How to construct a nonholonomic tetrad basis Suppose I have some body with spacetime position $x^{\mu}$ and 4-velocity $u^{\mu}$. The position is defined with respect to some coordinate (holonomic) basis.
How would I construct the position defined w.r.t to some non-holonomic tetrad basis, $x^{\hat{\mu}}$?
Thanks
 A: The relation between any two basis on a manifold is always a transformation matrix. If the two basis are holonomic (coordinate basis) this transformation matrix is actually the Jacobian matrix between the two coordinate systems. Otherwise the matrix is a generic non singular matrix ${L^\alpha}_{\hat{\beta}}$.
In simpler words: you express your anholomic basis $\mathbf{e}_{\hat{\beta}}$ vectors as a linear combination of the holonomic basis vectors $\mathbf{e}_{\alpha}$.
A simple example: suppose you use polar coordinates $(r,\theta)$ in $\mathbf{R}^2$. The basis vectors are $(\frac{\partial }{\partial r},\frac{\partial }{\partial \theta})$. Vector $\frac{\partial }{\partial \theta}$ does not have unit module though, so if you want to use an orthonormal basis you choose $(\frac{\partial }{\partial r},\frac{1}{r}\frac{\partial }{\partial \theta})$. This basis is anholomic and is expressed as a very simple linear combination of the coordinate basis vectors. ${L^\alpha}_{\hat{\beta}}$ in this case is simply \begin{pmatrix}
1 & 0\\ 
0 & \frac{1}{r}
\end{pmatrix}
In your question you ask how to express "position" in the anholomic basis. This is an ill-conceived question. "Position" is not a vector, it is a point on a manifold and is represented as a n-tuple in a coordinate system. Since it is not a vector, it is not expressible as a linear combination of basis vectors. 
The velocity vector $u^\alpha$ on the other hand, transforms as a covariant vector via the matrix   ${L^{\hat{\beta}}}_\alpha$ which is the inverse of the matrix used to transform the basis vectors.
