Einstein-Cartan Theory: Dynamical Definition of Spin Tensor

In summary, the conversation discusses the topic of the theory of relativity of Einstein-Cartan and specifically focuses on the article "General relativity with spin and torsion: Foundations and prospects" by Hehl from 1976. The individual is struggling to understand equations (3.7) on page 399, which involve the spin tensor and pseudo spin tensor, and is trying to find a solution using previous equations and the chain rule. However, they encounter a mistake and are unsure how to demonstrate the correct formula without the spin tensor's antisymmetry in the last two indices. They ask for help in finding the mistake and reaching the correct answer.
  • #1
lapo
4
0
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!
 
  • #3
i haven't yet reach the right answer..
 

1. What is the Einstein-Cartan Theory?

The Einstein-Cartan Theory is a mathematical framework that combines the principles of General Relativity and Quantum Mechanics to describe the behavior of gravity and the structure of space-time.

2. How does the Einstein-Cartan Theory differ from Einstein's Theory of General Relativity?

The main difference between the two theories is that the Einstein-Cartan Theory includes the concept of spin, which is the intrinsic angular momentum of a particle. This allows for a more complete description of the behavior of matter in space-time.

3. What is the significance of the spin tensor in the Einstein-Cartan Theory?

The spin tensor is a mathematical quantity that describes the distribution of spin in a given space-time. It plays a crucial role in the theory as it determines the curvature of space-time in the presence of matter and contributes to the overall dynamics of the system.

4. How is the spin tensor defined in the Einstein-Cartan Theory?

In this theory, the spin tensor is defined as the covariant derivative of the spin density tensor, which is a measure of the density of spin in a given region of space-time. It is represented by a mathematical symbol called the spin connection.

5. What are some current applications of the Einstein-Cartan Theory?

The Einstein-Cartan Theory has been applied in various areas of physics, including cosmology, astrophysics, and particle physics. It has also been used to study the behavior of matter in extreme conditions, such as black holes and the early universe. Additionally, it has been incorporated into some attempts to unify gravity with the other fundamental forces of nature.

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