\input robmacro
\documentstyle{article}
%\texwidth
%\oddsidemargin=1-(8.5-\texwidth)
\begin{document}

$R_{abcd}=C_{abcd}+2^2\left( \frac {1}{n-2}\right) g_{[a[c}L_{d]b]}$

$L_{ab}=R_{ab}-\frac{1}{2(n-1)}Rg_{ab}$
\begin{eqnarray*}
                  L &=&R-\frac{n}{2(n-1)}R                  \\
                    &=&\left(\frac{2n-2-n}{2(n-1)}\right)R  \\
                    &=&\left(\frac{n-2}{2(n-1)}\right)R
\end{eqnarray*}

\begin{eqnarray*}
R_{ab}&=&L_{ab}+\frac{1}{2(n-1)}\left(\frac{2(n-1)}{n-2}L\right)g_{ab}\\
                        &=&L_{ab}+\frac{1}{n-2}Lg_{ab}
\end{eqnarray*}


$G_{ab}=R_{ab}-\frac{1}{2}Rg_{ab}$
\begin{eqnarray*}
                  G &=&R-\frac{n}{2}R                  \\
                    &=&\left(\frac{2-n}{2}\right)R  \\
                    &=&-\left(\frac{n-2}{2}\right)R
\end{eqnarray*}

\begin{eqnarray*}
R_{ab}&=&G_{ab}+\frac{1}{2}\left(-\frac{2}{n-2}G\right)g_{ab}\\
      &=&G_{ab}-\frac{1}{n-2}Gg_{ab}
\end{eqnarray*}

\begin{eqnarray*}
G_{ab}&=&\left(L_{ab}+\frac{1}{n-2}Lg_{ab}\right)
          -\frac{1}{2}\left(\frac{2(n-1)}{n-2}L\right)g_{ab}\\
      &=&L_{ab}+\frac{1}{n-2}\left(1-(n-1)\right)Lg_{ab} \\
      &=&L_{ab}-Lg_{ab}
         \qquad\mbox{or}\qquad
         \mbox{\bf G} =\mbox{\bf L}-(\mbox{tr \bf L})\mbox{\bf I }
\end{eqnarray*}



\begin{eqnarray*}
G&=&L-Ln=-(n-1)L
\end{eqnarray*}

\begin{eqnarray*}
L_{ab}&=&G_{ab}+\left(-\frac{1}{n-1}G\right)g_{ab}\\
      &=&G_{ab}-\frac{1}{n-1}Gg_{ab}
\end{eqnarray*}

Note that $0=\nabla^a G_{ab}=\nabla^a L_{ab} - g_{ab}\nabla^a L$
implies that
$$\nabla^a L_{ab}=\nabla_b L \qquad\mbox{or}\qquad
  \mbox{div \bf L}=\mbox{grad (tr \bf L)}$$

Consider the characteristic polynomial for $L^a{}_p$.
For convenience, let us define $\Delta^a{}_p=L^a{}_p-\lambda\delta^a{}_p$.
\begin{eqnarray*}
      0 &=&\det(\Delta^a{}_p)\\
        &=&n!\Delta^{a}{}_{[a}
             \Delta^{b}{}_{ b}\cdots
             \Delta^{n}{}_{n]}\\
        &=&n!(L^{a}{}_{[a}-\lambda\delta^{a}{}_{[a})
             (L^{b}{}_{ b}-\lambda\delta^{b}{}_{ b})\cdots
             (L^{n}{}_{n]}-\lambda\delta^{n}{}_{n]})\\
        &=&n!\sum_k {\textstyle\frac{n!}{k!(n-k)!}}
                    \delta^a{}_{[a}\cdots \delta^k{}_{ k}
                         L^l{}_{l}\cdots       L^n{}_{n]}
                      (-\lambda)^k
\end{eqnarray*}
By making use of the identities,
\begin{eqnarray*}
        i!\ \mbox{\boldmath$\alpha$}_{[a\cdots i]}\
        \mbox{\boldmath$\wedge$} \
        (n-i)!\ \mbox{\boldmath$\beta$}_{[j\cdots n]}
        &=&  n!\
             \mbox{\boldmath$\alpha$}_{[a\cdots i}
             \mbox{\boldmath$\beta$}_{j\cdots n]}
\end{eqnarray*}
and
\begin{eqnarray*}
        i!\ \mbox{\boldmath$\alpha$}_{[a\cdots i]}\
        \mbox{\boldmath$\wedge$} \
        (n-i)!\ \mbox{\boldmath$\beta$}_{[j\cdots n]}
        &=&  n!\
             \mbox{\boldmath$\alpha$}_{[a\cdots i}
             \mbox{\boldmath$\beta$}_{j\cdots n]}\\
\end{eqnarray*}

so that
\begin{eqnarray*}
        \delta^a{}_{[a}\cdots \delta^k{}_{\vphantom{[]}k}
            Q^{l \ldots n}_{l \ldots n]}
        &=&\frac{(n-k)!}{n!}Q^{l \ldots n}_{[l \ldots n]}
\end{eqnarray*}
we obtain
\begin{eqnarray*}
        0&=&\det(\Delta^a{}_p)\\
         &=&n!\sum_k {\textstyle\frac{n!}{k!(n-k)!}}
                    \delta^a{}_{[a}\cdots \delta^k{}_{ k}
                         L^l{}_{l}\cdots       L^n{}_{n]}
                      (-\lambda)^k\\
          &=&n!\sum_k {\textstyle\frac{n!}{k!(n-k)!}}
                    {\textstyle\frac{(n-k)!}{n!}}
                         L^l{}_{[l}\cdots       L^n{}_{n]}
                      (-\lambda)^k\\
          &=&n!\sum_k {\textstyle\frac{1}{k!}}
                         L^l{}_{[l}\cdots       L^n{}_{n]}
                      (-\lambda)^k\\
\end{eqnarray*}

The coefficient of $(-\lambda)^{n-k}$, namely,
$L^a{}_{[a}\cdots L^{k}{}_{k]}$,
is called the ``$k^{\rm th}$ principal invariant'' of $L^a{}_p$.
The $1^{\rm st}$ principal invariant is the trace, $L^a{}_{a}$ or
${\rm tr \bf L}$.
The $2^{\rm nd}$ principal invariant is $L^a{}_{[a}L^b{}_{b]}$ or
$\frac{1}{2}\left( ({\rm tr \bf L})^2-{\rm tr}{\bf L}^2\right)$.
The $n^{\rm th}$ principal invariant is the determinant,
$n!L^a{}_{[a}L^b{}_{b}\cdots L^n{}_{n]}$.
Geometrically, if $L^a{}_p$ is regarded as a linear transformation
of an $n$-dimensional vector space, the determinant is the relative
change of ``[$n$-]volume'' enclosed by parallelipeds whose sides
are the initial and final basis vectors.


Of particular interest to us is the $2^{\rm nd}$ principal invariant
$L^a{}_{[a}L^b{}_{b]}$.  Note that
\begin{eqnarray*}
      \frac{d}{dL^m{}_n}\left(L^a{}_{[a}L^b{}_{b]}\right)
      &=&\delta^a{}_m\delta^n{}_{[a}{} L^b{}_{b]}+
         L^a{}_{[a}\delta^b{}_{|m|}\delta^n{}_{b]}{}\\
      &=&\delta^n{}_{[m}{} L^b{}_{b]}+
         L^a{}_{[a}\delta^n{}_{m]}\\
      &=&2\delta^n{}_{[m}{} L^b{}_{b]}\\
      &=&2\ {\textstyle\frac{1}{2}}\left(
              \delta^n{}_{m} L^b{}_{b}-\delta^n{}_{b}
           L^b{}_{m}\right)\\
      &=&\left(\delta^n{}_{m} L^b{}_{b}-L^n{}_{m}\right)\\
      &=&-G^n{}_m
\end{eqnarray*}





\end{document}
\end


