- #1
lapo
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Hi, this is my first message on thi forum :D
I apologize in advance for my english.
I'm doing my thesis work on the theory of relativity of Einstein-Cartan.
I'm following the article of Hehl of 1976; it's title is "General relativity with spin and torsion: Foundations and prospects".
I can't understand the equations (3.7) at page 399:
$$\mu^{\lambda\nu\mu}=-\tau^{\lambda\nu\mu}+\tau^{\nu\mu\lambda}-\tau^{\mu\lambda\nu}\qquad \tau^{\nu\mu\lambda}= \mu^{[\mu\nu]\lambda} $$
Where $\tau$ (spin tensor) e $\mu$ ( pseudo spin tensor) are defined as:
$${\tau_{\lambda}}^{\nu\mu}=\frac{1}{\sqrt{-g}}\frac{\delta\left(\sqrt{-g} \mathcal{L}\right)}{\delta {K_{\mu\nu}}^{\lambda}}\qquad
{\mu_{\lambda}}^{\nu\mu}=\frac{1}{\sqrt{-g}}\frac{\delta\left(\sqrt{-g} \mathcal{L}\right)}{\delta {\Theta_{\mu\nu}}^{\lambda}} $$
Where $\Theta$ and $K$ are respectively torsion and contorsion. For them we have also:
$${K_{\mu\lambda}}^\nu=-{\Theta_{\mu\lambda}}^\nu+{{\Theta_{\lambda}}^{\nu}}_\mu-{\Theta^{\nu}}_{\mu\lambda}\qquad {\Theta_{\lambda\mu}}^\nu={K_{[\mu\lambda]}}^\nu $$
I managed to get the formulas in question in a way that implies the antisymmetry of the spin tensor in the last two indices; but i know that, in general, this is not true. On the other hand, i can't find my mistake; it's look like as all is good.
My reasoning is based on the previous equations and on the chain rule:
\begin{equation*}
\begin{split}
{\mu_{\lambda}}^{\nu\mu}&=\frac{1}{\sqrt{-g}}\frac{\delta\left(\sqrt{-g} \mathcal{L}\right)}{\delta {\Theta_{\mu\nu}}^{\lambda}}\\
&=\frac{1}{\sqrt{-g}}\frac{\partial\left(\sqrt{-g} \mathcal{L}\right)}{\partial {\Theta_{\mu\nu}}^{\lambda}}\\
&=\frac{1}{\sqrt{-g}}\frac{\partial\left(\sqrt{-g} \mathcal{L}\right)}{\partial {K_{\rho\sigma}}^{\epsilon}}\frac{\partial{K_{\rho\sigma}}^{\epsilon}}{\partial {\Theta_{\mu\nu}}^{\lambda}}\\
&={\tau_\epsilon}^{\sigma\rho}\left[-\frac{ \partial{\Theta_{\rho\sigma}}^{\epsilon}}{\partial {\Theta_{\mu\nu}}^{\lambda}}+\frac{\partial {\Theta_{\sigma}}^{\epsilon}\,_\rho}{\partial {\Theta_{\mu\nu}}^{\lambda}}-\frac{\partial {\Theta^{\epsilon}}_{\rho\sigma}}{\partial {\Theta_{\mu\nu}}^{\lambda}}\right]\\
&={\tau_\epsilon}^{\sigma\rho}\left[-\frac{\partial{\Theta_{\rho\sigma}}^{\epsilon}}{\partial {\Theta_{\mu\nu}}^{\lambda}}+g^{\epsilon\gamma}g_{\rho k}\frac{\partial {\Theta_{\sigma\gamma}}^{k}}{\partial {\Theta_{\mu\nu}}^{\lambda}}-g^{\epsilon\gamma}g_{\sigma k}\frac{\partial {\Theta_{\gamma\rho}}^{k}}{\partial {\Theta_{\mu\nu}}^{\lambda}}\right]\\
\end{split}
\end{equation*}
So:
\begin{equation*}{\mu_{\lambda}}^{\nu\mu}=-{\tau_{\lambda}}^{\nu\mu}+{\tau^{\nu\mu}}_\lambda-{\tau^\mu}_{\lambda}\,^\nu
\end{equation*}
Which is (Formula 1)
\begin{equation*}
\mu^{\lambda\nu\mu}=-\tau^{\lambda\nu\mu}+\tau^{\nu\mu\lambda}-\tau^{\mu\lambda\nu}
\end{equation*}
Up to this point all it's ok. The problem is that if i remember the definition of the spin tensor and follow a similar reasoning:
\begin{equation*}
\begin{split}
{\tau_{\lambda}}^{\nu\mu}&={\mu_\epsilon}^{\sigma\rho}\frac{\partial {\Theta_{\rho\sigma}}^\epsilon}{\partial {K_{\mu\nu}}^\lambda}\\
&={\mu_\epsilon}^{\sigma\rho}\frac{\partial K_{[\sigma\rho]}\,^\epsilon }{\partial {K_{\mu\nu}}^\lambda}\\
&={\mu_\epsilon}^{\sigma\rho}\frac{1}{2}\frac{\partial }{\partial {K_{\mu\nu}}^\lambda}\left[ {K_{\sigma\rho}}^\epsilon-{K_{\rho\sigma}}^\epsilon\right]
\end{split}
\end{equation*}
we find the following result:
\begin{equation*}
\tau^{\lambda\nu\mu}= \mu^{\lambda[\mu\nu]}
\end{equation*}
Where is the mistake?
Following the previous reasoning and using the antisymmetry of the spin tensor in the last two indices in the formula (1) we arrive easily to:
\begin{equation*}
\tau^{\lambda\nu\mu}= \mu^{[\mu\nu]\lambda}
\end{equation*}
Therefore we have demonstrated the antisymmetry of the spin tensor in the first two indices. Using this asymmetry another time in the formula (1) we obtain finally:
\begin{equation*}
\tau^{\nu\mu\lambda}= \mu^{[\mu\nu]\lambda}
\end{equation*}
Where is my mistake? And in which way i can demonstrate the last formula without the antisymmetry of $\tau$ in the last two indices?
thank you very much, bye!
I apologize in advance for my english.
I'm doing my thesis work on the theory of relativity of Einstein-Cartan.
I'm following the article of Hehl of 1976; it's title is "General relativity with spin and torsion: Foundations and prospects".
I can't understand the equations (3.7) at page 399:
$$\mu^{\lambda\nu\mu}=-\tau^{\lambda\nu\mu}+\tau^{\nu\mu\lambda}-\tau^{\mu\lambda\nu}\qquad \tau^{\nu\mu\lambda}= \mu^{[\mu\nu]\lambda} $$
Where $\tau$ (spin tensor) e $\mu$ ( pseudo spin tensor) are defined as:
$${\tau_{\lambda}}^{\nu\mu}=\frac{1}{\sqrt{-g}}\frac{\delta\left(\sqrt{-g} \mathcal{L}\right)}{\delta {K_{\mu\nu}}^{\lambda}}\qquad
{\mu_{\lambda}}^{\nu\mu}=\frac{1}{\sqrt{-g}}\frac{\delta\left(\sqrt{-g} \mathcal{L}\right)}{\delta {\Theta_{\mu\nu}}^{\lambda}} $$
Where $\Theta$ and $K$ are respectively torsion and contorsion. For them we have also:
$${K_{\mu\lambda}}^\nu=-{\Theta_{\mu\lambda}}^\nu+{{\Theta_{\lambda}}^{\nu}}_\mu-{\Theta^{\nu}}_{\mu\lambda}\qquad {\Theta_{\lambda\mu}}^\nu={K_{[\mu\lambda]}}^\nu $$
I managed to get the formulas in question in a way that implies the antisymmetry of the spin tensor in the last two indices; but i know that, in general, this is not true. On the other hand, i can't find my mistake; it's look like as all is good.
My reasoning is based on the previous equations and on the chain rule:
\begin{equation*}
\begin{split}
{\mu_{\lambda}}^{\nu\mu}&=\frac{1}{\sqrt{-g}}\frac{\delta\left(\sqrt{-g} \mathcal{L}\right)}{\delta {\Theta_{\mu\nu}}^{\lambda}}\\
&=\frac{1}{\sqrt{-g}}\frac{\partial\left(\sqrt{-g} \mathcal{L}\right)}{\partial {\Theta_{\mu\nu}}^{\lambda}}\\
&=\frac{1}{\sqrt{-g}}\frac{\partial\left(\sqrt{-g} \mathcal{L}\right)}{\partial {K_{\rho\sigma}}^{\epsilon}}\frac{\partial{K_{\rho\sigma}}^{\epsilon}}{\partial {\Theta_{\mu\nu}}^{\lambda}}\\
&={\tau_\epsilon}^{\sigma\rho}\left[-\frac{ \partial{\Theta_{\rho\sigma}}^{\epsilon}}{\partial {\Theta_{\mu\nu}}^{\lambda}}+\frac{\partial {\Theta_{\sigma}}^{\epsilon}\,_\rho}{\partial {\Theta_{\mu\nu}}^{\lambda}}-\frac{\partial {\Theta^{\epsilon}}_{\rho\sigma}}{\partial {\Theta_{\mu\nu}}^{\lambda}}\right]\\
&={\tau_\epsilon}^{\sigma\rho}\left[-\frac{\partial{\Theta_{\rho\sigma}}^{\epsilon}}{\partial {\Theta_{\mu\nu}}^{\lambda}}+g^{\epsilon\gamma}g_{\rho k}\frac{\partial {\Theta_{\sigma\gamma}}^{k}}{\partial {\Theta_{\mu\nu}}^{\lambda}}-g^{\epsilon\gamma}g_{\sigma k}\frac{\partial {\Theta_{\gamma\rho}}^{k}}{\partial {\Theta_{\mu\nu}}^{\lambda}}\right]\\
\end{split}
\end{equation*}
So:
\begin{equation*}{\mu_{\lambda}}^{\nu\mu}=-{\tau_{\lambda}}^{\nu\mu}+{\tau^{\nu\mu}}_\lambda-{\tau^\mu}_{\lambda}\,^\nu
\end{equation*}
Which is (Formula 1)
\begin{equation*}
\mu^{\lambda\nu\mu}=-\tau^{\lambda\nu\mu}+\tau^{\nu\mu\lambda}-\tau^{\mu\lambda\nu}
\end{equation*}
Up to this point all it's ok. The problem is that if i remember the definition of the spin tensor and follow a similar reasoning:
\begin{equation*}
\begin{split}
{\tau_{\lambda}}^{\nu\mu}&={\mu_\epsilon}^{\sigma\rho}\frac{\partial {\Theta_{\rho\sigma}}^\epsilon}{\partial {K_{\mu\nu}}^\lambda}\\
&={\mu_\epsilon}^{\sigma\rho}\frac{\partial K_{[\sigma\rho]}\,^\epsilon }{\partial {K_{\mu\nu}}^\lambda}\\
&={\mu_\epsilon}^{\sigma\rho}\frac{1}{2}\frac{\partial }{\partial {K_{\mu\nu}}^\lambda}\left[ {K_{\sigma\rho}}^\epsilon-{K_{\rho\sigma}}^\epsilon\right]
\end{split}
\end{equation*}
we find the following result:
\begin{equation*}
\tau^{\lambda\nu\mu}= \mu^{\lambda[\mu\nu]}
\end{equation*}
Where is the mistake?
Following the previous reasoning and using the antisymmetry of the spin tensor in the last two indices in the formula (1) we arrive easily to:
\begin{equation*}
\tau^{\lambda\nu\mu}= \mu^{[\mu\nu]\lambda}
\end{equation*}
Therefore we have demonstrated the antisymmetry of the spin tensor in the first two indices. Using this asymmetry another time in the formula (1) we obtain finally:
\begin{equation*}
\tau^{\nu\mu\lambda}= \mu^{[\mu\nu]\lambda}
\end{equation*}
Where is my mistake? And in which way i can demonstrate the last formula without the antisymmetry of $\tau$ in the last two indices?
thank you very much, bye!