Derivation of Dirac matrices in spherical polar coordinates How to derive Dirac $\gamma$ matrices in spherical polar coordinates?
 A: In flat space, you only need to rotate bases in the spacelike part, and leave $\gamma^0$ alone; it stays the same.
That is, you just express the spacelike dot product $\vec \gamma \cdot \nabla$ in spherical coordinates,
$$
\vec \gamma=  \hat{\mathbf r} \gamma_ r 
  + \hat{\boldsymbol\theta} \gamma_\theta 
  + \hat{\boldsymbol\varphi} \gamma _\varphi,   \\
\nabla  = \hat{\mathbf r}  \partial_ r
  + \hat{\boldsymbol\theta} {1 \over r}{\partial_\theta} 
  + \hat{\boldsymbol\varphi}{1 \over r\sin\theta}{  \partial _\varphi}, $$
where
$$
\begin{align}
 \hat{\mathbf r} &= \sin \theta \cos \varphi \,\hat{\mathbf x} +
  \sin \theta \sin \varphi \,\hat{\mathbf y} + \cos \theta \,\hat{\mathbf z}, \\
 \hat{\boldsymbol\theta} &= \cos \theta \cos \varphi \,\hat{\mathbf x} +
  \cos \theta \sin \varphi \,\hat{\mathbf y} - \sin \theta \,\hat{\mathbf z}, \\
 \hat{\boldsymbol\varphi} &= - \sin \varphi \,\hat{\mathbf x} +
  \cos \varphi \,\hat{\mathbf y}
\end{align} $$
an (orthogonal!) rotation ,
$$ \begin{pmatrix}  \hat{\mathbf r}   \\
 \hat{\boldsymbol\theta}   \\
 \hat{\boldsymbol\varphi}  \end{pmatrix}= R \begin{pmatrix}      \hat{\mathbf x} \\
 \hat{\mathbf y}\\  \hat{\mathbf z}\end{pmatrix}
 \\
R =\begin{pmatrix}
     \sin\theta\cos\varphi&\sin\theta\sin\varphi& \cos\theta\\
     \cos\theta\cos\varphi&\cos\theta\sin\varphi&-\sin\theta\\
    -\sin\varphi&\cos\varphi   &0
   \end{pmatrix},\\
$$
hence
$$
 \begin{pmatrix}  \gamma_  r   \\
 \gamma_ \theta   \\
 \gamma_\varphi  \end{pmatrix}= R \begin{pmatrix}      \gamma_ 1 \\
 \gamma_2\\  \gamma_3 \end{pmatrix},
$$
the very matrices (5) and (6) of Dzhunushaliev cited, (XV-16 of Vaudon), dependent on $\varphi$, alright.
As you might expect for all directional gradients in spherical coords,
$$
\vec \gamma \cdot \nabla=    
   \gamma_ r    \partial_ r
  +   \gamma_\theta  {1 \over r}{\partial_\theta} 
  +    \gamma _\varphi {1 \over r\sin\theta}{  \partial _\varphi}~~, $$
which probably addresses your tetrad puzzlement, in practice.
