- Stokes tétele:
$$
\oint A_i\;dx^i=\int df^{ki}\frac{\partial A^i}{\partial x^k}
=
\frac{1}{2}\int df^{ki}\left(\frac{\partial A^i}{\partial x^k}-\frac{\partial A^k}{\partial x^i}\right)
$$
ahol
$$df^{ki}=dx^{(1) k}dx^{(2) i}-dx^{(1) i}dx^{(2) k}$$
a $dx^{(1) i}$ és $dx^{(2) i}$ vektorok által kifeszített paralelogramma
területe.
- A Riemann-tenzor kifejezése a
Christoffel-szimbólumokkal:
$
\begin{eqnarray}
R^i_{\phantom{1}klm}A_i&=&
\frac{\partial \left(\Gamma^i_{\phantom{1}km} A_i\right)}{\partial x^l}-
\frac{\partial \left(\Gamma^i_{\phantom{1}kl} A_i\right)}{\partial x^m}\\
&=&\left(\frac{\partial \Gamma^i_{\phantom{1}km}}{\partial x^l}-
\frac{\partial \Gamma^i_{\phantom{1}kl} }{\partial x^m}\right) A_i
+\Gamma^i_{\phantom{1}km} \Gamma^n_{\phantom{1}il}A_n- \Gamma^i_{\phantom{1}kl}\Gamma^n_{\phantom{1}im}A_n\\
&=&\left(\frac{\partial \Gamma^i_{\phantom{1}km}}{\partial x^l}-
\frac{\partial \Gamma^i_{\phantom{1}kl} }{\partial x^m}
+\Gamma^n_{\phantom{1}km} \Gamma^i_{\phantom{1}nl}- \Gamma^n_{\phantom{1}kl}\Gamma^i_{\phantom{1}nm}
\right) A_i
\end{eqnarray}
$
Felhasználtuk, hogy
$$
\frac{\partial A_i}{\partial x^l}=\Gamma^n_{\phantom{1}il}A_n
$$
A fenti levezetés tetszőleges $A_i$ vektorra érvényes, ezért
$$R^i_{\phantom{1}klm}=\frac{\partial \Gamma^i_{\phantom{1}km}}{\partial x^l}-
\frac{\partial \Gamma^i_{\phantom{1}kl} }{\partial x^m}
+\Gamma^n_{\phantom{1}km} \Gamma^i_{\phantom{1}nl}- \Gamma^n_{\phantom{1}kl}\Gamma^i_{\phantom{1}nm}$$
- Kontravariáns vektorkomponens megváltozása zárt görbe
mentén:
$$\Delta \left(A^kB_k\right)=0$$
$
\begin{eqnarray}
\Delta \left(A^kB_k\right)&=&B_k\Delta A^k+A^k\Delta B_k\\
&=&B_k\Delta A^k+A^k\frac{1}{2}R^i_{\phantom{1}klm}B_i \Delta f^{lm}\\
&=&B_k\left(\Delta A^k+A^i\frac{1}{2}R^k_{\phantom{1}ilm} \Delta f^{lm}\right)
\end{eqnarray}
$
Mivel $B_k$ tetszőleges,
$$
\Delta A^k=-\frac{1}{2}R^k_{\phantom{1}ilm}A^i \Delta f^{lm}
$$
- Kovariáns vektorkomponens második kovariáns deriváltjai
és a görbületi tenzor:
$
\begin{eqnarray}
A_{i;k;l}&=&A_{i;k,l}-\Gamma^n_{\phantom{1}il}A_{n;k}-\Gamma^n_{\phantom{1}kl}A_{i;n}\\
&=&A_{i,k,l}-\frac{\partial }{\partial
x^l}\left(\Gamma^n_{\phantom{1}ik}A_{n}\right)-\Gamma^n_{\phantom{1}il}A_{n,k}+\Gamma^n_{\phantom{1}il}\Gamma^m_{\phantom{1}nk}A_m-\Gamma^n_{\phantom{1}kl}A_{i,n}+\Gamma^n_{\phantom{1}kl}\Gamma^m_{\phantom{1}in}A_m\\
&=&A_{i,k,l}-\frac{\partial \Gamma^n_{\phantom{1}ik}}{\partial
x^l}A_{n}-\Gamma^n_{\phantom{1}ik}A_{n,l}
-\Gamma^n_{\phantom{1}il}A_{n,k}-\Gamma^n_{\phantom{1}kl}A_{i,n}
+\Gamma^n_{\phantom{1}il}\Gamma^m_{\phantom{1}nk}A_m+\Gamma^n_{\phantom{1}kl}\Gamma^m_{\phantom{1}in}A_m\\
A_{i;l;k}&=&A_{i,l,k}-\frac{\partial \Gamma^n_{\phantom{1}il}}{\partial
x^k}A_{n}-\Gamma^n_{\phantom{1}il}A_{n,k}
-\Gamma^n_{\phantom{1}ik}A_{n,l}-\Gamma^n_{\phantom{1}lk}A_{i,n}
+\Gamma^n_{\phantom{1}ik}\Gamma^m_{\phantom{1}nl}A_m+\Gamma^n_{\phantom{1}lk}\Gamma^m_{\phantom{1}in}A_m\\
A_{i;k;l}-A_{i;l;k}&=&\left(\frac{\partial \Gamma^m_{\phantom{1}il}}{\partial
x^k}-\frac{\partial \Gamma^m_{\phantom{1}ik}}{\partial
x^l}+\Gamma^n_{\phantom{1}il}\Gamma^m_{\phantom{1}nk}-\Gamma^n_{\phantom{1}ik}\Gamma^m_{\phantom{1}nl}\right)A_m\\
&=&R^m_{\phantom{1}ikl}A_m
\end{eqnarray}
$
- Kontravariáns vektorkomponens második kovariáns deriváltjai
és a görbületi tenzor:
$
\begin{eqnarray}
A^i_{;k;l}&=&A^i_{;k,l}+\Gamma^i_{\phantom{1}nl}A^n_{;k}-\Gamma^n_{\phantom{1}kl}A^i_{;n}\\
&=&A^i_{,k,l}+\frac{\partial }{\partial
x^l}\left(\Gamma^i_{\phantom{1}nk}A^{n}\right)+\Gamma^i_{\phantom{1}nl}A^n_{,k}+\Gamma^i_{\phantom{1}nl}\Gamma^n_{\phantom{1}mk}A^m-\Gamma^n_{\phantom{1}kl}A^i_{,n}-\Gamma^n_{\phantom{1}kl}\Gamma^i_{\phantom{1}mn}A^m\\
&=&A^i_{,k,l}+\frac{\partial \Gamma^i_{\phantom{1}nk}}{\partial
x^l}A^{n}+\Gamma^i_{\phantom{1}nk}A^n_{,l}+\Gamma^i_{\phantom{1}nl}A^n_{,k}+\Gamma^i_{\phantom{1}nl}\Gamma^n_{\phantom{1}mk}A^m-\Gamma^n_{\phantom{1}kl}A^i_{,n}-\Gamma^n_{\phantom{1}kl}\Gamma^i_{\phantom{1}mn}A^m\\
A^i_{;l;k}&=&A^i_{,l,k}+\frac{\partial \Gamma^i_{\phantom{1}nl}}{\partial
x^k}A^{n}+\Gamma^i_{\phantom{1}nl}A^n_{,k}+\Gamma^i_{\phantom{1}nk}A^n_{,l}+\Gamma^i_{\phantom{1}nk}\Gamma^n_{\phantom{1}ml}A^m-\Gamma^n_{\phantom{1}lk}A^i_{,n}-\Gamma^n_{\phantom{1}lk}\Gamma^i_{\phantom{1}mn}A^m
\end{eqnarray}
$
$
\begin{eqnarray}
A^i_{;k;l}-A^i_{;l;k}&=&-\left(\frac{\partial \Gamma^i_{\phantom{1}ml}}{\partial
x^k}-\frac{\partial \Gamma^i_{\phantom{1}mk}}{\partial
x^l}+\Gamma^i_{\phantom{1}nk}\Gamma^n_{\phantom{1}ml}
-\Gamma^i_{\phantom{1}nl}\Gamma^n_{\phantom{1}mk}
\right)A^m\\
&=&-R^i_{\phantom{1}mkl}A^m
\end{eqnarray}
$
- A görbületi tenzor átalakítása a szimmetriái
vizsgálatához:
$
\begin{eqnarray}
R_{iklm}&=&g_{in}R^n_{\phantom{1}klm}=
g_{in}\left(\frac{\partial \Gamma^n_{\phantom{1}km} }{\partial x^l}
-\frac{\partial \Gamma^n_{\phantom{1}kl} }{\partial x^m}
+\Gamma^n_{\phantom{1}pl}\Gamma^p_{\phantom{1}km}-\Gamma^n_{\phantom{1}pm}\Gamma^p_{\phantom{1}kl}\right)\\
&=&\frac{\partial \Gamma_{ikm} }{\partial x^l}
-\frac{\partial \Gamma_{ikl} }{\partial x^m}-\Gamma^n_{\phantom{1}km}g_{in,l}+
\Gamma^n_{\phantom{1}kl}g_{in,m}+g_{in}\left(\Gamma^n_{\phantom{1}pl}\Gamma^p_{\phantom{1}km}-\Gamma^n_{\phantom{1}pm}\Gamma^p_{\phantom{1}kl}\right)
\end{eqnarray}
$
Felhasználjuk, hogy
$
\begin{eqnarray}
\Gamma_{ikm}&=&\frac{1}{2}\left(g_{ik,m}+g_{im,k}-g_{km,i}\right)\\
\Gamma_{ikl}&=&\frac{1}{2}\left(g_{ik,l}+g_{il,k}-g_{kl,i}\right)\\
g_{in,l}&=&\Gamma^p_{\phantom{1}il}g_{pn}+\Gamma^p_{\phantom{1}nl}g_{ip}\\
g_{in,m}&=&\Gamma^p_{\phantom{1}im}g_{pn}+\Gamma^p_{\phantom{1}nm}g_{ip}
\end{eqnarray}
$
Ezzel
$
\begin{eqnarray}
R_{iklm}&=&\frac{1}{2}\left(g_{ik,ml}+g_{im,kl}-g_{km,il}-g_{ik,lm}-g_{il,km}+g_{kl,im}\right)\\
&-&\Gamma^n_{\phantom{1}km}\Gamma^p_{\phantom{1}il}g_{pn}-\Gamma^n_{\phantom{1}km}\Gamma^p_{\phantom{1}nl}g_{ip}+\Gamma^n_{\phantom{1}kl}\Gamma^p_{\phantom{1}im}g_{pn}+\Gamma^n_{\phantom{1}kl}\Gamma^p_{\phantom{1}nm}g_{ip}\\
&+&g_{in}\left(\Gamma^n_{\phantom{1}pl}\Gamma^p_{\phantom{1}km}-\Gamma^n_{\phantom{1}pm}\Gamma^p_{\phantom{1}kl}\right)
\end{eqnarray}
$
Azaz
$
\begin{eqnarray}
R_{iklm}&=&\frac{1}{2}\left(\frac{\partial^2 g_{im}}{\partial x^k\partial x^l}+
\frac{\partial^2 g_{kl}}{\partial x^i\partial x^m}-\frac{\partial^2 g_{km}}{\partial x^i\partial x^l}-\frac{\partial^2 g_{il}}{\partial x^k\partial x^m}\right)\\
&+&g_{pn}\left(\Gamma^n_{\phantom{1}kl}\Gamma^p_{\phantom{1}im}
-\Gamma^n_{\phantom{1}km}\Gamma^p_{\phantom{1}il}\right)
\end{eqnarray}
$
- A görbületi tenzor bármely három indexe szerint képzett
ciklikus összeg zérus:
$
\begin{eqnarray}
R_{iklm}+R_{imkl}+R_{ilmk}&=&
\frac{1}{2}\left(\frac{\partial^2 g_{im}}{\partial x^k\partial x^l}+
\frac{\partial^2 g_{kl}}{\partial x^i\partial x^m}-\frac{\partial^2
g_{km}}{\partial x^i\partial x^l}-\frac{\partial^2 g_{il}}{\partial x^k\partial
x^m}\right)\\
&+&\frac{1}{2}\left(\frac{\partial^2 g_{il}}{\partial x^m\partial x^k}+
\frac{\partial^2 g_{mk}}{\partial x^i\partial x^l}-\frac{\partial^2
g_{ml}}{\partial x^i\partial x^k}-\frac{\partial^2 g_{ik}}{\partial x^m\partial
x^l}\right)\\
&+&\frac{1}{2}\left(\frac{\partial^2 g_{ik}}{\partial x^l\partial x^m}+
\frac{\partial^2 g_{lm}}{\partial x^i\partial x^k}-\frac{\partial^2
g_{lk}}{\partial x^i\partial x^m}-\frac{\partial^2 g_{im}}{\partial x^l\partial
x^k}\right)\\
&+&g_{pn}\left(\Gamma^n_{\phantom{1}kl}\Gamma^p_{\phantom{1}im}
-\Gamma^n_{\phantom{1}km}\Gamma^p_{\phantom{1}il}\right)\\
&+&g_{pn}\left(\Gamma^n_{\phantom{1}mk}\Gamma^p_{\phantom{1}il}
-\Gamma^n_{\phantom{1}ml}\Gamma^p_{\phantom{1}ik}\right)\\
&+&g_{pn}\left(\Gamma^n_{\phantom{1}lm}\Gamma^p_{\phantom{1}ik}
-\Gamma^n_{\phantom{1}lk}\Gamma^p_{\phantom{1}im}\right)
\end{eqnarray}
$
- A Bianchi-azonosság bizonyítása:
$$
R^n_{\phantom{1}ikl;m}+R^n_{\phantom{1}imk;l}+R^n_{\phantom{1}ilm;k}=\frac{1}{2}R^n_{\phantom{1}ikl;m}E^{klmp}\sqrt{-g}
$$
Ahol $$E^{klmp}=\frac{1}{\sqrt{-g}}e^{klmp}$$
A kérdéses ciklikus összeg $1/\sqrt{-g}$-szerese tehát tenzor.
Áttérhetünk ezért lokálisan geodetikus rendszerre (ha a tenzor ott eltűnik,
akkor máshol is):
$$
R^n_{\phantom{1}ikl;m}=\frac{\partial R^n_{\phantom{1}ikl}}{\partial x^m}
=\frac{\partial^2 \Gamma^n_{\phantom{1}il}}{\partial x^m \partial x^k}
-\frac{\partial^2 \Gamma^n_{\phantom{1}ik}}{\partial x^m \partial x^l}
$$
Ezzel
$
\begin{eqnarray}
R^n_{\phantom{1}ikl;m}+R^n_{\phantom{1}imk;l}+R^n_{\phantom{1}ilm;k}
&=&\frac{\partial^2 \Gamma^n_{\phantom{1}il}}{\partial x^m \partial x^k}
-\frac{\partial^2 \Gamma^n_{\phantom{1}ik}}{\partial x^m \partial x^l}\\
&+&\frac{\partial^2 \Gamma^n_{\phantom{1}ik}}{\partial x^l \partial x^m}
-\frac{\partial^2 \Gamma^n_{\phantom{1}im}}{\partial x^l \partial x^k}\\
&+&\frac{\partial^2 \Gamma^n_{\phantom{1}im}}{\partial x^k \partial x^l}
-\frac{\partial^2 \Gamma^n_{\phantom{1}il}}{\partial x^k \partial x^m}
\end{eqnarray}
$
