\du {\star\, a} \r \cr 
& ~~~~~ ~~~~~ ~~~~~ ~ + \frac 12 e\du {\star\, a} \n  
       \star C\du{\m\n} a \star g_\star^{\m\r} \star 
       \Bar C_{\r\s \, b} \star e_\star^{b\s} \, \Big] \star
{\root\star\of{e_\star}} \cr  & + \frac 12
{\root\star\of{e_\star}}\,{}^\dagger 
     \star \Im \big( \G\du{\m\n}\r \big) \, 
     \star \L\ud{\m\n \s\t}{\r\l}
     \star \Im \big( \G\du{\s\t}\l \big) 
     \star {\root\star\of{e_\star}}  ~~, 
&(5.2) \cr } $$ 
paying attention to comply with the hermiticity mentioned above.   Here
$~e\du{\star a} \m \star e\du \m b = \d\du a b$~ and  
$~e\du\star{b \n} \equiv \big(e\du{\star b}\n\big)^\dagger$, {\it etc.},
similarly to \chamseddine.  Due to the $~\star$~ product, we need  the
subscript $~_\star$~ for the inverse vierbeins, such as $~e\du{\star
a}\m$~ and $~e\du\star {b \n}$.  Relevantly, $~e_\star$~ is the
$~\star\-$determinant of the vierbein
$~e\du\m a$: ~$~e_\star\equiv (1/4!) \e_{a b c d}\, 
\e^{\m\n\r\s} e\du\m a \star e\du\n b \star e\du\r c \star e\du\r d$.  The
operation $~\root\star\of{e_\star}$~ can be understood as the expansion
$~{\root\star\of{1+x}} \equiv 1 + \sum_{n=1}^\infty (1/n!)
(1/2) (1/2-1)
\cdots (3/2-n) \, \overbrace{x\star x\star \cdots \star x}^n$~ 
%%% 
\ref\gps{{\it See, e.g.,} N.~Grandi, R.L.~Pakman and F.A.~Schaposnik,
Nucl.~Phys.~{\bf B588} (2000) 508, hep-th/0004104.},   
while $~{\root\star\of{e_\star}}\,{}^\dagger$~ is its hermitian
conjugate.      
%%%
Since $~g\low{\m\n}$~ is hermitian, there is no problem with the product
for the definition of the square roots in 
$~\root\star\of{e_\star}\,{}^\dagger\,\star\root\star\of{e_\star}$~ as in
the commutative case.  The definition of $~C\du{\m\n}a$~ or $~\Bar C_{\m\n
a}$~ needs no 
$~\star\-$ symbol on itself like $~C\du{\star \,\m\n} a$, because it has
no product in its definition just as in the commutative case (3.8).  
Thanks to teleparallel complex gravity as the foundation,  we no longer
have the problem with the negative energy states in our system, while all
the unphysical components are either decoupled, or do no harm to the
physical components $~G_{\m\n} \equiv 
\Re\big( g\low{\m\n} \big) \equiv g\low{(\m\n)}$.    
As for the uniqueness of our lagrangian (5.2), we are aware that there
might be other possible forms of lagrangians depending on the order of the
terms in (5.2).  For example, the metrics and the 
anholonomy coefficients might be flipped around in the first term
in (5.2), as long as the hermiticity is satisfied.  However, we
take here the standpoint that any of these lagrangians shares the same
quadratic terms in the teleparallel gravity, and we gave (5.2) as an
explicit example, whose quadratic terms are shared by any of those other
examples.    


\bigskip\bigskip\bigskip

% \newpage 


\leftline{\bf 6.~~Concluding Remarks}

In this Letter, we have presented teleparallel complex gravity  as
the starting point for noncommutative gravity.  Since the introduction
of  the $~\star$~ product implies that the metric is to be complex, and 
the problem of the negative energy ghost is caused by 
the Lorentz connection \chamseddine, it is natural to consider the
formulation without manifest $~U(1,3)$~ local Lorentz symmetry equivalent
to teleparallelism, motivating the
combination of teleparallel complex gravity.  

We have studied the transformation properties under general coordinate 
transformations and global Lorentz transformations.  We 
have next presented a lagrangian (3.8) that contains not only
the usual  Einstein-Hilbert action for the symmetric part 
$~G_{\m\n} \equiv g\low{(\m\n)} \equiv \Re\big(g\low{\m\n}
\big)$~ but also the imaginary part $~B_{\m\n} \equiv \Im(g_{\m\n}) = 
-i g_{\[\m\n\]}$.  Our  lagrangian (3.8) has an improved property that
there is no kinetic term with negative energy states for 
$~B_{\m\n} \equiv - i g\low{\[\m\n\]}$.  This has been proven explicitly
with the lagrangian quadratic terms (3.16), which show that all the
kinetic terms for $~a_{\m\n}$~ and $~B_{\m\n}$~ disappear under the
condition (3.3), while the component $~b_{\m\n}$~ has a kinetic term as a
spin $~2$~ field with positive definite energy.  Thus there is no
negative energy ghost in our system.  We have next looked into the
field equation $~\calF^{\m\n}\eqdot 0$~ of the vierbein $~e\du\m a$, and
concluded that the real symmetric part $~\Re\big(\calF^{(\m\n)}\big)\eqdot
0$~ contains the Einstein gravitational field equation for the symmetric
component $~g\low{(\m\n)}$, while other remaining parts
$~\Re\big(\calF^{\[\m\n\]}\big)\eqdot 0,~
\Im\big(\calF^{\[ \m\n\] }\big)\eqdot 0$~ and 
$~ \Im\big(\calF^{(\m\n)}\big)\eqdot 0$~ yield $~6 + 6 + 10 = 22$~ 
component general covariant constraint equations respectively for the
extra $~6 + 6 + 10 = 22$~ auxiliary non-propagating components $~a_{\m\n},
~B_{\m\n}$~ and $~b_{\m\n}$, as  desired.  We have also seen that the
Lorentz non-covariant solution $~\theta^{\m\n} \equiv
\Im\big(g^{\m\n}\big) = 
\hbox{const.}$~ is also an acceptable solution, which is important for the
noncommutative gravity \noncomm.  

We have also confirmed that the covariant constraint $~\Im(\G\du{\m\n}\r)
=0$~ is enough to freeze the extra components $~B_{\m\n},~b_{\m\n}$~ in
such a way that the physical component $~h_{\m\n}$~ is not constrained for
its  propagation.  This is due to the only perturbatively possible
solutions $~B_{\m\n} = \calO(\varphi^2),~b_{\m\n} = \calO(\varphi^2)$~ for
the constraint $~\Im\big(\G\du{\m\n}\r \big) \eqdot 0$, which are one order
higher in terms of fields.  Based on this nice feature, we have confirmed
no disturbing effect on the physical components $~G_{\m\n}$~ from
these extra components up to cubic-order terms in the field equation,
equivalent to quartic-order terms in the lagrangian.   We emphasize that
our total lagrangian (3.8) + (3.21) `reproduces' general relativity to
this order, with no bad effect from the extra components.  

As an important test of classical gravitational theory, we have  analyzed
the possible effect of the extra components on the perihelion advance. 
Based on our point mass action $~I_{\rm M}$, we have concluded that there
is no disturbing effect on the perihelion advance, due to the absence of
the imaginary part of the metric $~\Im\big( g_{\m\n} \big)$~ in $~I_{\rm
M}$.  Our result has excluded a worry that some extra components of our
teleparallel complex gravity might interfere with such a high precision
observation as the perihelion advance of Mercury.  

Based on this result of teleparallel complex gravity, we have
presented a lagrangian for noncommutative gravity with no negative energy
ghosts.  Even though computations for teleparallelism
formulation seem to be more involved compared with \chamseddine, the
advantage here is that the problem with the negative energy states in
$~B_{\m\n}\-$field in \chamseddine\ has been now resolved.  

The main purpose of our present paper is to establish teleparallel 
complex gravity as the foundation for noncommutative gravity. 
However, our result also suggests the importance of teleparallel gravity
even {\it before} its complexification in string physics, M-theory or
D-branes.  This further indicates teleparallel
supergravity playing an important role in superstring or M-theory.   
As a matter of fact, a teleparallel superspace had already been considered
in 1970's in four-dimensions (4D) as a possible resolution
to renormalizability problem
%%% 
\ref\taylor{J.G.~Taylor, \pl{78}{78}{577}.}, 
by reducing the number of counter-terms in the absence of 
supercurvature.          
%%% 
Also in our recent paper 
%%% 
\ref\gnr{S.J.~Gates, Jr., H.~Nishino and S.~Rajpoot,
Phys.~Rev.~{\bf D65} (2002) 024013, hep-th/0107155.},   
%%% 
we have constructed a 11D teleparallel superspace
as a reformulation of 11D supergravity, not only
as a consistent but also as a more natural background for supermembranes. 
From these developments, we regard teleparallel supergravity or
teleparallel superspace as a more natural formulation suitable for the
description of (super)strings, M-theory or D-branes.  

Considering the fundamental fact that the constant tensor
$~\theta^{\m\n}$~ in  noncommutative gravity manifestly breaks Lorentz
symmetry, we also emphasize that teleparallelism is the most natural
candidate for the foundation of noncommutative gravity.    

We are grateful to A.~Chamseddine and S.J.~Gates, Jr.~for 
helpful discussions.  A special acknowledgment is due to the 
referee(s) of this paper who suggested to analyze the quadratic
lagrangian terms and the perihelion advance of Mercury.   


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