\( \newcommand{\be}{\begin{equation}} \newcommand{\ee}{\end{equation}} \newcommand{\ba}{\begin{array}} \newcommand{\ea}{\end{array}} \newcommand{\bea}{\begin{eqnarray}} \newcommand{\eea}{\end{eqnarray}} \newcommand{\bean}{\begin{eqnarray*}} \newcommand{\eean}{\end{eqnarray*}} \newcommand{\la}{\label} \newcommand{\nn}{\nonumber} \newcommand{\half}{{\scriptstyle \frac{1}{2}}} \newcommand{\third}{{\scriptstyle \frac{1}{3}}} \newcommand{\bli}[2]{\begin{list}{#1}{\itemsep=0.0cm \topsep=0.0cm \partopsep=0.0cm #2}} \newcommand{\eli}{\end{list}} \newtheorem{problem}{Problem}[chapter] \newcommand{\bprob}{\begin{problem}} \newcommand{\eprob}{\end{problem}}\)

Tetrads in General Relativity

IV. Spin Connection

Connection Relationships

   The coderivative of a tensor is given by its partial derivative plus connection terms. In the case of a tetrad (non-coordinate) basis, each Latin index gets a factor of the spin connection. For example, if ${\mathbf{A}}$ is a vector

4.6

\[  {D_\mu }{\mathbf{A}} = {D_\mu }({A^a}{{\mathbf{e}}_a}) = ({\partial _\mu }{A^a} +\Sigma {{_\mu}{^b}}{_a} {A^b}){{\mathbf{e}}_a}: = (D_\mu ^\Sigma {A^a}){{\mathbf{e}}_a}  \]

Here $D_\mu ^\Sigma$ is just a short-hand notation.

   Since a vector is a geometric object, it is independent of the way it is written. So, equation (4.6) must be equivalent to the coordinate representation (1.17). The relationship between connections is given by the vierbeins. From the definitions (1.15) and (4.2) of the connections one derives the equivalent relationships

4.7

\[  \Gamma _{\mu \nu }^\lambda : = {{\mathbf{g}}^\lambda } \cdot ({D_\mu }{{\mathbf{g}}_\nu }) = {e^\lambda }_a({\partial _\mu }{e_\nu }^a + \Sigma {{_\mu}{^a}}{_b}\,{e_\nu }^b): = {e^\lambda }_a {D_{( \mu} ^\Sigma} {e_{\nu)} }^a   \]

4.8

\[{\Sigma {_\mu }}{^a}{_b}: = {{\mathbf{e}}^a} \cdot ({D_\mu }{{\mathbf{e}}_b}) = {e_\nu }^a({\partial _\mu }{e^\nu }_b + \Gamma _{\mu \lambda }^\nu {e^\lambda }_b): = {e_\nu }^a {D_{\mu}^\Gamma} {e^\nu }_b \]

   The Levi-Civita connection coefficients are symmetric in their lower indices; so, in (4.7) these indices on the right-hand side need to be symmetrized. This also implies that relation (4.8) should be taken as defining the torsion-free spin connection, since the connection coefficients are derived from the Levi-Civita connection, which is the unique metric compatible, torsion-free connection on a Lorentzian (pseudo-Riemannian) manifold. In general, there is no restriction: the spin connection may also contain torsion.

   Equation (4.8) is covariant under manifold diffeomorphisms (MDs), but not local Lorentz transformations (LLTs) ${e_\mu }^a(x) \to {{ e'}_\mu }^a(x) = {\Lambda ^a}_b(x){e_\mu }^b(x)$. Under the latter, the spin connection transforms in-homogeneously as

4.9

\[{\Sigma '}{_\mu }{^a}{_b} = {\Lambda ^a}_c\,{\Sigma _\mu }{^c}{_d}\,{\Lambda _b}^d + {\Lambda ^a}_c \,{\partial _\mu }\, {\Lambda _b}^c\]

with two distinct contributions: (first term) non-inertial effects induced by the new frame; (second term) inertial contributions due to the rotation of the new frame with respect to the previous one. This results in the proper transformation of the coderivative. On the other hand, equation (4.7) is covariant under LLTs, but not MDs.