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Tetrads in General Relativity

VI. Physical Geometry

Lorentz Invariance

   Since local Lorentz invariance is a fundamental symmetry in GRT, the Dirac equation (6.19) should be invariant under the combined local transformations (6.12) and:

6.20

\[{\mathcal{D}_a}\psi (x) \to {({\Lambda ^{ - 1}})^b}_a{\mathcal{D}'_b}\psi '(x)\]

The tetrad index of the coderivative transforms according to the Lorentz group representation ${\Lambda _a}^b(x)$, whereas the spin representation generates the spinor transformation $S[\Lambda ]$.

   The tetrad coderivatives in (6.20) are ${e^\mu }_a{\mathcal{D}_\mu } := {\mathcal{D}_a} = {\partial _a} + {\Gamma _a}$ and ${\mathcal{D}'_a}: = {\partial _a} + {\Gamma '_a}$ with

6.21

\[{\Gamma _a}: = {e^\mu }_a{\Gamma _\mu } = \frac{1}{4}{\omega _{abc}}{\gamma ^b} \wedge {\gamma ^c}{\text{ }}\]

the tetrad spinor connection (6.18) expressed in terms of the Ricci rotation coefficients defined in (4.19). The primed spinor connection is obtained by substituting the transformed spin connection coefficients (4.9) into (6.18). With the fundamental relation (6.14) it is then found that the tetrad spinor connections are related by the (invertible) transformation

6.22

\[{\Gamma '_a} = S[\Lambda ]({\partial _a} + {\Gamma _a}){S^{ - 1}}[\Lambda ]\]

   The last property ensures Lorentz covariance of the coderivative:

6.23

\[{\mathcal{D}'_a}S[\Lambda ]\psi ({\Lambda ^{ - 1}}x) = S[\Lambda ]{\mathcal{D}_a}\psi ({\Lambda ^{ - 1}}x)\]

because the inhomogeneous derivative terms cancel out at the right-hand side. Analogously to (6.13,14), the Lorentz invariance of the Dirac equation in curved spacetime follows:

6.24

\[(i{\gamma ^a}{\mathcal{D}_a} - m)\psi (x) \to S(\Lambda )(i{\gamma ^a}{\mathcal{D}_a} - m)\psi ({\Lambda ^{ - 1}}x)\]

   The covariant Dirac equation may be viewed as a result of the gravitational minimal coupling prescription, that is, that the coupling of matter fields to gravitation may be achieved by replacing ordinary partial derivatives in the physical laws of special relativity by Fock-Ivanenko coderivatives.