Coordinate conditions in black hole solutions The most general metric for a static and spherically symmetric metric is given by:
$$
ds^2 = e^{2\gamma(u)}dt^2 - e^{2\alpha(u)}du^2 - e^{2\beta(u)}d\Omega^2
$$
I have the freedom of choosing Schwarzschild coordinates, in which:
$$
e^{2\beta(u)} = u^2,
$$
and now $u$ becomes the radius of the sphere.
However I have the freedom of choosing other coordinates systems, i.e. gauge freedom, like the harmonic gauge $\alpha(u) = 2\beta(u) + \gamma(u)$ or the quasi-global gauge given by $\alpha(u)=-\gamma(u)$ (those two are the ones I use the most).
My questions are: 


*

*How do I prove that those gauges may be used without any loss of generality? (As example, how do I show if $\alpha=-5\gamma + \beta^2$ is a gauge or not?)

*I see this as a coordinate freedom, but in my research it's mostly known as "gauge freedom", any differences or just nomenclature?
 A: What Erik has said is correct, essentially you write down an ODE (for your case) but PDE in general and see if there are solutions for which your 'gauge' choice holds. This is exactly a coordinate-freedom, perhaps this is where the confusion for question 2 comes in.
In GR there is a second kind of gauge freedom known as tetrad invariance, which is invariance under reference frame transformations. Essentailly you want the metric tensor to be invariant under a tetrad transformation. For example, in the Newman-Penrose formalism we write the metric in terms of the null tetrad $(\boldsymbol{l},\boldsymbol{n},\boldsymbol{m},\boldsymbol{\bar{m}})$
\begin{equation}
g_{ab}=-l_a  n_b - n_a  l_b +m_a  \bar{m}_b +\bar{m}_a  m_b,
\end{equation}
which we want to be invariant under a transformation of the basis (e.g. $m \rightarrow e^{i \vartheta} m$ and $\bar{m} \rightarrow e^{-i \vartheta} \bar{m}$ for some angle $\vartheta$). Note the absence of coordinates. The term `gauge invariant' is usually used to blanket both the coordinate and tetrad invariance together. A fantastic article which covers this in detail is Stewart and Walker (1974) (unfortunately pay-walled), where they discuss the importance of the distinction.
