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\begin{document}
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\title{
Constructing hyperbolic systems
in the Ashtekar formulation
of general relativity
}
\author{Gen Yoneda \cite{Emailyone}}
\address{
Department of Mathematical Science, Waseda University,
Okubo 341, Shinjuku, Tokyo 1698555, Japan}
\author{Hisaaki Shinkai \cite{Emailhis}}
\address{
Center for Gravitational Physics and Geometry,
104 Davey Lab., Department of Physics, \\
The Pennsylvania State University,
University Park, Pennsylvania 168026300, USA}
\date{November 5, 1999 (revised version,
to appear in Int. J. Mod. Phys. D., grqc/9901053)}
\maketitle
%
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\begin{abstract}
%\widetext
Hyperbolic formulations of the equations of motion are essential
technique for proving the wellposedness of the Cauchy problem of
a system, and are also helpful for implementing stable long time
evolution in numerical applications.
We, here, present three kinds of hyperbolic systems in the
Ashtekar formulation of general relativity for Lorentzian vacuum
spacetime.
We exhibit several (I) weakly hyperbolic, (II) diagonalizable
hyperbolic, and (III) symmetric hyperbolic systems, with each
their eigenvalues.
We demonstrate that Ashtekar's original equations form
a weakly hyperbolic system. We discuss
how gauge conditions and reality conditions are constrained
during each step toward constructing a symmetric hyperbolic system.
\end{abstract}
\pacs{PACS numbers: 04.20.Cv, 04.20.Ex, 04.20.Fy}
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\section{Introduction} \label{sec:intro}
Developing hyperbolic formulations of the Einstein equation
is growing into an important
research areas in general relativity \cite{Reula98}.
These formulations are used in the analytic proof of the existence, uniqueness
and stability (wellposedness)
of the solutions of the Einstein equation
\cite{Heldbook}.
So far, several first order hyperbolic formulations
have been proposed;
some of them
are flux conservative \cite{BonaMasso},
some of them are symmetrizable
or symmetric hyperbolic systems
\cite{FischerMarsden72,ChoquetBruhat,HF,fried96,VPE96,CBY,FR94,FR96}.
The recent interest in hyperbolic formulations
arises from their application to numerical relativity.
One of the most useful features is the existence of
characteristic speeds in hyperbolic systems.
We expect more stable evolutions
and expect imprements boundary conditions in their numerical
simulation.
Some numerical tests have been reported along this direction
\cite{miguel,SBCSThyper,cactus1}.
Ashtekar's formulation of general relativity \cite{Ashtekar}
has many
advantages.
By using his special pair of variables,
the constraint equations which appear in the theory become
loworder polynomials, and the theory has the
correct form for gauge theoretical interpretation. These features
suggest the possibility for developping a nonperturbative quantum
description of gravity.
Classical applications of the Ashtekar's formulation have also
been discussed by several authors. For
example, we \cite{yscon} discussed
the reality conditions for the metric and triad and proposed a new
set of variables for Lorentzian dynamics.
We \cite{ysndege}
also showed an example of passing a degenerate
point in 3space by locally relaxing the reality condition.
Although there is always a problem of reality
conditions in applying Ashtekar formulation to dynamics,
we think that this new approach is quite attractive, and broadens our
possibilities to attack dynamical issues.
A symmetric hyperbolic formulation of Ashtekar's variables
was first developped by Iriondo, Leguizam\'on and Reula (ILR)
\cite{Iriondo}.
They use {\it anti}Hermiticity of
the principal symbol for defining their {\it symmetric} system.
Unfortunately, in their first short paper \cite{Iriondo},
they did not discuss the consistency of
their system with the reality conditions, which are
crucial in the study of the Lorentzian dynamics using
the Ashtekar variables.
We considered this point in \cite{YShypPRL}, and found that
there are strict reality constraints
(alternatively they can be interpreted as gauge conditions).
Note that we primarily use the Hermiticity of the characteristic
matrix to define a symmetric hyperbolic system, which
we think the more conventional notation.
The difference between these definitions of symmetric hyperbolicity
is commented in Appendix C.
The dynamical equations in the Ashtekar formulation of
general relativity are themselves
quite close to providing a hyperbolic formulation.
As we will show in \S \ref{sec4},
the original set of equations of motion is a
firstorder (weakly) hyperbolic system.
One of the purposes of this paper is to develop
several hyperbolic systems based on the Ashtekar formulation
for Lorentzian vacuum spacetime, and discuss how gauge conditions
and reality conditions are to be implemented.
We categorize hyperbolic systems into three classes: (I) weakly
hyperbolic (system has
all real eigenvalues), (II) diagonalizable hyperbolic
(characteristic matrix is diagonalizable), and (III)
symmetric hyperbolic system. These three classes have the relation
(III) $\in$ (II) $\in$ (I), and
are defined in detail in \S \ref{sec:def}.
As far as we know, only a symmetric hyperbolic systems provide a fully
wellposed initial value formulation of partial
differential equations systems. However, there are two reasons to
consider the two other classes of hyperbolic systems, (I) and (II).
First, as we found in our
previous short paper \cite{YShypPRL}, the symmetric hyperbolic system
we obtained using Ashtekar's variables
has strict restrictions on the gauge conditions, while
the original Ashtekar
equations constitute a weakly hyperbolic system.
We are interested in
these differences, and show how additional constraints appear
during the steps toward constructing a symmetric hyperbolic system.
Second,
many numerical experiments show that there are
several advantages
if we apply a certain form of hyperbolic formulation.
%This is because we there know the characteristic
%speeds of the system.
Therefore, we think that presenting these three hyperbolic systems
%not only a symmetric hyperbolic system but also other two systems
is valuable to stimulate the studies in this field.
To aid in possibly applying these systems
% Concerning a chance to apply each system
in numerical applications,
we present characteristic speeds of each system we construct.
ILR, in their second paper \cite{ILRsecond}, expand their
previous discussion \cite{Iriondo} concerning
reality conditions during evolution.
They demand that the metric is
realvalued (metric reality condition),
and use the freedom of internal rotation
during the time evolution to set up
their soldering form so that it forms an antiHermitian
principle symbol,
which is their basis to characterize the system symmetric.
However, we adopt the view that redefining inner product of the
fundamental variables
introduces additional complications.
In our procedure, we first fix the inner product to construct
a symmetric hyperbolic system.
As we will describe in \S \ref{sec5},
our symmetric
hyperbolic system then requires a reality condition on the triad
(triad reality condition),
and in order to be consistent with its secondary condition
we need to impose further gauge conditions.
The lack of these constraints in ILR, we believe, comes from their
incomplete treatment of a new gauge freedom, socalled
{\it triad lapse} $\CA^a_0$
(discussed in \S \ref{sec:ash}), for dynamical evolutions in the
Ashtekar formulation.
In Appendix \ref{appC},
we show that ILR's proposal
to use internal rotation to reset triad reality does not work
if we adopt our conventional definition of hyperbolicity.
The layout of this paper is as follows:
In \S \ref{sec:def}, we define the three kinds of hyperbolic
systems which are considered in this paper.
In \S \ref{sec:ash}, we briefly review
Ashtekar's formulation and the way of handling reality conditions.
The following sections \S \ref{sec4} and \S \ref{sec5} are devoted to
constructing hyperbolic systems.
Summary and discussion are in \S \ref{sec:disc}.
Appendix \ref{appA} supplements our proof of the uniqueness of our
symmetric hyperbolic system.
Appendices \ref{appB} and \ref{appC} are comments
on ILR's treatment of the reality conditions.
%=================================================== section 2
\section{Three definitions of hyperbolic systems} \label{sec:def}
%=================================================== section 2
\def\cha{J}
\def\coe{K}
%\subsection{Three definitions}
We start by defining the hyperbolic
systems which are used in this paper.
\begin{Def}
%{\bf Definition}\quad
We assume
a certain set of
(complex) variables $u_\alpha$ $(\alpha=1,\cdots, n)$
forms a firstorder (quasilinear)
partial differential equation system,
\begin{equation}
\partial_t u_\alpha
= \cha ^{l}{}^{\beta}{}_\alpha (u) \, \partial_l u_\beta
+\coe_\alpha(u),
\label{def}
\end{equation}
where $\cha$ (the characteristic matrix) and
$\coe$ are functions of $u$
but do not include any derivatives of $u$.
We say that the system (\ref{def}) is:
\been
\def\theenumi{(\Roman{enumi})}
%=================
\item \label{weakhyp}
%{\bf real spectral hyperbolic},
{\bf weakly hyperbolic},
if all the eigenvalues of the
characteristic matrix are real {\rm \cite{weakhyperbolic}}.
%=================
\item \label{diaghyp}
%{\bf real diagonalizable hyperbolic},
{\bf diagonalizable hyperbolic},
if the characteristic matrix is
diagonalizable and has all real eigenvalues
{\rm \cite{mizodiaghyp}}.
%=================
\item \label{symhyp}
{\bf symmetric hyperbolic},
if the characteristic matrix is a
Hermitian matrix {\rm \cite{fried96,CH}}.
\enen
\end{Def}
Here we state each definition more concretely.
We treat $\cha^{l\beta}{}_\alpha$ as a $n \times n$ matrix
when the $l$index is fixed.
The following properties of these matrices are
for every basis of $l$index.
We say $\lambda^l$ is an
eigenvalue of $\cha^{l\beta}{}_\alpha$
when the characteristic equation,
$\det (\cha^{l\beta} {}_\alpha
\lambda^l \delta^\beta {}_\alpha)=0$,
is satisfied.
The eigenvectors, $p^\alpha$, are given by solving
$\cha^{l}{}^{\alpha} {}_\beta \, p^{l\beta}=\lambda^l p^{l\alpha}$.
The weakly hyperbolic system, \ref{weakhyp}, is
obtained when
$\cha^l$ has {\it real spectrum} for every $l$,
that is, when
this characteristic equation can be divided
by $n$ real firstdegree factors.
{}For any single equation system,
the Cauchy problem under weak hyperbolicity is not,
in general, $C^\infty$ wellposed, while it is solvable in the
class of the real analytic functions and in some
suitable Gevrey classes,
provided that the coefficients of the principal part
are sufficiently smooth.
%=================
The diagonalizable hyperbolic system, \ref{diaghyp},
is obtained when
$\cha$ is {\it real diagonalizable}, that is,
when
there exists complex regular matrix $P^l$
such that
$((P^l){}^{1}) {}^\alpha {}_\gamma \,
\cha^{l\gamma} {}_\delta \,
P^{l\delta} {}_\beta$
is real diagonal matrix for every $l$.
We can construct
characteristic curves if the system is in this class.
This system is often used as a model in the studies of
wellposedness in coupled linear hyperbolic system.
(This is the same as {\it strongly hyperbolic} system
as defined by some authors\cite{stewart,Gustaf}, but we use
the word {\it diagonalizable} since there exist other
definitions for {\it strongly hyperbolic}
systems\cite{taniguchisymp}.)
%with a purpose of generalization of the known facts in the
%symmetric hyperbolic system.
%=================
In order to define
the symmetric hyperbolic system, \ref{symhyp},
we need to declare an inner product
$\langle uu \rangle$
to judge whether $\cha^{l\beta} {}_\alpha$ is Hermitian.
In other words, we are
required to define the way of lowering the index
$\alpha$ of $u^\alpha$.
We say $\cha^{l\beta} {}_\alpha$ is Hermitian
with respect to this index rule,
when
$\cha^l {}_{\beta\alpha}=\bar{\cha}^l {}_{\alpha\beta}$
for every $l$,
where the overhead bar denotes complex conjugate.
Any Hermitian matrix is real diagonalizable,
%and the eigenvalues of real
%diagonalizable matrix are all real;
so that \ref{symhyp} $\in$ \ref{diaghyp}
$\in$ \ref{weakhyp}.
There are other
definitions of hyperbolicity; such as
{\it strictly hyperbolic} or {\it effectively hyperbolic},
if all eigenvalues of the characteristic
matrix are real and distinct (and nonzero for the latter).
These definitions are stronger than
\ref{diaghyp}, but exhibit no inclusion relation
with \ref{symhyp}. In this paper, however, we only consider
\ref{weakhyp}\ref{symhyp} above.
The symmetric system gives us the energy
integral inequalities,
which are the primary tools for studying
wellposedness of the system.
As was discussed by Geroch \cite{Geroch},
most physical systems
can be expressed as symmetric hyperbolic systems.
%Mathematically,
%the wellposedness of the system is proven only for the
%{\it linear} symmetric hyperbolic system, and not yet
%for the {\it quasilinear} symmetric hyperbolic system
%\cite{taniguchisymp}.
%The purpose of this paper is not to prove wellposedness
%strictly, but to construct hyperbolic systems \ref{weakhyp}\ref{symhyp}
%in the Ashtekar formulation.
%=================================================== section 3
\section{Ashtekar formulation}\label{sec:ash}
%=================================================== section 3
\subsection{Variables and Equations}
The key feature of Ashtekar's formulation of general relativity
\cite{Ashtekar} is the introduction of a selfdual
connection as one of the basic dynamical variables.
Let us write
the metric $g_{\mu\nu}$ using the tetrad
$E^I_\mu$, with $E^I_\mu$ satisfying the gauge condition $E^0_a=0$.
Define its inverse, $E^\mu_I$, by
$g_{\mu\nu}=E^I_\mu E^J_\nu \eta_{IJ}$ and
$E^\mu_I:=E^J_\nu g^{\mu\nu}\eta_{IJ}$,
where we use
$\mu,\nu=0,\cdots,3$ and
$i,j=1,\cdots,3$ as spacetime indices, while
$I,J=(0),\cdots,(3)$ and
$a,b=(1),\cdots,(3)$ are $SO(1,3)$, $SO(3)$ indices respectively.
We raise and lower
$\mu,\nu,\cdots$ by $g^{\mu\nu}$ and $g_{\mu\nu}$
(the Lorentzian metric);
$I,J,\cdots$ by $\eta^{IJ}={\rm diag}(1,1,1,1)$ and $\eta_{IJ}$;
$i,j,\cdots$ by $\gamma^{ij}$ and $\gamma_{ij}$ (the 3metric);
$a,b,\cdots$ by $\delta^{ab}$ and $\delta_{ab}$.
We also use volume forms $\epsilon_{abc}$:
$\epsilon_{abc} \epsilon^{abc}=3!$.
We define SO(3,C) selfdual and anti selfdual
connections
\be
{}^{\pm\!}{\cal A}^a_{\mu}
:= \omega^{0a}_\mu \mp ({i / 2}) \epsilon^a{}_{bc} \, \omega^{bc}_\mu,
\en
where $\omega^{IJ}_{\mu}$ is a spin connection 1form (Ricci
connection), $\omega^{IJ}_{\mu}:=E^{I\nu} \nabla_\mu E^J_\nu.$
Ashtekar's plan is to use only a selfdual part of
the connection
$^{+\!}{\cal A}^a_\mu$
and to use its spatial part $^{+\!}{\cal A}^a_i$
as a dynamical variable.
Hereafter,
we simply denote $^{+\!}{\cal A}^a_\mu$ as ${\cal A}^a_\mu$.
The lapse function, $N$, and shift vector, $N^i$, both of which we
treat as realvalued functions,
are expressed as $E^\mu_0=(1/N, N^i/N$).
This allows us to think of
$E^\mu_0$ as a normal vector field to $\Sigma$
spanned by the condition $t=x^0=$const.,
which plays the same role as that of ArnowittDeserMisner (ADM) formulation.
Ashtekar treated the set ($\tilde{E}^i_{a}$, ${\cal A}^a_{i}$)
as basic dynamical variables, where
$\tilde{E}^i_{a}$ is an inverse of the densitized triad
defined by
\be
\tilde{E}^i_{a}:=e E^i_{a},
\en
where $e:=\det E^a_i$ is a density.
This pair forms the canonical set.
In the case of pure gravitational spacetime,
the Hilbert action takes the form
\begin{eqnarray}
S&=&\int {\rm d}^4 x
[ (\partial_t{\cal A}^a_{i}) \tilde{E}^i_{a}
+(i/2) \null \! \mathop {\vphantom {N}\smash N}
\limits ^{}_{^\sim}\!\null \tilde{E}^i_a
\tilde{E}^j_b F_{ij}^{c} \epsilon^{ab}{}_{c}

\den^2
\Lambda \null \! \mathop {\vphantom {N}\smash N}
\limits ^{}_{^\sim}\!\null
%\det\tilde{E}
N^i F^a_{ij} \tilde{E}^j_a
+{\cal A}^a_{0} \, {\cal D}_i \tilde{E}^i_{a} ],
\label{action}
\end{eqnarray}
where
$\null \! \mathop {\vphantom {N}\smash N}
\limits ^{}_{^\sim}\!\null := e^{1}N$,
${F}^a_{\mu\nu} := (d {\cal A}^a)_{\mu\nu}
(i/2){\epsilon^a}_{bc}({\cal A}^b
\wedge {\cal A}^c)_{\mu\nu}
= \partial_\mu {\cal A}^a_\nu
 \partial_\nu {\cal A}^a_\mu
 i \epsilon^{a}{}_{bc} \, {\cal A}^b_\mu{\cal A}^c_\nu
$
is the curvature 2form,
$\Lambda$
is the cosmological constant,
${\cal D}_i \tilde{E}^j_{a}
:=\partial_i \tilde{E}^j_{a}
i \epsilon_{ab}{}^c \, {\cal A}^b_{i}\tilde{E}^j_{c}$,
and
$\den^2=\det\tilde{E}^i_a
=(\det E^a_i)^2$ is defined to be
$\det\tilde{E}^i_a=
(1/6)\epsilon^{abc}
\null\!\mathop{\vphantom {\epsilon}\smash \epsilon}
\limits ^{}_{^\sim}\!\null_{ijk}\tilde{E}^i_a \tilde{E}^j_b
\tilde{E}^k_c$, where
$\epsilon_{ijk}:=\epsilon_{abc}E^a_i E^b_j E^c_k$
and $\null\!\mathop{\vphantom {\epsilon}\smash \epsilon}
\limits ^{}_{^\sim}\!\null_{ijk}:=e^{1}\epsilon_{ijk}$
[When $(i,j,k)=(1,2,3)$,
we have
$\epsilon_{ijk}=e$,
$\null\!\mathop{\vphantom {\epsilon}\smash \epsilon}
\limits ^{}_{^\sim}\!\null_{ijk}=1$,
$\epsilon^{ijk}=\den^{1}$, and
$\tilde{\epsilon}^{ijk}=1$.].
Varying the action with respect to the nondynamical variables
$\null \!
\mathop {\vphantom {N}\smash N}\limits ^{}_{^\sim}\!\null$,
$N^i$
and ${\cal A}^a_{0}$ yields the constraint equations,
\begin{eqnarray}
{\cal C}_{H} &:=&
(i/2)\epsilon^{ab}{}_c \,
\tilde{E}^i_{a} \tilde{E}^j_{b} F_{ij}^{c}
\Lambda \, \det\tilde{E}
\approx 0, \label{cham} \\
{\cal C}_{M i} &:=&
F^a_{ij} \tilde{E}^j_{a} \approx 0, \label{cmom}\\
{\cal C}_{Ga} &:=& {\cal D}_i \tilde{E}^i_{a}
\approx 0. \label{cg}
\end{eqnarray}
The equations of motion for the dynamical variables
($\tilde{E}^i_a$ and ${\cal A}^a_i$) are
\begin{eqnarray}
\partial_t {\tilde{E}^i_a}
&=&i{\cal D}_j( \epsilon^{cb}{}_a \, \null \!
\mathop {\vphantom {N}\smash N}\limits ^{}_{^\sim}\!\null
\tilde{E}^j_{c}
\tilde{E}^i_{b})
+2{\cal D}_j(N^{[j}\tilde{E}^{i]}_{a})
+i{\cal A}^b_{0} \epsilon_{ab}{}^c \, \tilde{E}^i_c, \label{eqE}
\\
\partial_t {\cal A}^a_{i} &=&
i \epsilon^{ab}{}_c \,
\null \! \mathop {\vphantom {N}\smash N}
\limits ^{}_{^\sim}\!\null \tilde{E}^j_{b} F_{ij}^{c}
+N^j F^a_{ji} +{\cal D}_i{\cal A}^a_{0}+\Lambda \null \!
\mathop {\vphantom {N}\smash N}\limits ^{}_{^\sim}\!\null
\tilde{E}^a_i,
\label{eqA}
\end{eqnarray}
\noindent
where
${\cal D}_jX^{ji}_a:=\partial_jX^{ji}_ai
\epsilon_{ab}{}^c {\cal A}^b_{j}X^{ji}_c,$
for $X^{ij}_a+X^{ji}_a=0$.
In order to construct metric variables from the variables
$(\tilde{E}^i_a, {\cal A}^a_i, \null \!
\mathop {\vphantom {N}\smash N}\limits ^{}_{^\sim}\!\null, N^i)$,
we first prepare
tetrad $E^\mu_I$ as
$E^\mu_{0}=({1 / e \null \! \mathop {\vphantom {N}\smash N}
\limits ^{}_{^\sim}\!\null}, {N^i / e \null \!
\mathop {\vphantom {N}\smash N}\limits ^{}_{^\sim}\!\null})$ and
$E^\mu_{a}=(0, \tilde{E}^i_{a} /e).$
Using them, we obtain metric $g^{\mu\nu}$ such that
\begin{equation}
g^{\mu\nu}:=E^\mu_{I} E^\nu_{J} \eta^{IJ}. \label{recmet}
\end{equation}
\subsection{Reality conditions}
The metric (\ref{recmet}), in general, is not realvalued in the Ashtekar
formulation.
To ensure that the metric is realvalued,
we need to impose real lapse and shift vectors together with
two conditions (the metric reality condition);
\begin{eqnarray}
\im (\tilde{E}^i_a \tilde{E}^{ja} ) &=& 0, \label{wreality1} \\
\re (\epsilon^{abc}
\tilde{E}^k_a \tilde{E}^{(i}_b {\cal D}_k \tilde{E}^{j)}_c)
&=& 0,
\label{wreality2final}
\end{eqnarray}
where the latter comes from the secondary condition of reality
of the metric
$\im \{ \partial_t(\tilde{E}^i_a \tilde{E}^{ja} ) \} = 0$
\cite{AshtekarRomanoTate}, and
we assume
$\det\tilde{E}>0$ (see \cite{yscon}).
These metric reality conditions,
(\ref{wreality1}) and
(\ref{wreality2final}), are automatically preserved during the evolution
if the variables satisfy the conditions on the initial data
\cite{AshtekarRomanoTate,yscon}.
For later convenience, we also prepare
stronger reality conditions.
These conditions are
\begin{eqnarray}
\im (\tilde{E}^i_a ) &=& 0
\label{sreality1} \\
{\rm and~~}
\im ( \partial_t {\tilde{E}^i_a} ) &=& 0,
\label{sreality2}
\end{eqnarray}
\noindent
and we call them the ``primary triad reality condition" and the
``secondary triad
reality condition", respectively.
Using the equations of motion of $\tilde{E}^i_{a}$,
the gauge constraint (\ref{cham})(\ref{cg}),
the metric reality conditions
(\ref{wreality1}), (\ref{wreality2final})
and the primary condition (\ref{sreality1}),
we see that (\ref{sreality2}) is equivalent to \cite{yscon}
\begin{equation}
\re({\cal A}^a_{0})=
\partial_i( \null \! \mathop {\vphantom {N}\smash N}
\limits ^{}_{^\sim}\!\null )\tilde{E}^{ia}
+(1 /2e) E^b_i \null \! \mathop {\vphantom {N}\smash N}
\limits ^{}_{^\sim}\!\null
\tilde{E}^{ja} \partial_j\tilde{E}^i_b
+N^{i} \re({\cal A}^a_i), \label{sreality2final}
\end{equation}
or with undensitized variables,
\begin{equation}
\re({\cal A}^a_{0})=
\partial_i( N)
{E}^{ia}
+N^{i} \, \re({\cal A}^a_i).
\label{sreality2final2}
\end{equation}
{}From this expression we see that
the secondary triad reality condition
restricts the three components of ``triad lapse" vector
${\cal A}^a_{0}$.
Therefore (\ref{sreality2final}) is
not a restriction on the dynamical variables
($\tilde{E}^i_a $ and ${\cal A}^a_i$)
but on the slicing, which we should impose on each hypersurface.
Thus the secondary triad reality condition does not restrict the
dynamical variables any
further than the secondary metric condition does.
Throughout this paper, we basically impose metric
reality condition. We assume that initial data of
$(\dtri^i_a, \CA^a_i)$ for evolution are solved so as
to satisfy all three constraint equations and metric
reality condition (\ref{wreality1}) and (\ref{wreality2final}).
Practically, this is
obtained, for
example, by solving ADM constraints and by transforming a
set of initial data to Ashtekar's notation.
\subsection{Characteristic matrix}
We shall see how the
definitions of hyperbolic systems in \S \ref{sec:def}
can be applied for
Ashtekar's equations of motion
(\ref{eqE}) and (\ref{eqA}).
Since both dynamical variables,
$\dtri^i_a$ and $\CA^a_i$, have 9 components each
(spatial index: $i=1,2,3$ and
SO(3) index: $a=(1),(2),(3)$),
the combined set of variables, $u^\alpha =
(\dtri^i_a, \CA^a_i) $, has 18 components.
Ashtekar's formulation itself is
in the firstorder (quasilinear)
form in the sense of (\ref{def}),
but is not in a symmetric hyperbolic form.
We start by writing the
principal part of the Ashtekar's evolution equations as
\begin{equation}
\partial_t \left[ \begin{array}{l}
\tilde{E}^i_a \\
\CA^a_i
\end{array} \right] \cong
\left[ \begin{array}{cc}
A^l {}_a {}^{bi}{}_j &
B^l{}_{ab}{}^{ij} \\
C^{lab}{}_{ij} &
D^{la}{}_{bi}{}^j
\end{array} \right]
\partial_l
\left[ \begin{array}{l}
\tilde{E}^j_b \\
\CA^b_j
\end{array} \right],
\label{matrixform}
\end{equation}
where
$\cong$ means that we have extracted only the terms which
appear in the principal part of the system. We name
these components as $A,B,C$ and $D$ for later convenience.
The characteristic equation becomes
\be
\det \left( \begin{array}{cc}
A^l {}_a {}^{bi}{}_j
\lambda^l \delta^b_a\delta^i_j &
B^l{}_{ab}{}^{ij} \\
C^{lab}{}_{ij} &
D^{la}{}_{bi}{}^j
\lambda^l\delta^a_b\delta^j_i
\end{array} \right)=0.
\en
If $B^l{}_{ab}{}^{ij}$ and
$C^{lab}{}_{ij}$ vanish, then
the characteristic matrix is diagonalizable
if $A$ and $D$ are diagonalizable, since the spectrum of the
characteristic matrix is composed of those of $A$ and $D$.
The eigenvectors for every $l$index, $(p^l{}^i_a, q^l {}^a_i)$, are given by
\be
\left( \begin{array}{cc}
A^l {}_a {}^{bi}{}_j &
B^l{}_{ab}{}^{ij} \\
C^{lab}{}_{ij} &
D^{la}{}_{bi}{}^j
\end{array} \right)
\left( \matrix{p^l {}^j_b \cr q^l {}^b_j }\right)
=\lambda^l
\left( \matrix{p^l {}^i_a \cr q^l {}^a_i }\right)
\mbox{ for every }l.
\en
The lowering rule for the $\alpha$ of $u^\alpha$
follows those of the spacetime or internal indices.
% If $u^\alpha=(\dtri^i_a,\CA^b_j)$,
% then $u_\alpha=(\dtri^a_i,\CA^j_b)$.
The corresponding inner product takes the form
%form, we naturally define
$\langle u u \rangle :=u_\alpha \bar{u}^\alpha$.
%\be
%\langle u_2  u_1\rangle :=
%\langle u_2  u_1
%=
%\langle(\dtri^{1i}_a,\CA^{1b}_j),(\dtri^{2i}_a,\CA^{2b}_j)
%\rangle
%=
%\dtri^{1i}_a \bar{\dtri}^{2j}_b \gamma_{ij}\delta^{ab}
%+\CA^{1a}_i\bar{\CA}^{2b}_j \gamma^{ij}\delta_{ab}
%\en
According to this rule,
we say the characteristic matrix is a Hermitian when
\begin{eqnarray}
0&=&
A^{labij}\bar{A}^{lbaji}, \label{cond1}
\\
0&=&
D^{labij}
\bar{D}^{lbaji}, \label{cond2}
\\
0 &=&
B^{labij}\bar{C}^{lbaji}. \label{cond3}
\end{eqnarray}
%=================================================== section 4
\section{Constructing
hyperbolic systems with original equations of motion}
\label{sec4}
%=================================================== section 4
In this section, we consider which form of hyperbolicity applies to
% system is constituted by taking
the original equations of motion,
(\ref{eqE}) and (\ref{eqA}),
under the metric reality condition (\S \ref{sec41}) or
under the triad reality condition (\S \ref{sec42}).
% Another possibilities of modifying equations of motion
% will be discussed in the next section.
\subsection{
%Oringinal EOM
Under metric reality condition (system Ia and IIa)}
\label{sec41}
As the first approach, we take the
%original
equations of motion
(\ref{eqE}) and (\ref{eqA}) with metric reality conditions
(\ref{wreality1}) and (\ref{wreality2final}).
The principal term of (\ref{eqE}) and (\ref{eqA})
become
\beas
\partial_t \tilde{E}^i_a
&=&
i{\cal D}_j( \epsilon^{cb}{}_{a} \,
\null \! \mathop {\vphantom {N}\smash N}
\limits ^{}_{^\sim}\!\null
\tilde{E}^j_c\tilde{E}^i_b)
+2{\cal D}_j(N^{[j}\tilde{E}^{i]}_a)
+i{\cal A}^b_0 \epsilon_{ab}{}^{c} \, \tilde{E}^i_c
\nonumber\\&\cong&
i\epsilon^{cb}{}_{a} \,
\null \! \mathop {\vphantom {N}\smash N}
\limits ^{}_{^\sim}\!\null
(\partial_j\tilde{E}^j_c)\tilde{E}^i_b
i\epsilon^{cb}{}_{a} \,
\null \! \mathop {\vphantom {N}\smash N}
\limits ^{}_{^\sim}\!\null
\tilde{E}^j_c(\partial_j\tilde{E}^i_b)
+{\cal D}_j(N^j\tilde{E}^i_a)
{\cal D}_j(N^i\tilde{E}^j_a)
\nonumber\\&\cong&
[i\epsilon^{bc}{}_{a} \,
\null \! \mathop {\vphantom {N}\smash N}
\limits ^{}_{^\sim}\!\null
\delta^l_j \tilde{E}^i_c
i\epsilon^{cb}{}_{a} \,
\null \! \mathop {\vphantom {N}\smash N}
\limits ^{}_{^\sim}\!\null
\tilde{E}^l_c \delta^i_j
+N^l\delta^i_j \delta^b_a
N^i\delta^l_j \delta^b_a ]
(\partial_l\tilde{E}^j_b),
\\
\partial_t {\cal A}^a_i &=&
i \epsilon^{ab}{}_{c} \,
\null \! \mathop {\vphantom {N}\smash N}
\limits ^{}_{^\sim}\!\null
\tilde{E}^j_b F^c_{ij}
+N^j F^a_{ji}
+{\cal D}_i{\cal A}^a_0+\Lambda \,
\null \! \mathop {\vphantom {N}\smash N}
\limits ^{}_{^\sim}\!\null
\tilde{E}^a_i
\nonumber\\&\cong&
i \epsilon^{ab}{}_{c} \,
\null \! \mathop {\vphantom {N}\smash N}
\limits ^{}_{^\sim}\!\null
\tilde{E}^j_b (\partial_i{\cal A}^c_j
\partial_j{\cal A}^c_i)
+N^j (\partial_j{\cal A}^a_i\partial_i{\cal A}^a_j)
% \partial_l{\cal A}^b_j
\nonumber\\&\cong&
[
+i \epsilon^a{}_b {}^c \,
\null \! \mathop {\vphantom {N}\smash N}
\limits ^{}_{^\sim}\!\null
\tilde{E}^j_c \delta^l_i
i \epsilon^a{}_b{}^c \,
\null \! \mathop {\vphantom {N}\smash N}
\limits ^{}_{^\sim}\!\null
\tilde{E}^l_c \delta^j_i
+N^l \delta^a_b \delta^j_i
N^j \delta^a_b \delta^l_i
](\partial_l{\cal A}^b_j).
\enas
%where
%$\cong$ means that we extracted
%%only {\it principal part}, that is
%the terms which
%contribute to characteristic matrix.
The principal terms in the notation of (\ref{matrixform}) become
\bear
A^{labij}&=&
i\ut N \ep^{abc} \dtri^i_c \gamma^{lj}
+i\ut N \ep^{abc} \dtri^l_c \gamma^{ij}
+N^l \delta^{ab} \gamma^{ij}
N^i\delta^{ab} \gamma^{lj},
\label{oriA}
%+P^{iab} \gamma^{lj}
\\
B^{labij}&=& C^{labij}=0,
\label{oriBC}
\\
D^{labij}&=&
+i\ut N \ep^{abc} \dtri^j_c \gamma^{li}
i\ut N \ep^{abc} \dtri^l_c \gamma^{ij}
+N^l \delta^{ab} \gamma^{ij}
N^j \delta^{ab} \gamma^{li}.
\label{oriD}
%+S^{liajb}
%iQ^{ai} \ep^{bcd} \dtri^j_c \dtri^l_d
%R^{ila} \dtri^{jb}
%+R^{ija} \dtri^{lb}.
\enar
We get the 18 eigenvalues of the characteristic matrix,
all of which are independent of the choice of triad:
\beas
0 ~(\mbox{multiplicity=} 6), ~
N^l ~(4), ~
N^l \pm N\sqrt{\gamma^{ll}} ~(4 \mbox{ each}),
\enas
where we do {\it not} take the sum in $\gamma^{ll}$
(and we maintain this notation
hereafter for eigenvalues and related discussions).
Therefore we can say that
this system is weakly hyperbolic, of type \ref{weakhyp}.
We note that this system is not type \ref{diaghyp} in general,
because this is not diagonalizable, for example, when
$N^l=0$.
We classify this system as type \ref{weakhyp},
and call this {\it system Ia}, hereafter.
%IIa (oringinial with metric reality and gauge cond.
The necessary and sufficient
conditions to make
this system diagonalizable, type \ref{diaghyp}, are that
the gauge conditions
\be
N^l \neq 0
\mbox{~nor~}
\pm N \sqrt{\gamma^{ll}}
,
\qquad
\mbox{and~}
\gamma^{ll} >0
\label{IIa_1}
\en
where the last one is the positive definiteness of $\gamma^{ll}$.
This can be proved as follows.
Suppose that $(\ref{IIa_1})$ is satisfied.
Then $0,N^l,N^l\pm \sqrt{\gamma^{ll}}$
are four distinct eigenvalues
and we see
$\rank(\cha^l)=12$,
$\rank(\cha^lN^l I)=14$,
$\rank(\cha^l(N^l\pm N \sqrt{\gamma^{ll}})I)=14$.
Therefore the characteristic matrix is diagonalizable.
Conversely suppose that
$N^l=0$ or
$N^l=\pm N \sqrt{\gamma^{ll}}$, then we see
the characteristic matrix is not diagonalizable
in each case.
The components of the characteristic matrix are
the same as {\it system Ia},
so all eigenvalues are equivalent with {\it system Ia}.
We can also show that this system is not Hermitian
hyperbolic.
Therefore we classify the system
[{\it Ia} + (\ref{IIa_1})] to real
diagonalizable hyperbolic, type \ref{diaghyp}, and
call this set as {\it system IIa}.
However, we will show
in the next section that real diagonalizable hyperbolic
system can also be constructed with less strict gauge
conditions by modifying righthandside of equations of
motion ({\it system IIb}).
\subsection{Under triad reality condition (system Ib)}\label{sec42}
Next, we consider systems of the original
equations of motion,
(\ref{eqE}) and (\ref{eqA}), with the triad reality
condition.
Since
this reality condition
requires the additional
(\ref{sreality2final}) or (\ref{sreality2final2})
as the secondary condition (that is, to preserve the reality of
triad during time evolution),
in order to be consistent with this requirement and to
avoid the system becoming second order in
fundamental variables, we need to set
$\ptl_iN=0$. This fixes the real part of the {\it triad lapse} gauge as
$\re(\CA^a_0)=\re(\CA^a_iN^i)$.
We naturally define its imaginary part as
$\im(\CA^a_0)=\im(\CA^a_iN^i)$.
Thus the triad lapse is fixed as
$\CA^a_0=\CA^a_iN^i$.
This gauge restriction does not affect principal
part of the evolution equation
for $\dtri^i_a$, but requires us to add the term
\beas
{\cal D}_i{\cal A}^a_0
&\cong&
\ptl_i{\cal A}^a_0
=
\ptl_i(\CA^a_jN^j)
\cong
N^j(\ptl_i\CA^a_j)
=
N^j\delta^a_b\delta^l_i(\ptl_l\CA^b_j)
\enas
to the righthandside of the equation of $\CA^a_i$.
That is,
we need to add
$N^j\delta^a_b\delta^l_i$
to $D^{la}{}_{bi}{}^j$ in (\ref{oriD}),
\beas
D^{labij}
&=&
i \epsilon^{abc} \ut N \tilde{E}^j_c \gamma^{li}
i \epsilon^{abc} \ut N \tilde{E}^l_c \gamma^{ji}
+N^l \delta^{ab} \gamma^{ji}.
\enas
The other components of the characteristic matrix
remain the same [(\ref{oriA}) and (\ref{oriBC})].
We find that the set of eigenvalues of this system is
\beas
0 \ (\mbox{multiplicity =}3), \qquad
N^l \ (7), \qquad
N^l\pm N \sqrt{\gamma^{ll}}\ (4 \mbox{ each}).
\enas
Therefore the system is again, type \ref{weakhyp}.
This system is not
real diagonalizable because $D^l$ is not.
So we classify this system as type \ref{weakhyp}
and call this set as {\it system Ib}.
We note that
this system is not
real diagonalizable
for any choice of gauge.
Therefore we cannot construct a system of type \ref{diaghyp}
using the same technique of constructing {\it system IIa}.
However, as we will show in the next section,
the system becomes diagonalizable (and symmetric) hyperbolic
under the triad reality condition
if we modify the equations of motion.
%=================================================== section 5
\section{Constructing a symmetric hyperbolic system}
\label{sec5}
%=================================================== section 5
{}From the analysis of the previous section, we found that the
original set of
equations of motion in the Ashtekar formulation constitute a
weakly hyperbolic system, type \ref{weakhyp},
or a diagonalizable hyperbolic system, type \ref{diaghyp},
under appropriate gauge conditions, but we
also found that we could not obtain a symmetric hyperbolic system,
type \ref{symhyp}.
In this section, we show that type \ref{symhyp} is obtained
if we modify the equations of motion.
% We add terms from constraint equations to
% equations of motion, which is the same technique used by ILR.
We begin by describing our approach without considering reality
conditions, but we will soon show that the triad reality
condition is required for making the characteristic matrix
Hermitian.
%  
We first prepare the constraints
(\ref{cham})(\ref{cg}) as
\begin{eqnarray}
{\cal C}_{H} &\cong&
i\epsilon^{ab}{}_{c} \, \tilde{E}^i_a \tilde{E}^j_b
\partial_i{\cal A}^c_j
=
i\epsilon^{dc}{}_{b} \, \tilde{E}^l_d \tilde{E}^j_c
(\partial_l{\cal A}^b_j)
=
i\epsilon_b{}^{cd} \, \tilde{E}^j_c \tilde{E}^l_d
(\partial_l{\cal A}^b_j),
\\
{\cal C}_{M k} &=&
F^a_{kj} \tilde{E}^j_a
\cong
(\partial_k{\cal A}^a_j\partial_j {\cal A}^a_k) \tilde{E}^j_a
=
[\delta^l_k \tilde{E}^j_b
+\delta^j_k \tilde{E}^l_b
](\partial_l{\cal A}^b_j),
\\
{\cal C}_{Ga} &=&
{\cal D}_i \tilde{E}^i_a
\cong
\partial_l\tilde{E}^l_a.
\end{eqnarray}
We apply the same technique as used by ILR to modify the
equation of motion of $\tilde{E}^i_a$ and ${\cal A}^a_i$; by
adding the constraints which weakly produce
${\cal C}_{H} \approx 0, \,
{\cal C}_{M k} \approx 0,$ and
${\cal C}_{Ga} \approx 0$. (Indeed, this technique has also been used
for constructing symmetric hyperbolic systems for the original
Einstein equations \cite{CBY,FR96}.)
We also assume the triad lapse ${\cal A}^a_0$ is
\begin{equation}
\partial_i{\cal A}^a_0 \cong
T^{lab} {}_{ij} \, \partial_l \tilde{E}^j_b
+S^{la} {}_{bi} {}^j \, \partial_l {\cal A}^b_j, \label{A0katei}
\end{equation}
where $T$ and $S$ are parameters which do not include derivatives
of the fundamental variables. This assumption is general for our
purpose of studying the principal part of the system.
One natural way to construct a symmetric hyperbolic system is
to keep $B=C=0$ and modify the $A$ and $D$ terms in (\ref{matrixform}),
so that we modify (\ref{eqE}) using ${\cal C}_{G}$, and
modify (\ref{eqA}) using ${\cal C}_{H}$ and ${\cal C}_{M}$.
That is, we add the following terms to the equations of motion:
\begin{eqnarray}
%\lefteqn{
\mbox{modifying term for }\partial_t \tilde{E}^i_a %}
%\nonumber \\
&=&
P^i{}_{ab} \, {\cal C}_G^b
\cong
P^i{}_a{}^b \, \partial_j\tilde{E}^j_b
=
(P^i{}_a{}^b \, \delta^l_j)(\partial_l\tilde{E}^j_b),
\label{eqE2}
\\
%\lefteqn{
\mbox{modifying term for }
\partial_t {\cal A}^a_i %}
%\nonumber \\
&=&
{\cal D}_i{\cal A}^a_0
+Q^a_i {\cal C}_H
+R_i{}^{ja} \, {\cal C}_{Mj}
\nonumber \\ &\cong&
T^{lab} {}_{ij} \, \partial_l \tilde{E}^j_b
+S^{la} {}_{bi} {}^j \, \partial_l {\cal A}^b_j
i Q^a_i \epsilon_b{}^{cd} \,
\tilde{E}^j_c \tilde{E}^l_d (\partial_l{\cal A}^b_j)
+R_i{}^{ka} [\delta^l_k \tilde{E}^j_b
+\delta^j_k \tilde{E}^l_b ]
\partial_l{\cal A}^b_j
\nonumber\\&\cong&
[
S^{la} {}_{bi} {}^j
iQ^a_i \epsilon_b{}^{cd} \, \tilde{E}^j_c \tilde{E}^l_d
R_i{}^{la} \, \tilde{E}^j_b
+R_i{}^{ja} \, \tilde{E}^l_b
](\partial_l{\cal A}^b_j)
+T^{lab} {}_{ij} \, \partial_l \tilde{E}^j_b, \label{eqA2}
\end{eqnarray}
where $P, Q$, and $R$ are parameters and will be
fixed later.
In Appendix \ref{appA2}, we show that the modifications to
the offdiagonal blocks $B$ and $C$, i.e.
modifying (\ref{eqE}) using ${\cal C}_{H}$ and ${\cal C}_{M}$ and
modify (\ref{eqA}) using ${\cal C}_{G}$,
will not affect the final conclusion at all.
Note that
we truncated ${\cal A}^a_0$ in (\ref{eqE2}),
while it remains in (\ref{eqA2}), since
only the derivative of ${\cal A}^a_0$ effects the principal
part of the system. The terms in (\ref{matrixform}) become
\begin{eqnarray}
A^{labij}&=&
i\epsilon^{bca}
\null \! \mathop {\vphantom {N}\smash N}\limits ^{}_{^\sim}\!\null
\gamma^{lj}\tilde{E}^i_c
i\epsilon^{cba}
\null \! \mathop {\vphantom {N}\smash N}\limits ^{}_{^\sim}\!\null
\tilde{E}^l_c \gamma^{ij}
+N^l\gamma^{ij} \delta^{ab}
N^i\gamma^{lj} \delta^{ab}
+P^{iab} \gamma^{lj},
\\
B^{labij}&=&0,
\\
C^{labij}&=&
T^{labij},
\\
D^{labij}&=&
i \epsilon^{abc}
\null \! \mathop {\vphantom {N}\smash N}\limits ^{}_{^\sim}\!\null
\tilde{E}^j_c \gamma^{li}
i \epsilon^{abc}
\null \! \mathop {\vphantom {N}\smash N}\limits ^{}_{^\sim}\!\null
\tilde{E}^l_c \gamma^{ji}
+N^l \delta^{ab} \gamma^{ji}
N^j \delta^{ab} \gamma^{li}
+S^{labij}
\nonumber \\ &&
iQ^{ai} \epsilon^{bcd} \tilde{E}^j_c \tilde{E}^l_d
R^{ila} \tilde{E}^{jb}
+R^{ija} \tilde{E}^{lb}. \label{D}
\end{eqnarray}
The condition (\ref{cond3}) immediately shows $T^{labij}=0$.
The condition (\ref{cond1}) is
written as
\begin{eqnarray}
0&=&
i\ep^{abc} \ut N \gamma^{lj} \dtri^i_c
+i\ep^{abc} \ut N \gamma^{li} \bar{\dtri}{}^j_c
2\ep^{abc} \ut N \gamma^{ij} \, \im(\dtri^l_c)
\nonumber \\&&
N^i\gamma^{lj} \delta^{ab}
+N^j\gamma^{li} \delta^{ab}
+P^{iab} \gamma^{lj}
\bar{P}^{jba} \gamma^{li} := \dagger^{labij}. \label{aa}
\end{eqnarray}
By contracting $\dagger^l{}_{ab}{}^{ij}$,
we get
$\re(\ep_{abc}\dagger^{labik}\gamma_{li}
2\ep_{abc}\dagger^{kabij}\gamma_{ij})
=20 \, \ut N \, \im(\dtri^k_c)$.
This suggests that we should impose $ \im( \tilde{E}^l_c) =0$,
in order to get $\dagger^l{}_{ab}{}^{ij}=0$.
This means that the triad reality condition is required
for making the characteristic matrix Hermitian.
%Therefore, we
%Because the third term in the righthandside
%cannot be eliminated using $P$, we
%assume the triad reality condition
%which requires the secondary condition,
%in the next subsection \S \ref{sym}.
\subsection{Under triad reality condition (system IIIa)}
\label{sym}
In this subsection,
we assume the triad reality condition hereafter.
In order to be consistent with the secondary triad reality condition
(\ref{sreality2final2}) during time evolution, and in order to
avoid the system becoming second order, we
need to specify the lapse function as $\partial_i N=0$.
This lapse condition reduces to
\bear
\re(\CA^a_0)&=&
N^{i} \, \re({\cal A}^a_i),
\label{A0}
\\
\ptl_i \, \re(\CA^a_0)&\cong&
N^j\ptl_i \, \re({\cal A}^a_j).
\label{A0bibun}
\enar
By comparing these with the real and
imaginary components of (\ref{A0katei}), i.e.,
\bear
%\partial_i{\cal A}^a_0 &\cong&
%+S^{la} {}_{bi} {}^j \partial_l {\cal A}^b_j
%\\
\partial_i \, \re({\cal A}^a_0) &\cong&
\re(S^{la} {}_{bi} {}^j) \, \partial_l \, \re({\cal A}^b_j)
\im(S^{la} {}_{bi} {}^j) \, \partial_l \, \im({\cal A}^b_j),
\label{A0re}
\\
\partial_i \, \im({\cal A}^a_0) &\cong&
\im(S^{la} {}_{bi} {}^j) \, \partial_l \, \re({\cal A}^b_j)
+\re(S^{la} {}_{bi} {}^j) \, \partial_l \, \im({\cal A}^b_j),
\label{A0im}
\enar
we obtain
\beas
\re(S^{la} {}_{bi} {}^j)=N^j \delta^l_i \, \delta^{a}_b
\qquad \mbox{\rm and} \qquad
\im(S^{la} {}_{bi} {}^j)=0.
\enas
Thus $S$ is determined as
\be
S^{la} {}_{bi} {}^j=N^j \delta^l_i \, \delta^{a}_b.
\label{S}
\en
This value of $S$, and ${T}=0$,
determine the form of the triad lapse as
\begin{equation}
{\cal A}^a_0 = {\cal A}^a_i N^i+\mbox{nondynamical~terms}.
\label{shifttriadrelation}
\end{equation}
ILR do not discuss consistency of their system with the
reality condition (especially with the secondary reality condition).
However, since ILR assume ${\cal A}^a_0 = {\cal A}^a_iN^i$,
we think that ILR also need to impose a similar restricted lapse
condition in order to preserve reality of their system.
By decomposing $\dagger$, that is (\ref{cond1}), into its
real and imaginary parts, we
get
\beas
0
&=&
N^i\gamma^{lj} \delta^{ab}
+N^j\gamma^{li} \delta^{ba}
+ \gamma^{lj} \, \re(P)^{iab}
 \gamma^{li} \, \re(P)^{jba},
\\
0
&=&
\epsilon^{bca} \ut N
\gamma^{lj}\tilde{E}^i_c
\epsilon^{acb} \ut N
\gamma^{li}\tilde{E}^j_c
+ \gamma^{lj} \, \im(P)^{iab}
+ \gamma^{li} \, \im(P)^{jba}.
\enas
By multiplying $\gamma_{li}$ to these and
taking symmetric and antisymmetric components on the indices $ab$,
we have
\beas
0
&=&
2 N^j\delta^{(ba)}
+\re(P)^{j(ab)}
3\re(P)^{j(ba)}
=
2 N^j\delta^{ba}
2\re(P)^{j(ab)},
\\
0
&=&
2 N^j\delta^{[ba]}
+\re(P)^{j[ab]}
3\re(P)^{j[ba]}
=
4\re(P)^{j[ab]},
\\
0
&=&
\im(P)^{j(ab)}
+3\im(P)^{j(ba)}
=
4\im(P)^{j(ab)},
\\
0
&=&
2\epsilon^{acb}\ut N \tilde{E}^j_c
+\im(P)^{j[ab]}
+3\im(P)^{j[ba]}
=
2\epsilon^{acb}\ut N \tilde{E}^j_c
2\im(P)^{j[ab]}.
\enas
These imply
\begin{equation}
P^{iab}=
N^i \delta^{ab}+i\ut N \epsilon^{abc}\tilde{E}^i_c.
\label{paraP}
\end{equation}
Our task is finished when we specify the parameters
$Q$ and $R$.
By substituting (\ref{S}) into (\ref{D}),
the condition (\ref{cond2})
becomes
\begin{eqnarray}
0
&=&
i \epsilon^{abc} \ut N \dtri^j_c \gamma^{li}
+i \epsilon^{bac} \ut N \dtri^i_c \gamma^{lj}
iQ^{ai} \epsilon^{bcd} \tilde{E}^j_c \tilde{E}^l_d
i\bar{Q}^{bj} \epsilon^{acd} \tilde{E}^i_c \tilde{E}^l_d
R^{ila} \tilde{E}^{jb}
+R^{ija} \tilde{E}^{lb} \nonumber \\ &&
+\bar{R}^{jlb} \tilde{E}^{ia}
\bar{R}^{jib} \tilde{E}^{la}.
\label{dd}
\end{eqnarray}
We found that a combination of the choice
\begin{eqnarray}
Q^{ai}&=&
\den^{2}
\null \! \mathop {\vphantom {N}\smash N}\limits ^{}_{^\sim}\!\null
\tilde{E}^{ia},
\mbox{~and~}
R^{ila}=
i \den^{2}
\null \! \mathop {\vphantom {N}\smash N}\limits ^{}_{^\sim}\!\null
\epsilon^{acd} \tilde{E}^i_d \tilde{E}^l_c,
\label{paraQR}
\end{eqnarray}
satisfies the condition (\ref{dd}).
We show in Appendix \ref{appA1}
that this pair of
$Q$ and $R$ satisfies (\ref{dd})
and that this choice is unique.
The final equations of motion are
\bear
A^{labij}&=&
i\ep^{abc} \ut N
\dtri^l_c \gamma^{ij}
+N^l\gamma^{ij} \delta^{ab},
\label{fmA}
\\
B^{labij}&=&C^{labij}=0,
\label{fmBC}
\\
D^{labij}&=&
i\ut N(\ep^{abc} \dtri^j_c \gamma^{li}
 \ep^{abc} \dtri^l_c \gamma^{ji} \nonumber \\ &~&
\den^{2} \dtri^{ia} \ep^{bcd} \dtri^j_c \dtri^l_d
\den^{2}\ep^{acd} \dtri^i_d \dtri^l_c \dtri^{jb}
+\den^{2}
\ep^{acd} \dtri^i_d \dtri^j_c \dtri^{lb}
)
+N^l \delta^{ab} \gamma^{ij}.
\label{fmD}
\enar
%We note that in the modifying process of characteristic
%matrix (\ref{fmA}),(\ref{fmBC}),(\ref{fmD})
%with parameter (\ref{paraP}) and (\ref{paraQR}),
%we do not use any facts from imposing triad reality
%condition.
To summarize, we obtain a symmetric hyperbolic system,
type \ref{symhyp}
by modifying the equations of motion,
restricting the gauge to: $\CA^a_0=\CA^a_iN^i$, $\ptl_i N=0$,
and assuming the triad reality condition.
We name this set {\it system IIIa}.
%The characteristic speeds of this system
The eigenvalues of this system are
%are given by finding eigenvalues
%of the characteristic matrix $A$ of (\ref{def}).
%Since $A$ is a Hermitian, eigenvalues of $A$ are all real.
%Then it is again clear that this system is symmetric hyperbolic.
%Actually the eigenvalues of the $18 \times 18$ matrix $A^l$
%for $x^l$direction are:
\be
N^l ~\mbox{(multiplicity = 6)}, \qquad
N^l \pm \sqrt{\gamma^{ll}} N ~\mbox{(5~each)} \qquad
\mbox{and} \qquad
N^l \pm 3\sqrt{\gamma^{ll}} N~\mbox{(1~each)}.
\label{symhypIIIa_eigen}
\en
These speeds are again independent of the choice of (real) triad.
\subsection{Under metric reality condition (system IIb)}
Using this technique, we can also construct another example of
diagonalizable hyperbolic system.
Since the parameters $S$ and $T$ specify {\it triad
lapse}, a gauge variable for time evolutions,
it is possible to change our interpretation
that we take the evolution of the system within the
{\it metric} reality condition.
Of course, the characteristic matrix is no longer Hermitian.
{}From the fact that
we do not use the triad reality condition in the
process of modifying the characteristic matrix using parameters
%$P, Q$ and $R$
(\ref{paraP}), (\ref{paraQR})
nor in the process of
deriving the eigenvalues, this system has the same components
in its characteristic matrix and has the same eigenvalues.
The process of examining diagonalizability is
independent of the reality conditions. Therefore
this system is classified as a diagonalizable hyperbolic
system, type \ref{diaghyp}.
To summarize,
we gain another diagonalizable hyperbolic system
by modifying the equations of motion using terms from constraint
equations, with characteristic matrix (\ref{fmA})(\ref{fmD})
under metric reality condition. The eigenvalues are
(\ref{symhypIIIa_eigen}), and this system is restricted only by
a condition on triad lapse, $\CA^a_0=\CA^a_iN^i$, and not on
lapse and shift vector like {\it system IIa}. We call this
system {\it system IIb}.
%=================================================== section 6
\setcounter{section}{5}
\section{Discussion} \label{sec:disc}
%=================================================== section 6
We have constructed several hyperbolic systems
based on the Ashtekar formulation of general relativity,
together with discussions of the required gauge conditions and
reality conditions. We summarize their features in Table
\ref{thetable}.
\begin{table}[p]
%\begin{table}
%\begin{center}
\begin{tabular}{cccccccc}
%\hline
system & Eqs of & reality & gauge conditions &
first & all real & diagonal & sym.
\\
& motion & condition & required &
order & eigenvals & izable & matrix
\\
\hline \hline
{\it Ia} & original & metric &  & yes & yes & no & no
\\ \hline
{\it Ib} & original & triad &
$\CA^a_0=\CA^a_iN^i$, $\ptl_i N=0$
& yes & yes & no & no
\\ \hline
{\it IIa} & original & metric &
$N^l \neq 0, \, \pm N \sqrt{\gamma^{ll}}$ ($\gamma^{ll}\neq 0$)
& yes & yes & yes & no
\\ \hline
{\it IIb} & modified
& metric & $\CA^a_0=\CA^a_iN^i$ & yes & yes & yes & no
\\ \hline
{\it IIIa} & modified & triad & $\CA^a_0=\CA^a_iN^i$, \, $\ptl_i N=0$
& yes & yes & yes & yes
\\ %\hline
\end{tabular}
\caption{List of obtained hyperbolic systems. The system {\it I, II}
and {\it III} denote weakly hyperbolic, diagonalizable
hyperbolic and symmetric hyperbolic systems, respectively. }
%\end{center}
\label{thetable}
\end{table}
The original dynamical equations in the Ashtekar formulation
are classified as a weakly hyperbolic system.
If we further assume a set of gauge conditions or reality
conditions or both, then the system can be either
a diagonalizable or a symmetric
hyperbolic system.
We think such a restriction process
%towards a symmetric hyperbolic system
helps in understanding the structure of this dynamical system,
and also that of the original Einstein equations.
{}From the point of view of numerical applications,
weakly and diagonalizable hyperbolic
systems are still good candidates to describe the spacetime dynamics
since they have much more gauge freedom than the
obtained symmetric hyperbolic system.
The symmetric hyperbolic system we obtained, is constructed by
modifying the righthandside of the dynamical equations using
appropriate combinations of the
constraint equations. This is a modification of
somewhat popular technique used also
by Iriondo, Leguizam\'on and Reula.
We exhibited the process of determining coefficients,
%in more detail than our previous paper,
showing how uniquely they are determined
(cf Appendix \ref{appA}).
In result,
this symmetric hyperbolic formulation requires a triad reality condition,
which we suspect that Iriondo {\it et al} implicitly
assumed in their system.
As we demonstrated in \S \ref{sec5}, in order to keep the system first
order, and to be consistent with the secondary triad reality
condition, the lapse function is strongly restricted in form;
it must be constant.
The shift vectors and triad lapse ${\cal A}^a_0$ should have the
relation
(\ref{shifttriadrelation}). This can be interpreted as the shift
being free and the triad lapse determined.
This gauge restriction sounds tight, but this
arises from our general assumption of (\ref{A0katei}).
ILR propose to use the internal rotation to reduce this reality
constraint, however this proposal does not work in our notation
(see Appendices \ref{appB} and \ref{appC}).
There might be a possibility to improve the situation
by renormalizing the shift and triad lapse terms into
the lefthandside of the
equations of motion like the case of
general relativity \cite{CBY}.
Or this might be
because our system is constituted by Ashtekar's original
variables.
We are now trying to relax this gauge restriction and/or
to simplify the characteristic speeds
by other gauge choices and also
by introducing new dynamical variables.
This effort will be reported elsewhere.
%
%
\vspace{0.4cm}
We thank John Baker for his useful comments on the initial draft.
We thank Abhay Ashtekar for his comments on our draft.
We also thank Matt Visser for careful reading the manuscript.
A part of this work was done when HS was at Dept. of Physics,
Washington University, St. Louis, Missouri.
HS was partially supported by NSF PHYS 9600049, 9600507,
and NASA NCCS 5153 when he was at WashU.
HS was supported by the Japan Society for the Promotion of Science.
%
% APPENDIX
%
%\newpage
\appendix
%
\section{Detail processes of deriving the symmetric hyperbolic
system IIIa}
\label{appA}
%
In this Appendix, we show several detail calculations
for obtaining the symmetric hyperbolic system {\it IIIa}.
%
\subsection{Determining $Q$ and $R$}\label{appA1}
%
We show here that
the choice $Q$ and $R$ of (\ref{paraQR})
satisfies (\ref{dd}). That is, the
final $D$ in (\ref{fmD}) satisfies (\ref{cond2}),
and that this choice $Q$ and $R$ is unique.
First we show that
$D$ in (\ref{fmD}) satisfies Hermiticity, (\ref{cond2}).
{}From the direct calculation, we get
\beas
D^{labij}

\bar{D}^{lbaji}
&=&
i \, \ut N (
\ep^{abc} \dtri^j_c \gamma^{li}
\ep^{abc} \dtri^l_c \gamma^{ji}
\den^{2}\ep^{bcd} \dtri^{ia}\dtri^j_c\dtri^l_d
+\den^{2}\ep^{acd} \dtri^{jb}\dtri^i_c\dtri^l_d
\den^{2}\ep^{acd} \dtri^{lb}\dtri^i_c\dtri^j_d )
\\&&
+i \, \ut N (
\ep^{bac} \dtri^i_c \gamma^{lj}
\ep^{bac} \dtri^l_c \gamma^{ij}
\den^{2}\ep^{acd} \dtri^{jb}\dtri^i_c\dtri^l_d
+\den^{2}\ep^{bcd} \dtri^{ia}\dtri^j_c\dtri^l_d
\den^{2}\ep^{bcd} \dtri^{la}\dtri^j_c\dtri^i_d )
\\&=&
i \, \ut N (
\ep^{abc} \dtri^j_c \gamma^{li}
\ep^{abc} \dtri^i_c \gamma^{lj}
\den^{2}\ep^{acd} \dtri^{lb}\dtri^i_c\dtri^j_d
\den^{2}\ep^{bcd} \dtri^{la}\dtri^j_c\dtri^i_d
) =: i \, \ut N \dagger^{labij}.
\enas
(This $\dagger^{labij}$ definition is used only within
this subsection \ref{appA1}.)
Hermiticity,
$\dagger^{labij}=0$, can be shown from the fact
\beas
2\dagger^{l(ab)ij}
&=&
\den^{2}\ep^{acd} \dtri^{lb}\dtri^i_c\dtri^j_d
\den^{2}\ep^{acd} \dtri^{lb}\dtri^j_c\dtri^i_d
\den^{2}\ep^{bcd} \dtri^{la}\dtri^j_c\dtri^i_d
\den^{2}\ep^{bcd} \dtri^{la}\dtri^i_c\dtri^j_d
=0,
\enas
and its antisymmetric part $\dagger^{l[ab]ij}=0$, which is
derived from
\beas
\ep_{abe}\dagger^{labij}
&=&
\ep_{abe}\ep^{abc} \dtri^j_c \gamma^{li}
\ep_{abe}\ep^{abc} \dtri^i_c \gamma^{lj}
\den^{2}\ep_{abe}\ep^{acd} \dtri^{lb}\dtri^i_c\dtri^j_d
\den^{2}\ep_{bea}\ep^{bcd} \dtri^{la}\dtri^j_c\dtri^i_d
\\&=&
2\dtri^j_e \gamma^{li}
2\dtri^i_e \gamma^{lj}
\den^{2}\dtri^{lb}\dtri^i_b\dtri^j_e
+\den^{2}\dtri^{lb}\dtri^i_e\dtri^j_b
\den^{2}\dtri^{la}\dtri^j_e\dtri^i_a
+\den^{2}\dtri^{la}\dtri^j_a\dtri^i_e
\\&=&
2\dtri^j_e \gamma^{li}
2\dtri^i_e \gamma^{lj}
\dtri^j_e \gamma^{il}
+\dtri^i_e \gamma^{lj}
\dtri^j_e \gamma^{li}
+\dtri^i_e \gamma^{lj}
=0.
\enas
Next we show that
the choice $Q$ and $R$ of (\ref{paraQR})
is unique in order to satisfy (\ref{dd}).
Suppose we have two pairs of ($Q, R$), say
($Q_1, R_1$) and ($Q_2, R_2$), as solutions of (\ref{dd}).
Then the pair ($Q_1Q_2, R_1R_2$) should satisfy
a truncated part of (\ref{dd}),
\begin{eqnarray}
\ddagger^{labij}
:=
i \, Q^{ai} \ep^{bcd} \dtri^j_c \dtri^l_d
i \, \bar{Q}^{bj} \ep^{acd} \dtri^i_c \dtri^l_d
R^{ila} \dtri^{jb}
+R^{ija} \dtri^{lb}
+\bar{R}^{jlb} \dtri^{ia}
\bar{R}^{jib} \dtri^{la}=0.
\label{katei}
\end{eqnarray}
Now we show that the equation $\ddagger^{labij}=0$ has only the
trivial solution $Q=R=0$.
By preparing
\bear
\ddagger^{labij}\gamma_{li}&=&
% iQ^{ai} \ep^{bcd} \dtri^j_c \dtri^d_i
% R^{ila}\gamma_{li} \dtri^{jb}
% +R^{ija} \dtri^b_i
% +\bar{R}^{jlb} \dtri^a_l
% \bar{R}^{jib} \dtri^a_i
% \nonumber \\&=&
iQ^{ai} \ep^{bcd} \dtri^j_c \dtri_{di}
R^{ila}\gamma_{li} \dtri^{jb}
+R^{ija} \dtri^b_i,
\label{1}
\\
\ddagger^{labij}\gamma_{li}\tilde{E}_{jb}
&=&
3\den^2 R^{ila}\gamma_{li}
+\den^2 R^{ija} \gamma_{ij}
=
2\den^2 R^{ija}\gamma_{ij},
\nonumber
\enar
we get $
R^{ija}\gamma_{ij}=0
$. By substituting this into (\ref{1}), we can express $R$ by $Q$ as
\be
%0&=&
%iQ^{ai} \ep^{bcd} \dtri^j_c \dtri^d_i
%+R^{ija} \dtri^b_i
%\nonumber\\
%R^{ija} \dtri^b_i
%&=&
%iQ^{ai} \ep^{bcd} \dtri^j_c \dtri^d_i
%\nonumber\\
%R^{ija} \dtri^b_i \dtri^k_b
%&=&
%iQ^{ai} \ep^{bcd} \dtri^j_c \dtri^d_i\dtri^k_b
%\nonumber\\
%\den^2 R^{kja}
%&=&
%iQ^{ai} \ep^{bcd} \dtri^j_c \dtri^d_i\dtri^k_b
%\nonumber\\
R^{ija}
=
i \, \den^{2} Q^{ak} \ep^{bcd} \dtri^j_c \tilde{E}_{kd}\dtri^i_b
=
i \, Q^{ak} \tilde{\ep}^{ij}{}_{k}.
\label{RQ}
\en
%
Therefore (\ref{katei}) becomes
%\bear
$$
\ddagger^{labij}
=
%&=&
% iQ^{ai} \ep^{bcd} \dtri^j_c \dtri^l_d
% i\bar{Q}^{bj} \ep^{acd} \dtri^i_c \dtri^l_d
% (iQ^{ak} \tilde{\ep}^{il}{}_{k}) \dtri^{jb}
% +(iQ^{ak} \tilde{\ep}^{ij}{}_{k}) \dtri^{lb}
% +(i\bar{Q}^{bk} \tilde{\ep}^{jl}{}_{k}) \dtri^{ia}
% (i\bar{Q}^{bk} \tilde{\ep}^{ji}{}_{k}) \dtri^{la}
% \nonumber \\&=&
i \, Q^{ai} \ep^{bcd} \dtri^j_c \dtri^l_d
i \, \bar{Q}^{bj} \ep^{acd} \dtri^i_c \dtri^l_d
i \, Q^{ak} \tilde{\ep}^{il}{}_{k} \, \dtri^{jb}
+i \, Q^{ak} \tilde{\ep}^{ij}{}_{k} \, \dtri^{lb}
i \, \bar{Q}^{bk} \tilde{\ep}^{jl}{}_{k} \, \dtri^{ia}
+i \, \bar{Q}^{bk} \tilde{\ep}^{ji}{}_{k} \, \dtri^{la}.
%\enar
$$
{}From this equation, we get the following contracted relations:
\bear
\den^{2}\ddagger^{labij}
&=&
% iQ^{ai} \ep^{bcd} E^j_c E^l_d
% i\bar{Q}^{bj} \ep^{acd} E^i_c E^l_d
% iQ^{ak} \ep^{il}{}_{k} E^{jb}
% +iQ^{ak} \ep^{ij}{}_{k} E^{lb}
% i\bar{Q}^{bk} \ep^{jl}{}_{k} E^{ia}
% +i\bar{Q}^{bk} \ep^{ji}{}_{k} E^{la} \nonumber
% \\&=&
i \, Q^{ai} \ep^{bjl}
i \, \bar{Q}^{bj} \ep^{ail}
i \, Q^{ak} \ep^{il}{}_{k} \, E^{jb}
+i \, Q^{ak} \ep^{ij}{}_{k} \, E^{lb}
i \, \bar{Q}^{bk} \ep^{jl}{}_{k} \, E^{ia}
+i \, \bar{Q}^{bk} \ep^{ji}{}_{k} \, E^{la},
\nonumber
\\
\den^{2}\ddagger^{labij}E_{ia}
&=&
% iQ^a{}_a \ep^{bjl}
% iQ^{ak} \ep^{al}{}_{k} \, E^{jb}
% +iQ^{ak} \ep^{aj}{}_{k} \, E^{lb}
% 3i\bar{Q}^{bk} \ep^{jl}{}_{k}
% +i\bar{Q}^{bk} \ep^{jl}{}_{k}
% \nonumber\\&=&
i \, Q^a{}_a \ep^{bjl}
+2i \, Q^{ak} \ep_a{}^{[j}{}_{k} \, E^{l]b}
2i \, \bar{Q}^{bk} \ep^{jl}{}_{k},
\nonumber\\
\den^{2}\ddagger^{labij}E_{ia}\ep_{ljc}
&=&
% +iQ^a{}_a \ep^{ljb} \ep_{ljc}
% +2iQ^{ak} E^{lb} \ep^j {}_{ka} \ep_{jcl}
% +2i\bar{Q}^{bk} \ep^{lj}{}_{k} \ep_{ljc}
% \nonumber\\&=&
% +2iQ^a{}_a \delta^b_c
% +2iQ^{lc} E^{lb}
% 2iQ^{cl} E^{lb}
% +4i\bar{Q}^{bc}
% =
2i \, Q^a{}_a \delta^b_c
+2i \, Q^{bc}
2i \, Q^{cb}
+4i \, \bar{Q}^{bc},
\label{ljc}
\\
\den^{2}\ddagger^{labij}E_{ia}\ep_{ljc}\delta^c_b
&=&
% +6iQ^a{}_a
% +2iQ^a{}_a
% 2iQ^a{}_a
% +4i\bar{Q}^{aa}
% =
6i \, Q^a{}_a
+4i \, \bar{Q}^a{}_a
%\nonumber\\&=&
% =
% +6i[\re(Q^a{}_a)+i\im(Q^a{}_a)]
% +4i[\re(Q^a{}_a)i\im(Q^a{}_a)]
=
10 i \, \re(Q^a{}_a)
2 \im(Q^a{}_a).
\nonumber
\enar
where $Q^{ab}:=Q^{ai}E^b_i$
and $\ep^{bjl}:=\ep^{ijl}E^b_i$.
{}From the last one, we get $Q^a{}_a=0$.
By substituting this into
(\ref{ljc}), we get
\be
\den^{2}\ddagger^{labij} \, E_{ia} \ep_{lj}{}^c
=
2i \, Q^{bc}
2i \, Q^{cb}
+4i \, \bar{Q}^{bc}
=
4i \, Q^{[bc]}
+4i \, \bar{Q}^{bc}
\label{kore}
\en
The symmetric part of (\ref{kore})
%$
%\den^{2}\ddagger^{labij}E^a_i\ep_{ljc}(bc)
%=
%+4i\bar{Q}^{(bc)}
%$,
indicates
$Q^{(bc)}=0$,
%and also the antisymmetric part is
%pure imaginary
%expressed as
%\den^{2}\ddagger^{labij}E^a_i\ep_{ljc}[bc]
%&=&
%+2iQ^{[bc]}
%+4i\bar{Q}^{[bc]}
%=
%+2i[\re(Q^{[bc]})+i\im(Q^{[bc]})]
%+4i[\re(Q^{[bc]})i\im(Q^{[bc]})]
%\nonumber\\
%&=&
%8i\re(Q^{[bc]}),
%\nonumber
%\enas
%$ which
%gives
%$\re(Q^{[bc]})=0$.
and
\beas
\den^{2}\ddagger^{labij}E_{ja}E_{lb}
&=&
2i \, Q^{bc} \ep_{bc}{}^i
3i \, \bar{Q}^{bc} \ep_{bc}{}^i
=
\re(Q^{bc}\ep_{bc}{}^i)
+5i \, \im(Q^{bc}\ep_{bc}{}^i)
\enas
gives us $Q^{bc}\ep_{bc}{}^i=Q^{[bc]}=0$.
Therefore
$Q^{bc}=Q^{ai}=0$ is determined uniquely.
{}From (\ref{RQ}),
we also get $R^{ija}=0$.
%
\subsection{Modifications to offdiagonal blocks}\label{appA2}
%
On the starting point of the modifications to the equations of
motions (\ref{eqE2}) and (\ref{eqA2}), we assumed that offdiagonal
terms keep vanishing. In this subsection, we
show that the modifications to
the offdiagonal blocks $B$ and $C$ in the matrix notation of
(\ref{matrixform}), i.e.
modifying (\ref{eqE}) using ${\cal C}_{H}$ and ${\cal C}_{M}$ and
modify (\ref{eqA}) using ${\cal C}_{G}$,
does not affect the final conclusion at all.
Suppose we have a symmetric hyperbolic system (\ref{fmA})(\ref{fmD}),
and suppose we additionally modify the equations of motion (\ref{eqE})
and (\ref{eqA}) as
\bear
\mbox{modifying term for }\partial_t \tilde{E}^i_a
&=&
G^i_a \CC_H +H^{ij}_a \CC_{Mj} \nonumber \\
&\cong&
G^i_a
(i\ep_b{}^{cd} \, \dtri^j_c \dtri^l_d) (\ptl_l\CA^b_j)
+
H^{ik}_a
(\delta^l_k \dtri^j_b
+\delta^j_k \dtri^l_b
)(\ptl_l\CA^b_j)
\nonumber
\\&=&
(iG^i_a\ep_b{}^{cd} \, \dtri^j_c \dtri^l_d
H^{il}_a \dtri^j_b
+H^{ij}_a \dtri^l_b ) (\ptl_l\CA^b_j),
\\
\mbox{modifying term for }\ptl_t \CA^a_i&=&
I^{ab}{}_i \, \CC_{Gb}
\cong
(I^{ab}{}_i \, \delta^l_j)
(\ptl_l\dtri^j_b),
\enar
where $G^i_a$, $H^{ij}_a$ and $I^{ab}{}_i$ are parameters to be determined.
In the matrix notation, these can be written as
\bear
B^l{}_{ab}{}^{ij} &=&
iG^i_a\ep_b{}^{cd} \, \dtri^j_c \dtri^l_d
H^{il}_a \dtri^j_b
+H^{ij}_a \dtri^l_b,
\\
C^{lab}{}_{ij}&=&
I^{ab}{}_i \, \delta^l_j.
\enar
The Hermitian condition (\ref{cond3}) becomes
\bear
0&=&
iG^i_a\ep_b{}^{cd} \, \dtri^j_c \dtri^l_d
H^{il}_a \dtri^j_b
+H^{ij}_a \dtri^l_b
\bar{I}_{ba}{}^j \, \gamma^{li} =: \dagger^l{}_{ab}{}^{ij} \label{daggerA2}
\enar
(We use this $\dagger^l{}_{ab}{}^{ij} $ definition only inside of this
subsection \ref{appA2}.)
If there exists a nontrivial combination of
$G^i_a$, $H^{ij}_a$ and $I^{ab}{}_i$
which satisfy this relation, then it will constitute
alternative symmetric hyperbolic system.
However, we see only the trivial solution is allowed
for (\ref{daggerA2})
as follows.
{}From the relations of
$\dagger^{kabij}\gamma_{ij}+\dagger^{labik}\gamma_{li}
=4\bar{I}^{bak}$,
we obtain $I^{ab}{}_i=0$. With this $I^{ab}{}_i=0$, we obtain
$\dagger^l{}_{ab}{}^{ij} \, \dtri^b_j=2\den^2 H^{il}_a$,
which determine $H^{ij}_a=0$.
Similarly, from $I^{ab}{}_i=0$ and $H^{ij}_a=0$,
we get
$\dagger^{l}{}_a{}^{bij} \, \ut\ep_{jlk} \dtri^k_b=6i\den^2 G^i_a$,
which determine $G^i_a=0$.
%================================================= Appendix B
\section{Internal rotation and Ashtekar equations}
\label{appB}
%================================================= Appendix B
In this Appendix,
we consider the effect of a $SO(3)$ rotation on the triad,
which corresponds to a $SU(2)$ rotation on the soldering form.
The equations that we derive here
will be applied in the discussion in Appendix \ref{appC}.
%========== B.1 ========== B.1 ========== B.1 ========== B.1
\subsection{Primary and secondary conditions of internal rotation}
The $SO(3)$ internal transformation only affects inner space, and
not the spacetime quantities. Let us write $U$ for such a rotation.
$U$ should satisfy the condition
\be
U^a{}_c \, U^{bc} = \delta^{ab}.
%,\quad \det U= \pm 1,
\label{unitary}
\en
This comes from the transformation of $\delta^{ab}$ to
$\delta^{\ast ab}:=U^a{}_c \, U^b{}_d \, \delta^{cd}$, which should satisfy
$\delta^{\ast ab}=\delta^{ab}$.
The determinant $\det U$ must be $\pm 1$, and
we choose $\det U=1$ for later convenience.
The transformation $\delta^a{}_b\to \delta^\ast{}^a{}_b$
is naturally defined by
$\delta^\ast{}^a{}_b:=U^a{}_c \, U_b{}^d \, \delta^c{}_d$.
{}From (\ref{unitary}), we get the fundamental relations:
$\delta^\ast{}^a{}_b=\delta^a{}_b$,
$\delta^\ast{}_{ab}=\delta_{ab}$,
and $\ep^\ast{}^{abc}=\ep^{abc}$.
%and other raised or lowered $\ep^{abc}$
%are transformed identically.
Now we define the transformation of the triad $E^i_a$
and of the inverse triad $E^a_i$ as
\bear
%E^i_a \to
E^\ast {}^i_a &:=& U_a{}^b \, E^i_b. \label{triadH} \\
%E^a_i \to
E^\ast {}^a_i &:=& U^a{}_b \, E^b_i.
\enar
The 3metric, $\gamma^{ij}$,
is preserved under this transformation, since
$\gamma^{ij}=E^i_a E^{ja}=E^\ast {}^i_a E^\ast {}^{ja}$.
We note that this secondary condition,
$\ptl_t \gamma^{ij}=\ptl_t (E^i_a E^{ja})=
\ptl_t (E^\ast {}^i_a E^\ast {}^{ja})$, will not give us
further conditions. This is equivalent with the time
derivative of (\ref{unitary}).
%========== B.2 ========== B.2 ========== B.2 ========== B.2
\subsection{Internal rotation of Ashtekar variables}
Using $\det U=1$,
the transformation of the densitized triad becomes
\bear
%\dtri^i_a &\to&
\dtri^\ast {}^i_a =
U_a{}^b \, \dtri^i_b,
\label{tansdtri}
\enar
and straightforward calculation shows
\bear
%\CA^a_i &\to&
\CA^\ast{}^a_i
=
U^a{}_b \, \CA^b_i
\itwo \ep^{ab}{}_{c} \, U_b{}^{d} \, (\ptl_i U^c{}_d),
\label{tansCA}
\enar
where we also note that
$
\omega^\ast {}^{0a}_i = U^a{}_b \, \omega^{0b}_i,
$ and $
\omega^\ast{}^{bc}_i =
U^a{}_e(\ep^e{}_{bc} \, \omega^{bc}_i)
 \ep^a{}_{bc}(\ptl_i U^{bd})U^c{}_d.
$
We remark that the second term in (\ref{tansCA}) arises because
$\CA^a_i$ includes the spatial derivative of the triad.
%{}From the straightforward calculation
% with help of identities (\ref{id71}) and (\ref{id72}),
The relations of
triad lapse and curvature 2form become
\bear
%\CA^a_0 &\to&
\CA^\ast {}^a_0 &=&
U^a{}_b \, \CA^b_0
\itwo \ep^{ab}{}_c \, U_b{}^d(\ptl_t U^c{}_d)
, \\
%F^a_{ij} &\to&
F^\ast {}^a_{ij} &=& U^a{}_b \, F^b_{ij},
\enar
% We remark that $F^a_{ij}$ includes
% the spatial derivative of triad, but extra terms offsets.
and constraints (\ref{cham})(\ref{cg}) are transformed into
\bear
%\CC_H &\to&
\CC^\ast {}_H &=& \CC_H, \\
%\CC_{Mi} &\to&
\CC^\ast {}_{Mi} &=& \CC_{Mi}, \\
%\CC_{Ga} &\to&
\CC^\ast {}_{Ga} &=& U_a{}^b \, \CC_{Gb}.
\enar
The Hilbert action (\ref{action}) will be preserved ($S^\ast=S$)
under $U$, which is demonstrated by the ``cancellation relation"
\be
(\ptl_t \CA^\ast{}^a_i)\dtri^\ast{}^i_a
+\CA^\ast{}^a_0 \, \CC^\ast{}_{Ga}
=
(\ptl_t \CA^a_i)\dtri^i_a
+\CA^a_0 \, \CC_{Ga}.
\en
Therefore the equations of motion for
$\dtri^\ast{}^i_a$ and $\CA^\ast{}^a_i$
are equivalent with the
original ones, (\ref{eqE}) and (\ref{eqA}), putting a
$\ast$ on all terms.
The secondary metric reality condition (\ref{wreality2final}),
$W^{ij}:=\re (\ep^{abc}\dtri^\ast{}^k_a
\dtri^\ast{}^{(i}_b \CD^\ast{}_k \dtri^\ast{}^{j)}_c)$,
retains its form,
\beas
W^\ast{}^{ij}=W^{ij},
\enas
while the
secondary triad reality condition
(\ref{sreality2final2}),
$Y^a:=\re(\CA^a_0)+\ptl_i(N) E^{ia}+N^{i}\re(\CA^a_i)$,
is transformed as
\bear
Y^\ast{}^a&=&
\re(U^a{}_b) Y^b
i\ptl_i(N) \re(U^a{}_b) \im(E^i_b)
+\im(U^a{}_b)
[\im(\CA^b_0)
\ptl_i(N) \im(E^{ib})
N^{i} \, \im(\CA^b_i)] \nonumber
\\&&
+\half N^{i} \ep^a{}_{bc} \, \im(U^{bd}) (\ptl_i \, \im(U^c{}_d)).
\label{YScond_viaU}
\enar
This equation has many unexpected terms, even if we assume the
triad reality, $\im(E^i_a)=0$, before the transformation.
To summarize, under triad transformations,
$\CA^a_i$, $\CA^a_0$, and $Y^a$ are not
transformed covariantly, while the
other variables are
transformed covariantly.
%========== B.3 ========== B.3 ========== B.3 ========== B.3
\subsection{Make triad real using internal rotation}\label{appB3}
Suppose all the variables satisfy the metric reality conditions, that
is, $\dtri^i_a$ satisfies $\im(\dtri^i_a\dtri^{ja})=0$.
Can we obtain the triad which satisfies the triad reality
condition, $\im(\dtri^\ast{}^i_a)=0$, by an internal rotation?
The answer is affirmative. However,
such a rotation $U$ must satisfy
\bear
0 =
\im(\dtri^\ast{}^i_a) =
\im(U_a{}^b \, \dtri^i_b) =
\re(U_a{}^b) \, \im(\dtri^i_b)
+\im(U_a{}^b) \, \re(\dtri^i_b),
\enar
and its secondary condition
\be
0 =
\im(\ptl_t \dtri^\ast{}^i_a) =
\im[(\ptl_t U_a{}^b)\dtri^i_b
+U_a{}^b \, (\ptl_t \dtri^i_b)].
\label{YScond_viaU2}
\en
The application of this technique will be discussed in
\S \ref{appC2}. Before ending this section, we remark
two points.
First,
$\CA^a_i$ is not transformed covariantly by this rotation $U$.
Second,
when we consider the evolution of $\dtri^\ast{}^i_a$,
the evolution should be consistent with the
secondary triad reality condition (\ref{sreality2final}).
%\newpage
%================================================= Appendix C
\section{Consideration of ILR's treatment of reality conditions}
\label{appC}
%================================================= Appendix C
The symmetric hyperbolic system (system {\it IIIa}) that we
obtained in \S \ref{sec5} is strictly restricted by the
triad reality condition.
ILR (in their second paper \cite{ILRsecond}) propose to use an
internal rotation to deconstrain this situation.
Here we comment on this possibility.
\subsection{Difference of definition of symmetric
hyperbolic system}
First of all, we should point out again that there is a fundamental
difference in the definition used to characterize the system
as {\it symmetric}.
As we discussed in \S \ref{sec:def}, we define symmetry
using the fact that
the characteristic matrix is Hermitian, while ILR
\cite{Iriondo,ILRsecond} define it when
the principal symbol of the system $iB^l{}_j{}^a k_a$
($i\cha^{l\beta}{}_\alpha k_l$ in our notation)
is antiHermitian.
We suspect that these two definitions are equivalent
when the vector $k_a$ ($k_l$ in our notation)
is arbitrary real.
Actually, ILR have advanced a suggestion
that our definition and
their `modern' version are equivalent.
The judgement which is conventional or not, however,
we would like to leave to the reader.
Concerning our definition of symmetric hyperbolicity,
we think that the readers can quite easily compare our system with
other proposed symmetric hyperbolic systems in general
relativity:
all eigenvalues (in the system we presented) are all realvalued,
while ILR's are all pure imaginary.
(Even if the distinction of real and pure imaginary is ignored,
the eigenvalues calculated by us (\ref{symhypIIIa_eigen})
and by ILR are different.)
We note that, in addition, this fundamental difference will
lead to different conclusions regarding the treatment of the
reality condition (see the proceeding discussion).
%However, ILR commented even the case of complex $k_a$, so that
%we think two definitions are different more than the notations.
%=================================================== section C2
\subsection{Can we obtain a symmetric hyperbolic system by internal
rotation?}
\label{appC2}
What ILR proposed is the following: Suppose the system satisfies
the reality condition on the metric, but not on the triad.
By using the freedom of making an internal rotation, we can transform
the soldering form to satisfy the triad reality condition, in such
a way it forms symmetric hyperbolic system.
(In their terminologies,
they seek a ``rotated" scalar product that is to
find a more general symmetrizer.)
Therefore we can remove
the additional constraints of the triad reality.
This procedure, however, includes changing inner product
of dynamical variables, which might cause the topology
of wellposedness of the initial value formulation to change.
Here, we examine whether such a redefinition of the inner product is
acceptable in our definition of symmetric hyperbolicity.
%First we comment that
%changing inner product from canonical one
%is not preferable because it causes
%changing topology of wellposedness of initial value formulation.
%Hereafter discussion is in supposition that
%changing inner product is recognized.
%Let's see whether this procedure works in our definition.
Suppose we have a system which
satisfies the constraints, and the metric reality condition,
but not the triad reality conditions. As we commented in
\S \ref{sec:ash},
metric reality will be preserved automatically by the dynamical
equations (\ref{matrixform}) and (\ref{fmA})(\ref{fmD}).
Now we apply a $SO(3)$ rotation
$E^i_a \to E^\ast {}^i_a:= U_a{}^b \, E^i_b$ to the system.
We summarized the transformations of Ashtekar's variables and
equations by $U$ in Appendix \ref{appB}. In the new variables
$(\dtri^\ast{}^i_a,\CA^\ast{}^a_i)$, transformed via $U$,
the equations of motions are written covariantly.
As discussed in \S \ref{appB3}, it is possible to construct the
real triad by using $U$. However, we always should verify the triad
reality condition, both its primary condition (\ref{sreality1}),
and its
secondary condition (\ref{sreality2}).
The latter is expressed as
(\ref{YScond_viaU}) or (\ref{YScond_viaU2}).
If we interpret this secondary condition as a restriction
on the gauge variables, lapse $N$, shift
$N^i$, and triad lapse $\CA^a_0$, then
we only need to solve the primary condition in order to obtain
triad reality on 3hypersurface.
This is indeed solvable. For example, ILR explain a way to get a real
triad using orthonormality of the basis
in their Appendix A in \cite{ILRsecond}.
Next, let us see whether a symmetric hyperbolic system is obtained
by the new pair of variables $(\dtri^\ast{}^i_a,\CA^\ast{}^a_i)$.
We define the equations of motion similarly as
\be
\partial_t \left[ \begin{array}{l}
\dtri^\ast{}^i_a \\
\CA^\ast{}^a_i
\end{array} \right] =
\left[ \begin{array}{cc}
A^\ast{}^l{}_a{}^{bi}{}_j & B^\ast{}^l{}_{ab}{}^{ij} \\
C^\ast{}^{lab}{}_{ij} & D^\ast{}^{la}{}_{bi}{}^j
\end{array} \right]
\partial_l
\left[ \begin{array}{l}
\dtri^\ast{}^j_b \\
\CA^\ast{}^b_j
\end{array} \right]
+ \mbox{terms~with~no~} \partial_l \dtri^\ast{}^j_b
\mbox{~nor~} \partial_l \CA^\ast{}^a_i.
\label{matrixform2}
\en
By applying the same modifications as those in
\S \ref{sec5}, we get
\bear
A^\ast{}^{labij}&=&
i\ep^{abc} \ut N
\dtri^\ast{}^l_c \gamma^{ij}
+N^l\gamma^{ij} \delta^{ab},
\label{fmA2}
\\
B^\ast{}^{labij}&=&C^\ast{}^{labij}=0,
\label{fmBC2}
\\
D^\ast{}^{labij}&=&
i\ut N(\ep^{abc} \dtri^\ast{}^j_c \gamma^{li}
 \ep^{abc} \dtri^\ast{}^l_c \gamma^{ji}
\nonumber \\ &~&
e^{2} \dtri^\ast{}^{ia}
\ep^{bcd} \dtri^\ast{}^j_c \dtri^\ast{}^l_d
e^{2}\ep^{acd} \dtri^\ast{}^i_d
\dtri^\ast{}^l_c \dtri^\ast{}^{jb}
+e^{2}
\ep^{acd} \dtri^\ast{}^i_d \dtri^\ast{}^j_c
\dtri^\ast{}^{lb}
)
+N^l \delta^{ab} \gamma^{ij}.
\label{fmD2}
\enar
These equations are related to (\ref{fmA})(\ref{fmD}). We note that,
in the modification here, we added the terms
$
(N^i \delta^{ab}+
i\ut N \epsilon^{abc}\dtri^\ast{}^i_c)
\CC^\ast_{Gb}$
coming from the terms of the gauge constraint. This corresponds to the
relation $A^\ast{}^{labij}=
U^a{}_c \, U^b{}_d \, A^{lcdij}$.
Equations (\ref{fmA2})(\ref{fmD2}) forms a Hermitian matrix
in the principal part of (\ref{matrixform2}), but it contradicts
the consistent evolution with triad reality.
That is, for example, the lefthandside of dynamical
equation $\ptl_t \dtri^\ast{}^i_a=\cdots$ [upper half of
(\ref{matrixform2})]
is realvalued since we impose $\im(\dtri^\ast{}^i_a)=0$, while in the
righthandside includes complex value in the nonprincipal
part. To explain this in another words, the system
(\ref{matrixform2})(\ref{fmD2}) will not preserve the triad
reality. Therefore we again need to control gauge variables through
the secondary triad reality condition, and this discussion again
returns the same gauge restrictions with those in \S \ref{sec5}.
We also point out that the inner product of the fundamental variables
in our notation does not form Hermitian like in the case of ILR.
The inner product before the rotation $U$ can be written
\be
\langle
(\dtri^i_a,\CA^a_i) (\dtri^i_a,\CA^a_i) \rangle
:=
\delta^{ab}\gamma_{ij}\dtri^i_a\bar{\dtri}{}^j_b
+\delta_{ab}\gamma^{ij}\CA^a_i\bar{\CA}{}^b_j,
\en
which is common to ours and ILR's, while after the rotation the
inner product becomes
\bear
\langle
(\dtri^\ast{}^i_a,\CA^\ast{}^a_i)
(\dtri^\ast{}^i_a,\CA^\ast{}^a_i) \rangle
&=&
U_c{}^a \, \bar{U}^{cb} \dtri^i_a \bar{\dtri}{}^j_b
+
U_c{}^a \, \bar{U}^{cb} \gamma^{ij}\CA^a_i \bar{\CA}{}^b_j
\nonumber \\
&&
\itwo\gamma^{ij} \left(
\ep_{agf}\bar{U}{}^{gh}(\ptl_j \bar{U}{}^f{}_h)
U^a{}_c \, \CA^c_i
+\ep^a{}_{ec} \, U^{ed}(\ptl_i U^c{}_d)
\bar{U}{}_{af} \, \bar{\CA}{}^f_j
\right)
\nonumber \\
&&
{1\over 4}U_e{}^d(\ptl_i U_{cd})
\bar{U}{}^{eh}(\ptl_j \bar{U}{}^c{}_h)
+{1\over 4}U_e{}^d(\ptl_i U_{cd})
\bar{U}{}^{ch}(\ptl_j \bar{U}{}^e{}_h),
\label{inner_product}
\enar
%which is not appropriate form to take as the basic inner product.
which is not Hermitian, and can not be used as
the inner product of the
original variable $(\dtri^i_a,\CA^a_i)$ as in the ILR's
proposal.
As the final remark, we would like to comment that both
the variables to evolve by the equations, and the variables used
to confirm the Hermiticity of the system
should be common
throughout all evolutions. Otherwise, we cannot apply the
energy inequality for the evolution of that system.
{}From this point of view, we think it necessary to consider
the secondary triad
reality condition throughout
evolution of this system.
To summarize, we tried to follow ILR's procedure to remove the
restriction of the triad reality condition in our system,
which casts on our definition of symmetric hyperbolicity,
and which is based on the fixed inner product as of its Hermitian
form.
We, however, see that ILR's procedure does not work in our system
since it requires the restriction of the secondary reality conditions
of the triad.
Therefore we conclude that we cannot deconstrain
restrictions any further.
%Using
%the internal rotation, we obtain a real triad
%for constructing symmetric hyperbolic equations
%at every time step within the real metric submanifold.
%We show the same scalar product in our
%version, (\ref{inner_product}), together with
%transformations of equations of motion, which is required in our
%definition of symmetric hyperbolicity.
%However, we see that ILR's procedure does not
%work as far as we are keeping the inner product as of
%its Hermitian form.
%Therefore we conclude that we cannot remove
%the restriction of triad secondary reality conditions
%as far as we cast on our definition of symmetric hyperbolicity.
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\end{document}