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Tetrads in General Relativity

VI. Physical Geometry

Spinor Connection

   A formally covariant Dirac equation in GRT is most easily constructed by introducing spacetime-dependent gamma matrices by way of Einstein’s vierbein fields:

6.15

\[ {\gamma ^\mu }(x) : = {\operatorname{e} ^\mu }_a(x){\gamma ^a}, \qquad{\gamma _\mu }(x) : = {\operatorname{e} _\mu }^a(x){\gamma _a} \]

On account of the basic vierbein equality (2.3), their anti-commutator is seen to be equal to the metric tensor:

6.16

\[{\gamma ^\mu } \cdot {\gamma ^\nu } : = \frac{1}{2}({\gamma ^\mu }{\gamma ^\nu } + {\gamma ^\nu }{\gamma ^\mu }) = {g^{\mu \nu }}I\]

The derivative term of the Dirac equation (6.10) now can be put in the familiar form $i{\gamma ^\mu }{\partial _\mu }\psi (x)$, but with spacetime-dependent gamma matrices.

   In curved spacetime, also the spinor transformation matrix depends on spacetime location: $S(\Lambda ) \to S[\Lambda ]: = S[\Lambda (x)]$. As a consequence, the spacetime derivative of a spinor does not transform as a spinor. The remedy is to replace the derivative by the appropriate FI-coderivative (6.8):

6.17

\[{\partial _\mu }\psi (x) \to {\mathcal{D}_\mu }\psi (x) = ({\partial _\mu } + {\Gamma _\mu })\psi (x)\]

The spinor connection $\Gamma _\mu$ only acts on spinor indices and has the explicit form, see (6.9c):

6.18

\[\Gamma_\mu := \frac{1}{4}{\Sigma _{\mu ab}}{\gamma ^a} \wedge {\gamma ^b}\]

   The corresponding Dirac equation in curved space then simply reads

6.19

\[(i{\gamma ^\mu }{\mathcal{D}_\mu } - m)\psi (x) = 0\]

The coefficient matrices $\gamma^\mu$ are not unique. It rests to be shown that all representations are equivalent and that the Dirac equation (6.19) is invariant under local Lorentz transformations.