I was reading the following paper


<http://scitation.aip.org/docserver/fulltext/aip/journal/jmp/4/7/1.1704018.pdf?expires=1460721373&id=id&accname=2112043&checksum=607EDBF852A9E209F384DB6DA228B416>


about the Taub-NUT metric, and I don't understand the origin of equation (2.55). Here is the (more or less self-contained) description of the issue:


the authors have constructed an orthonormal tetrad $(l,n,m,\overline{m})$, in the coordinate system $(x^{1},r,x^{3},x^{4})$. At some point the vector field $m$ has the form


$m=\overline{\rho}\hskip 1pt(Y\_{1}\: , \: 0 \: , \: Y\_{2} \: , \: iY\_{2})$


where $\rho$ is a function of $r$, and $Y\_{2}=Y\_{2}(x^{3},x^{4})$ is a real function. The coordinate freedom still available is the coordinate transformation of the form


$\zeta'=g(\zeta)$


where $\zeta:=x^{3}+ix^{4}$.


QUESTION: the authors say, "it is possible to choose such a coordinate transformation $g$ so that $Y\_{2}=\frac{1}{\sqrt{2}}+\frac{M}{2\sqrt{2}}((x^{3})^{2}+(x^{4})^{2})$", where $M$ is some constant. How do they know that? How could I see that? In general, what kind of calculation do I have to do for $Y\_{2}$ to see what kind of $g$ I could pick?


Thank you for any hints



Choosing a coordinate transformation
I will add here some more details to expand my comment. Any two functions $Y\_1(x^3,x^4)$ and $Y\_2(x^3,x^4)$ give local coordinates on any open domain of the $(x^3,x^4)$-plane where their Jacobian determinant is non-vanishing, $\frac{\partial(Y\_1,Y\_2)}{\partial(x^3,x^4)}$. Moreover, if you already have the function $Y\_2$ given to you, you can always complete it by another function $Y\_1$ to a local coordinate system, at least on some open subset that contains no critical points of $Y\_2$.


Next, let $r = \sqrt{\frac{2\sqrt{2}}{M} Y\_2 - \frac{2}{M}}$ and $\theta = Y\_1$. This transformation from $(Y\_1,Y\_2)$ coordinates to $(r,\theta)$ coordinates is a local diffeomorphism on any open domain that avoids the values $Y\_2 \le 1/\sqrt{2}$.


Finally, let $(y^3,y^4) = (r\cos\theta, r\sin\theta)$, which is again a local diffeomorphism on any open domain that avoids the values $r \le 0$. Then, as desired, $(y^3,y^4)$ form a local coordinate system on some open subset of $Y\_2 > 1/\sqrt{2}$ and $Y\_2 = \frac{1}{\sqrt{2}} + \frac{M}{2\sqrt{2}} ((y^3)^2+(y^4)^2)$.


