Hi, this is my first message on thi forum :D(adsbygoogle = window.adsbygoogle || []).push({});

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:

$${K_{\mu\lambda}}^\nu=-{\Theta_{\mu\lambda}}^\nu+{{\Theta_{\lambda}}^{\nu}}_\mu-{\Theta^{\nu}}_{\mu\lambda}\qquad {\Theta_{\lambda\mu}}^\nu={K_{[\mu\lambda]}}^\nu $$

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.I managed to get

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 arriveeasilyto:

\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!!!

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# Dynamical definition of spin tensor- Einstein Cartan theory

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