\( \newcommand{\bs}{\boldsymbol}
\newcommand{\be}{\begin{equation}} \newcommand{\ee}{\end{equation}}
\newcommand{\ba}{\begin{array}} \newcommand{\ea}{\end{array}}
\newcommand{\bea}{\begin{eqnarray}} \newcommand{\eea}{\end{eqnarray}}
\newcommand{\bean}{\begin{eqnarray*}} \newcommand{\eean}{\end{eqnarray*}}
\newcommand{\la}{\label} \newcommand{\nn}{\nonumber}
\newcommand{\half}{{\scriptstyle \frac{1}{2}}}
\newcommand{\third}{{\scriptstyle \frac{1}{3}}}
\newcommand{\bli}[2]{\begin{list}{#1}{\itemsep=0.0cm \topsep=0.0cm
\partopsep=0.0cm #2}} \newcommand{\eli}{\end{list}}
\newtheorem{problem}{Problem}[chapter]
\newcommand{\bprob}{\begin{problem}} \newcommand{\eprob}{\end{problem}}\)
XIII. ECKS Theory
The contorsion tensor in differential geometry is the
difference between a connection with and without torsion, e.g. the
difference between the general asymmetric affine connection $\Gamma
_{\mu \nu }^\lambda $ and the symmetric Levi-Civita connection
(1.29). Specifically, the contorsion tensor ${C_{abc}}$ in
Riemann-Cartan space is defined as the difference between the spin
connection (Ricci coefficient) (4.20) and the torsion free
connection (4.30), here indicated by the overhead symbol $ \circ $:
\[{\omega _{abc}} = {\mathop
{\omega} \limits^ \circ }_{abc} + {C_{abc}}\]
Metric compatibility is assumed, i.e. both connections and the
contorsion tensor are anti-symmetric in their last two
indices
The explicit form of ${C_{abc}}$ may be obtained from the tensor
form (4.24) of the first Cartan equation which reads in tetrad base
components:
\[2{\omega _{[ab]c}} =
{f_{abc}} - {T_{cab}}\]
It differs from (4.30) by the components of the torsion 2-form:
\[{{\mathbf{T}}^a} =
\frac{1}{2}{T^a}_{bc}{{\mathbf{e}}^b} \wedge {{\mathbf{e}}^c} \quad
{T^a}_{bc} = 2{\Gamma ^a}_{[bc]}\]
Combining three index permutations of (13.31) in the same way as
described under (4.30), one derives for the contorsion tensor the
invertible relation:
\[{C_{abc}} =
\frac{1}{2}\left( {{T_{abc}} + {T_{cba}} - {T_{bca}}} \right) =
\frac{1}{2}{T_{abc}} + {T_{[cb]a}}\]
A contraction with ${{\mathbf{e}}^a}$ on the first index of (13.30)
yields the Maurer-Cartan connection as the sum of the Levi-Civita
and contorsion 1-forms:
\[{{\bs{\omega }}^a}_b =
{\mathop {{\bs{\omega}}^a} \limits^ \circ }_b + {{\mathbf{C}}^a}_b
\quad {{\mathbf{C}}^a}_b: = {{\mathbf{e}}^c}{{C_c}{^a}_b}\]
This decomposition of the connection is unique, because the
torsion-free spin connection satisfies the torsion constraint
(4.27). This is easily shown by wedging with ${{\mathbf{e}}^b}$ and
using the identity ${{\mathbf{C}}^a}_b \wedge {{\mathbf{e}}^b} =
{{\mathbf{T}}^a}$.