curvature

%LaTeX2.09
%Kiyoshi Shiraishi Aug. 1997
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\begin{document}

We assume the metric:
\bea
ds^2&=&-N^2 dt^2+\Lambda^2 (d\chi+\beta dt)^2+r^2 d\Omega^{2}_{2} \\
&=&\eta_{AB}e^A\otimes e^B,
\eea
where $A, B=0, 1, 2, 3$ and $\eta_{AB}=diag.(-1,1,1,1)$.
$N, \Lambda$ and $r$ are functions of $t$ and $\chi$. Here we can see
\bea
e^0&=&N dt, \\
e^1&=&\Lambda (d\chi +\beta dt), \\
e^a&=&r \tilde{e}^a ,
\eea
where $a=2, 3$ and $d\tilde{e}^a+\tilde{\omega}^{a}{}_{b}\wedge\tilde{e}^b=0$.

Taking the exterior derivative of these, we get
\bea
de^0&=&N'd\chi\wedge dt=\frac{N'}{N\Lambda}e^1\wedge e^0, \\
de^1&=&\dot{\Lambda}dt\wedge d\chi+\Lambda'd\chi\wedge \beta dt
       +\Lambda \beta'd\chi\wedge dt, \nn
    &=& \frac{(\dot{\Lambda}-(\Lambda\beta)')}{N\Lambda}e^0\wedge e^1 \\
de^a&=&\dot{r}dt\wedge \tilde{e}^a+r'd\chi\wedge \tilde{e}^a+rd\tilde{e}^a \nn
    &=&\frac{\dot{r}}{rN}e^0\wedge e^a+\frac{r'}{r\Lambda}e^1\wedge e^a-
       \frac{r'\beta}{rN}e^0\wedge e^a-\tilde{\omega}^{a}{}_{b}\wedge e^b .
\eea


Comparing these expressions with $de^A+\omega^{A}{}_{B}\wedge e^B=0$,
we find
\bea
\omega^{0}{}_{1}&=&\frac{N'}{N\Lambda}e^0-K^{\chi}_{\chi}e^1, \\
\omega^{0}{}_{a}&=&-K^{\theta}_{\theta}e^a, \\
\omega^{1}{}_{a}&=&-\frac{r'}{r\Lambda}e^a, \\
\omega^{a}{}_{b}&=&\tilde{\omega}^{a}{}_{b} ,
\eea
where
\bea
K^{\chi}_{\chi}&=&\frac{(-\dot{\Lambda}+(\Lambda\beta)')}{N\Lambda}, \\
K^{\theta}_{\theta}&=&\frac{-\dot{r}+r'\beta}{Nr}.
\eea

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Using these, we can calculate the curvature 2-form
\be
\Theta^{A}{}_{B}=d\omega^{A}{}_{B}+\omega^{A}{}_{C}\wedge\omega^{C}{}_{B}
\ee
as follows:
\bea
\Theta^{0}{}_{1}&=&d\omega^{0}{}_{1}+\omega^{0}{}_{a}\wedge\omega^{a}{}_{1}, 
\label{01}\\
\Theta^{0}{}_{a}&=&d\omega^{0}{}_{a}+\omega^{0}{}_{1}\wedge\omega^{1}{}_{a} 
                   +\omega^{0}{}_{b}\wedge\omega^{b}{}_{a}, \label{0a}\\
\Theta^{1}{}_{a}&=&d\omega^{1}{}_{a}+\omega^{1}{}_{0}\wedge\omega^{0}{}_{a} 
                   +\omega^{1}{}_{b}\wedge\omega^{b}{}_{a}, \label{1a}\\
\Theta^{a}{}_{b}&=&d\omega^{a}{}_{b}+\omega^{a}{}_{0}\wedge\omega^{0}{}_{b}
      +\omega^{a}{}_{1}\wedge\omega^{1}{}_{b}+
       \omega^{a}{}_{c}\wedge\omega^{c}{}_{b}. \label{ab}
\eea

{}From (\ref{01}), we obtain
\bea
\Theta^{0}{}_{1}&=&d\omega^{0}{}_{1} \nn
                &=&\left(\frac{N'}{N\Lambda}\right)'d\chi\wedge e^0
       +\frac{N'}{N\Lambda}de^0
       -\dot{K}^{\chi}_{\chi}dt\wedge \Lambda d\chi 
       -{(K^{\chi}_{\chi})}'d\chi\wedge \Lambda\beta dt
       -K^{\chi}_{\chi}de^1 \nn
       &=&\frac{1}{\Lambda}\left(\frac{N'}{N\Lambda}\right)'e^1\wedge e^0
       +\left(\frac{N'}{N\Lambda}\right)^2 e^1\wedge e^0 \nn
    & &-\frac{1}{N}\dot{K}^{\chi}_{\chi}e^0\wedge e^1
       -\frac{\beta}{N}{(K^{\chi}_{\chi})}'e^1\wedge e^0
       +\left(K^{\chi}_{\chi}\right)^2 e^0\wedge e^1 \nn
   &=&\left[-\frac{1}{\Lambda}\left(\frac{N'}{N\Lambda}\right)'
       -\left(\frac{N'}{N\Lambda}\right)^2  
       -\frac{1}{N}\dot{K}^{\chi}_{\chi}
       +\frac{\beta}{N}{(K^{\chi}_{\chi})}'
       +\left(K^{\chi}_{\chi}\right)^2  \right] e^0\wedge e^1.
\eea

{}From (\ref{0a}), we obtain
\bea
\Theta^{0}{}_{a}&=&d\omega^{0}{}_{a}+\omega^{0}{}_{1}\wedge\omega^{1}{}_{a} 
                   +\omega^{0}{}_{b}\wedge\omega^{b}{}_{a} \nn
                &=&-\dot{K}^{\theta}_{\theta}dt\wedge e^a
      -\left(K^{\theta}_{\theta}\right)'d\chi\wedge e^a
       -K^{\theta}_{\theta}de^a \nn
     & &+\left(\frac{N'}{N\Lambda}e^0-K^{\chi}_{\chi}e^1\right)\wedge
       \left(-\frac{r'}{r\Lambda}\right)e^a 
       +\left(-K^{\theta}_{\theta}e^b\right)\wedge 
      \tilde{\omega}^{b}{}_{a} \nn
       &=&-\frac{1}{N}\dot{K}^{\theta}_{\theta}e^0\wedge e^a
          -\left(K^{\theta}_{\theta}\right)'
          \left(\frac{1}{\Lambda}e^1-\frac{\beta}{N}e^0\right)\wedge e^a \nn
       & &-K^{\theta}_{\theta}\left(-K^{\theta}_{\theta}e^0\wedge e^a+
         \frac{r'}{r\Lambda}e^1\wedge e^a-\tilde{\omega}^{a}{}_{b}\wedge e^b
         \right) \nn
 & &+\left(\frac{N'}{N\Lambda}e^0-K^{\chi}_{\chi}e^1\right)\wedge
       \left(-\frac{r'}{r\Lambda}\right)e^a 
       +\left(-K^{\theta}_{\theta}e^b\right)\wedge 
      \tilde{\omega}^{b}{}_{a} \nn
    &=&\left[-\frac{1}{N}\dot{K}^{\theta}_{\theta}+
        \frac{\beta}{N}\left(K^{\theta}_{\theta}\right)'+
      \left(K^{\theta}_{\theta}\right)^2-
      \frac{r'}{r\Lambda}\frac{N'}{N\Lambda}\right]e^0\wedge e^a \nn
   & &+\left[-\frac{1}{\Lambda}\left(K^{\theta}_{\theta}\right)'
     -(K^{\theta}_{\theta}-K^{\chi}_{\chi})\frac{r'}{r\Lambda}
      \right]e^1\wedge e^a
\eea

{}From (\ref{1a}), we obtain
\bea
\Theta^{1}{}_{a}&=&d\omega^{1}{}_{a}+\omega^{1}{}_{0}\wedge\omega^{0}{}_{a} 
                   +\omega^{1}{}_{b}\wedge\omega^{b}{}_{a} \nn
       &=&-\left(\frac{r'}{r\Lambda}\right)^{.}dt\wedge e^a
         -\left(\frac{r'}{r\Lambda}\right)'d\chi\wedge e^a
         -\frac{r'}{r\Lambda}de^a \nn
       & &+\left(\frac{N'}{N\Lambda}e^0-K^{\chi}_{\chi}e^1\right)\wedge
            \left(-K^{\theta}_{\theta}e^a\right)+
          \left(-\frac{r'}{r\Lambda}e^b\right)\wedge \tilde{\omega}^{b}{}_{a} \nn
     &=&-\frac{1}{N}\left(\frac{r'}{r\Lambda}\right)^{\cdot}e^0\wedge e^a
         -\left(\frac{r'}{r\Lambda}\right)'
    \left(\frac{1}{\Lambda}e^1-\frac{\beta}{N}e^0\right)\wedge e^a \nn
    & &-\frac{r'}{r\Lambda}\left(-K^{\theta}_{\theta}e^0\wedge e^a+
         \frac{r'}{r\Lambda}e^1\wedge e^a-\tilde{\omega}^{a}{}_{b}\wedge e^b
         \right) \nn
     & &+\left(\frac{N'}{N\Lambda}e^0-K^{\chi}_{\chi}e^1\right)\wedge
            \left(-K^{\theta}_{\theta}e^a\right)+
          \left(-\frac{r'}{r\Lambda}e^b\right)\wedge \tilde{\omega}^{b}{}_{a} \nn
     &=&\left[-\frac{1}{\Lambda}\left(\frac{r'}{r\Lambda}\right)'
         -\left(\frac{r'}{r\Lambda}\right)^2+K^{\chi}_{\chi}K^{\theta}_{\theta}
\right]e^1\wedge e^a \nn
     & &+\left[-\frac{1}{N}\left(\frac{r'}{r\Lambda}\right)^{\bullet}
     +\frac{\beta}{N}\left(\frac{r'}{r\Lambda}\right)'+
   K^{\theta}_{\theta}\left(\frac{r'}{r\Lambda}-\frac{N'}{N\Lambda}
\right)\right]e^0\wedge e^a .
\eea


{}From (\ref{ab}), we obtain
\bea
\Theta^{a}{}_{b}&=&d\omega^{a}{}_{b}+\omega^{a}{}_{0}\wedge\omega^{0}{}_{b} 
                   +\omega^{a}{}_{1}\wedge\omega^{1}{}_{b} 
                   +\omega^{a}{}_{c}\wedge\omega^{c}{}_{b}\nn
       &=&d\tilde{\omega}^{a}{}_{b}+\tilde{\omega}^{a}{}_{c}\wedge
           \tilde{\omega}^{c}{}_{b} \nn
  & &+\left(K^{\theta}_{\theta}\right)^2 e^a\wedge e^b
     -\left(\frac{r'}{r\Lambda}\right)^2 e^a\wedge e^b \nn
  &=&\tilde{\Theta}^{a}{}_{b}
+\left[\left(K^{\theta}_{\theta}\right)^2 
     -\left(\frac{r'}{r\Lambda}\right)^2\right] e^a\wedge e^b .
\eea

$\{\tilde{e}^a\}$'s form a unit sphere, thus $\tilde{\Theta}^a{}_b=
\tilde{e}^a\wedge \tilde{e}^b$.

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Comparing the coefficients with
\be
\Theta^{A}{}_{B}=\frac{1}{2}R^{A}{}_{BCD}e^{C}\wedge e^{D},
\ee
we find
\bea
R^{01}{}_{01}&=&-\frac{1}{\Lambda}\left(\frac{N'}{N\Lambda}\right)'
       -\left(\frac{N'}{N\Lambda}\right)^2  
       -\frac{1}{N}\dot{K}^{\chi}_{\chi}
       +\frac{\beta}{N}{(K^{\chi}_{\chi})}'
       +\left(K^{\chi}_{\chi}\right)^2 \\
R^{0a}{}_{0b}&=&\delta^{a}_{b}\left[-\frac{1}{N}\dot{K}^{\theta}_{\theta}+
        \frac{\beta}{N}\left(K^{\theta}_{\theta}\right)'+
      \left(K^{\theta}_{\theta}\right)^2-
      \frac{r'}{r\Lambda}\frac{N'}{N\Lambda}\right] \\
R^{1a}{}_{1b}&=&\delta^{a}_{b}\left[-\frac{1}{\Lambda}\left(\frac{r'}{r\Lambda}\right)'
         -\left(\frac{r'}{r\Lambda}\right)^2+K^{\chi}_{\chi}K^{\theta}_{\theta}
\right] \\
R^{0a}{}_{1b}&=&-\frac{1}{\Lambda}\left(K^{\theta}_{\theta}\right)'
     -(K^{\theta}_{\theta}-K^{\chi}_{\chi})\frac{r'}{r\Lambda} \nn
 &=&-\left[-\frac{1}{N}\left(\frac{r'}{r\Lambda}\right)^{\bullet}
     +\frac{\beta}{N}\left(\frac{r'}{r\Lambda}\right)'+
   K^{\theta}_{\theta}\left(\frac{r'}{r\Lambda}-\frac{N'}{N\Lambda}
\right)\right] \\
R^{ab}{}_{cd}&=&(\delta^{a}_{c}\delta^{b}_{d}-\delta^{a}_{d}\delta^{b}_{c})
\left[\frac{1}{r^2}+\left(K^{\theta}_{\theta}\right)^2 
     -\left(\frac{r'}{r\Lambda}\right)^2\right]
\eea

The Ricci curvatures ($R^{A}_{B}=R^{AC}{}_{BC}$) are given as
\bea
R^{0}_{0}&=&-\frac{1}{\Lambda}\left(\frac{N'}{N\Lambda}\right)'
       -\left(\frac{N'}{N\Lambda}\right)^2  
       -\frac{1}{N}\dot{K}^{\chi}_{\chi}
       +\frac{\beta}{N}{(K^{\chi}_{\chi})}'
       +\left(K^{\chi}_{\chi}\right)^2 \nn
& &+2\left[-\frac{1}{N}\dot{K}^{\theta}_{\theta}+
        \frac{\beta}{N}\left(K^{\theta}_{\theta}\right)'+
      \left(K^{\theta}_{\theta}\right)^2-
      \frac{r'}{r\Lambda}\frac{N'}{N\Lambda}\right] \\
R^{1}_{1}&=&-\frac{1}{\Lambda}\left(\frac{N'}{N\Lambda}\right)'
       -\left(\frac{N'}{N\Lambda}\right)^2  
       -\frac{1}{N}\dot{K}^{\chi}_{\chi}
       +\frac{\beta}{N}{(K^{\chi}_{\chi})}'
       +\left(K^{\chi}_{\chi}\right)^2 \nn
& &+2\left[-\frac{1}{\Lambda}\left(\frac{r'}{r\Lambda}\right)'
         -\left(\frac{r'}{r\Lambda}\right)^2+K^{\chi}_{\chi}K^{\theta}_{\theta}
\right] \\
R^{a}_{b}&=&\delta^{a}_{b}\left\{
\left[\frac{1}{r^2}+\left(K^{\theta}_{\theta}\right)^2 
     -\left(\frac{r'}{r\Lambda}\right)^2\right] \right.\nn
& &+\left[-\frac{1}{N}\dot{K}^{\theta}_{\theta}+
        \frac{\beta}{N}\left(K^{\theta}_{\theta}\right)'+
      \left(K^{\theta}_{\theta}\right)^2-
      \frac{r'}{r\Lambda}\frac{N'}{N\Lambda}\right] \nn
& &\left.+\left[-\frac{1}{\Lambda}\left(\frac{r'}{r\Lambda}\right)'
         -\left(\frac{r'}{r\Lambda}\right)^2+K^{\chi}_{\chi}K^{\theta}_{\theta}
\right] \right\}\\
R^{0}_{1}&=&-\frac{2}{\Lambda}\left(K^{\theta}_{\theta}\right)'
     -(K^{\theta}_{\theta}-K^{\chi}_{\chi})\frac{2r'}{r\Lambda} 
\eea

The scalar curvature is
\bea
R&=&2\left[-\frac{1}{\Lambda}\left(\frac{N'}{N\Lambda}\right)'
       -\left(\frac{N'}{N\Lambda}\right)^2  
       -\frac{1}{N}\dot{K}^{\chi}_{\chi}
       +\frac{\beta}{N}{(K^{\chi}_{\chi})}'
       +\left(K^{\chi}_{\chi}\right)^2 \right]\nn
& &+4\left[-\frac{1}{N}\dot{K}^{\theta}_{\theta}+
        \frac{\beta}{N}\left(K^{\theta}_{\theta}\right)'+
      \left(K^{\theta}_{\theta}\right)^2-
      \frac{r'}{r\Lambda}\frac{N'}{N\Lambda}\right] \nn
& &+4\left[-\frac{1}{\Lambda}\left(\frac{r'}{r\Lambda}\right)'
         -\left(\frac{r'}{r\Lambda}\right)^2+K^{\chi}_{\chi}K^{\theta}_{\theta}
\right] \nn
& &+2
\left[\frac{1}{r^2}+\left(K^{\theta}_{\theta}\right)^2 
     -\left(\frac{r'}{r\Lambda}\right)^2\right] 
\eea


Then the components of the Einstein tensor
($G^{A}_B{}=R^{A}_{B}-\frac{1}{2}\delta^{A}_{B}R$) are written as
\bea
G^{0}_{0}&=&-2\left[-\frac{1}{\Lambda}\left(\frac{r'}{r\Lambda}\right)'
         -\left(\frac{r'}{r\Lambda}\right)^2+K^{\chi}_{\chi}K^{\theta}_{\theta}
\right] \nn
& &-
\left[\frac{1}{r^2}+\left(K^{\theta}_{\theta}\right)^2 
     -\left(\frac{r'}{r\Lambda}\right)^2\right] , \\
G^{1}_{1}&=&-2\left[-\frac{1}{N}\dot{K}^{\theta}_{\theta}+
        \frac{\beta}{N}\left(K^{\theta}_{\theta}\right)'+
      \left(K^{\theta}_{\theta}\right)^2-
      \frac{r'}{r\Lambda}\frac{N'}{N\Lambda}\right] \nn
& &-
\left[\frac{1}{r^2}+\left(K^{\theta}_{\theta}\right)^2 
     -\left(\frac{r'}{r\Lambda}\right)^2\right] , \\
G^{a}_{b}&=&\delta^{a}_{b}\left\{
-\left[-\frac{1}{\Lambda}\left(\frac{N'}{N\Lambda}\right)'
       -\left(\frac{N'}{N\Lambda}\right)^2  
       -\frac{1}{N}\dot{K}^{\chi}_{\chi}
       +\frac{\beta}{N}{(K^{\chi}_{\chi})}'
       +\left(K^{\chi}_{\chi}\right)^2 \right]\right. \nn
& &-\left[-\frac{1}{N}\dot{K}^{\theta}_{\theta}+
        \frac{\beta}{N}\left(K^{\theta}_{\theta}\right)'+
      \left(K^{\theta}_{\theta}\right)^2-
      \frac{r'}{r\Lambda}\frac{N'}{N\Lambda}\right] \nn
& &\left.-\left[-\frac{1}{\Lambda}\left(\frac{r'}{r\Lambda}\right)'
         -\left(\frac{r'}{r\Lambda}\right)^2+K^{\chi}_{\chi}K^{\theta}_{\theta}
\right]\right\} , \\
G^{0}_{1}&=&-\frac{2}{\Lambda}\left(K^{\theta}_{\theta}\right)'
     -(K^{\theta}_{\theta}-K^{\chi}_{\chi})\frac{2r'}{r\Lambda} .
\eea

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We can rewrite these as
\bea
G^{0}_{0}&=&-K^{\theta}_{\theta}\left(K^{\theta}_{\theta}+2K^{\chi}_{\chi}\right)
     -\frac{1}{r^2r'}\left[r\left(1-\left(\frac{r'}{\Lambda}
      \right)^2\right)\right]' , \\
G^{1}_{1}&=&-3\left(K^{\theta}_{\theta}\right)^2+
       \frac{2}{N}\left[\dot{K}^{\theta}_{\theta}-
       \beta\left(K^{\theta}_{\theta}\right)'\right]+
      \frac{2r'N'}{rN\Lambda^2}-
         \frac{1}{r^2}\left[1-\left(\frac{r'}{\Lambda}\right)^2\right] , \\
G^{a}_{b}&=&\delta^{a}_{b}\left\{
           -\left(K^{\chi}_{\chi}\right)^2
           -K^{\chi}_{\chi}K^{\theta}_{\theta}
           -\left(K^{\theta}_{\theta}\right)^2+
           \frac{1}{N}\left[\dot{K}^{\chi}_{\chi}+
           \dot{K}^{\theta}_{\theta}-
           \beta\left({(K^{\chi}_{\chi})}'+\left(K^{\theta}_{\theta}\right)'
           \right)\right] \right.\nn
           & &\left.+\frac{1}{N\Lambda}\left(\frac{N'}{\Lambda}\right)'+
           \frac{r'N'}{rN\Lambda^2}+
           \frac{1}{r\Lambda}\left(\frac{r'}{\Lambda}\right)'
           \right\} , \\
G^{0}_{1}&=&-\frac{2}{r\Lambda}\left(rK^{\theta}_{\theta}\right)'
     +K^{\chi}_{\chi}\frac{2r'}{r\Lambda} .
\eea

\end{document}
–ß‚é