I am currently following Michael Stone's lecture notes (http://people.physics.illinois.edu/stone/torsion_review.pdf) on deriving the Belinfante-Rosenfeld Energy Momentum tensor in a variational setting.
However I simply seem to be unable to arrive at the same equations as he. Since there are some steps missing in his derivation, I was hoping to get some help here (perhaps even by himself, since he seems to be active in this community). I am especially stumped by the calculations on page 15.
Professor Stone reduces the variation of the spin connection coefficients to the variation of the anholonomy coefficients of the orthonormal basis in which he works. He defines: $\left[e_k, e_j\right] = e_i f^i_{\;\;kj}$. Which I have calculated to be: $f^i_{\;\;kj} = e_\alpha^{\;\;i}f^\alpha_{\;\;kj} = e_\alpha^{\;\;i}\left[e_j\left(e^\alpha_{\;\;k}\right) - e_k\left(e^\alpha_{\;\;j}\right)\right]$
Varying this expression with respect to the vielbein would then by my calculation yield: $\delta f^i_{\;\;kj} = \left(\delta e_\alpha^{\;\;i}\right)f^\alpha_{\;\;kj} + e_\alpha^{\;\;i} \delta f^\alpha_{\;\;kj} $ with $\delta f^\alpha_{\;\;kj} = e_j\left(\delta e^\alpha_{\;\;k}\right) - e_k\left( \delta e^\alpha_{\;\;j}\right) $. My first question is therefore, whether my "filling the gaps" up until here is correct.
He then proposes the following equation: $\delta f_{ikj} = \eta_{ib}\left\lbrace\nabla_j\left( e_\alpha^{\;\;b}\delta e^\alpha_{\;\;k}\right) - \nabla_k\left( e_\alpha^{\;\;b}\delta e^\alpha_{\;\;j}\right) - \omega^b_{\;\;k\mu}\delta e^\mu_{\;\;j} + \omega^b_{\;\;j\mu}\delta e^\mu_{\;\;k}\right\rbrace$. This is the main thing I've been trying to understand. One "trick" I think he uses is that $\nabla_k\left( e_\alpha^{\;\;b}\delta e^\alpha_{\;\;j}\right) = \nabla_k\left(\delta e^b_{\;\;j}\right)$ since the tetrad "postulate" holds, allowing us to identify the term $e_\alpha^{\;\;b}\Gamma^\alpha_{\;\;\lambda k}\delta e^\lambda_{\;\;j}$ with the term $\omega^b_{\;\;mj}\delta e^m_{\;\;k}$. Am I perceiving that correctly?
My next problem, however, is that after using said trick we receive: $\nabla_j\left( e_\alpha^{\;\;b}\delta e^\alpha_{\;\;k}\right) = e_\alpha^{\;\;b} e_j\left(\delta e^\alpha_{\;\;k}\right) + \omega^b_{\;\;mj}\delta e^m_{\;\;k} - e_\alpha^{\;\;b}\omega^m_{\;\;kj}\delta e^\alpha_{\;\;m}$ if I now substract the same term with $j$ and $k$ interchanged I get: $\nabla_j\left( e_\alpha^{\;\;b}\delta e^\alpha_{\;\;k}\right) - \nabla_k\left( e_\alpha^{\;\;b}\delta e^\alpha_{\;\;j}\right) = e_\alpha^{\;\;b} e_j\left(\delta e^\alpha_{\;\;k}\right) + \omega^b_{\;\;mj}\delta e^m_{\;\;k} - e_\alpha^{\;\;b}\omega^m_{\;\;kj}\delta e^\alpha_{\;\;m} - e_\alpha^{\;\;b} e_k\left(\delta e^\alpha_{\;\;j}\right) - \omega^b_{\;\;mk}\delta e^m_{\;\;j} + e_\alpha^{\;\;b}\omega^m_{\;\;jk}\delta e^\alpha_{\;\;m} = e_\alpha^{\;\;b} \left(e_j\left(\delta e^\alpha_{\;\;k}\right) - e_k\left(\delta e^\alpha_{\;\;j}\right)\right) + e_\alpha^{\;\;b}\left(\omega^m_{\;\;jk} - \omega^m_{\;\;kj} \right)\delta e^\alpha_{\;\;m} + \omega^b_{\;\;mj}\delta e^m_{\;\;k} - \omega^b_{\;\;mk}\delta e^m_{\;\;j} = e_\alpha^{\;\;b}\delta f^\alpha_{\;\;jk} + e_\alpha^{\;\;b} f^m_{\;\;kj}\delta e^m_{\;\;k}+ + \omega^b_{\;\;mj}\delta e^m_{\;\;k} - \omega^b_{\;\;mk}\delta e^m_{\;\;j} = e_\alpha^{\;\;b}\delta f^\alpha_{\;\;jk} + f^\alpha_{\;\;kj}\delta e_\alpha^{\;\;b} + \omega^b_{\;\;mj}\delta e^m_{\;\;k} - \omega^b_{\;\;mk}\delta e^m_{\;\;j} = \delta f^b_{\;\;jk} + \omega^b_{\;\;mj}\delta e^m_{\;\;k} - \omega^b_{\;\;mk}\delta e^m_{\;\;j}$ The last two terms, however, do not coincide with the terms in the expression Stone gives for $\delta f_{ijk}$, and even if I go ahead anyway and assemble them into the expression for $\delta\omega_{ijk}$ it does not work out to the expression given by Stone.
So I am at my wits end and after a week of trying to fix these problems, I am turning to you in hopes of receiving help. Any guidance or help would be greatly appreciated!
