# Invariance of spin coefficients

I have a question about how spin coefficients (Newman Penrose formalism) transform.

I know that if we perform a tetrad rotation, say of Class III:

$(l,n,m,\overline{m})\mapsto \left(\frac{1}{A}l, An, e^{i\theta}m, e^{-i\theta}\overline{m}\right)$

then the spin coefficients transform also, as functions of $A,\theta$ and their derivatives.

Now, my question is: assume I'm working in a spacetime, and want to do a coordinate transformation (not a tetrad rescaling!) Would it make sense to say that the spin coefficients should be invariant under any coordinate transformation? Is it always true/not true?

Thank you for any hints.

## 1 Answer

Spin coefficients are indeed invariant under coordinate changes.

What changing coordinates does is that it changes the coordinate representation of the tetrad vectors (in terms of the coordinate vector fields $\partial^\alpha$), but the geometry (which is captured by the relationships between the spin coefficients) remains the same.

In fact, that's sort of the point of working with the tetrad formalism.