mathematical objects at different space time points, one first has to transform to the frame in which the metric tensor looks like the metric tensor $\eta^{\mu\nu}$ of flat Minkowski space time. Thus a specific Lorentz transformation at each space time point has to be performed. This is the reason why we have to consider local Lorentz transformations within the framework of general relativity. Thus, it is our next task to formulate the Dirac equation in a way that obeys local symmetry transformations

$$\psi\ \rightarrow\ U(x)\psi\quad,\quad\partial_\mu\ \rightarrow\ \Lambda_\mu^\nu(x)\partial_\nu.
\label{95}$$

A Lagrangian of the following shape satisfies invariance under local Lorentz transformations

$$\mathcal{L}=\bar \psi(i\gamma^\mu D_\mu-m)\psi,
\label{25}$$

where $D_\mu$ denotes a new derivative operator (covariant derivative) which transforms under local Lorentz transformations according to

$$D_\mu\rightarrow\Lambda_\mu^\nu U(x) D_\nu U^{\dagger}(x).
\label{26}$$

It is clear that such a covariant derivative obeys local Lorentz invariance, because the $U$ operators to the left and right of the derivative operator cancel the operators which act on the right and on the left state in the Lagrangian.

The covariant derivative $D_\mu$ applied to the metric tensor is zero. This means that $D_\mu$ is the derivative connecting different space time points endowed with a local pseudo-Euclidean coordinate system in which $g_{\mu\nu}=\eta_{\mu\nu}$. 