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Tetrads in General Relativity

IV. Spin Connection

Connection Coefficients

   The coderivative of a tensor in the coordinate basis is given by its partial derivative plus correction terms. The same procedure applies to the tetrad basis $\left\{ {{{\mathbf{e}}_a}} \right\} = \left\{ {{{\mathbf{e}}_0},{{\mathbf{e}}_1}, {{\mathbf{e}}_2},{{\mathbf{e}}_3}} \right\}$, but the $\Gamma $-connection coefficients (1.15) are replaced by the spin connection coefficients denoted $\Sigma {{_\mu}{^b}}{_a}$:

4.1

\[  {D_\mu }{{\mathbf{e}}_a} := \Sigma {{_\mu}{^b}}{_a} {{\mathbf{e}}_b}{\quad}{D_\mu }{{\mathbf{e}}^a} := - \Sigma {{_\mu}{^a}}{_b} {{\mathbf{e}}^b}  \]

4.2

\[  \Sigma {{_\mu}{^b}}{_a} = {{\mathbf{e}}^b} \cdot ({D_\mu }{{\mathbf{e}}_a}): = {{\mathbf{e}}^b} \cdot {\Sigma _\mu }({{\mathbf{e}}_a})  \]

The name ‘spin connection’ comes from the fact that this formalism can be used to take coderivatives of spinors. The generic name is ‘Lorentz connection’ in recognition of the underlying Lorentz symmetry of the tetrad basis. The index convention is taken from [Wikipedia: Spin Connection].

   By definition the spin connection coefficients are anti-symmetric, ${\Sigma _\mu }^{ab} = - {\Sigma _\mu }^{ba}$, in their internal indices. This can be understood as following from the requirement that the internal Minkowski metric be compatible with the coderivative:

4.3

\[0 = {D_\mu }{\eta ^{ab}}: = {D_\mu }({{\mathbf{e}}^a} \cdot {{\mathbf{e}}^b}) = - {\Sigma _\mu }^{ab} - {\Sigma _\mu }^{ba}\]

The anti-symmetry reflects the fact that the spin connection is the generator of a Lorentz transformation for each $\mu $.

   Given that tetrads, at any given point, are orthonormal by definition, the relationship between two tetrad frames is necessarily a transformation of the group $\text{SO}(1,3)$. Therefore, the basis vectors at two nearby events must be related to each other by an infinitesimal Lorentz transformation:

4.4

\[ {{\mathbf{e'}}_a}(x + dx) = {{\mathbf{e}}_a}(x) + d{x^\nu } {\Sigma {{_\nu}{^b}}{_a}}(x){{\mathbf{e}}_b}(x)\]

which can be thought of as a rotation in spacetime keeping lenths and angles the same: ${{\mathbf{e'}}_a}(x + dx) \cdot {{\mathbf{e'}}_b}(x + dx) = {\eta _{ab}}$.