smooth Kähler metric with respect to the unique complex structure on ${\mathbb}{CP}^2\#\,\overline{{\mathbb}{CP}}^{\,2}$, if and only if two conditions hold: i) $\phi_r>0$ everywhere, and ii) there are $b_0>a_0$ and $b_1<0$ such that $$\phi(r)=b_0+b_1 e^{-r} + b_2 e^{-2r}+ {\mathcal}O(e^{-3r})\quad\mbox{ as }\quad
r\rightarrow\infty.$$ Alternatively, one may obtain a complete Calabi metric on the noncompact space ${\mathcal}L^2_{-1}$ by imposing conditions at spatial infinity that guarantee completeness; see, e.g., [@FIK03]. As noted in equation (19) of that paper, any ${\mathrm}U(2)$-invariant metric on ${\mathbb}C^2\backslash(0,0)$ can be written in real coordinates on ${\mathbb}R^4\backslash(0,0,0,0)$ as $$\label{U2-real}
h_{\,{\mathbb}R^4\backslash(0,0,0,0)}
=\phi_r\big(\frac14{\mathrm}dr{\otimes}{\mathrm}dr+\omega^1{\otimes}\omega^1\big)
+\phi\big(\omega^2{\otimes}\omega^2+\omega^3{\otimes}\omega^3\big).$$

A coordinate transformation {#Diffeos}
---------------------------

A comparison of equations  and shows that a coordinate transformation is needed to write a Calabi metric in the $s$-coordinate system. We implement this as follows. Recalling that $s(x,t)$ denotes arclength from the ${\mathcal}S^3$ at the “interior” point $x=0$, and motivated by Calabi’s (fixed) $r$-coordinate introduced in Section \[CalabiConstruction\], we define here a function $$\label{magic}
\rho(s,t):=2\int_0^s\frac{{\mathrm}d\bar s}{f(\bar s,t)}.$$ The closing conditions then show that $\rho\rightarrow\pm\infty$ at ${\mathcal}S^2_\pm$. Moreover, one has $$\label{dsdr}
{{\mathrm}{d}s}=\frac12f\,{\mathrm}d\rho,$$ so that equation  may be re-expressed in the form $$\label{transformation}
 G = f^2\big(\frac14 {\mathrm}d\rho{\otimes}{\mathrm}d\rho+\omega^1{\otimes}\omega^1\big)
       + g^2\big(\omega^2{\otimes}\omega^2+ \omega^3{\otimes}\omega^3\big),$$ where we emphasize that the coordinate $\rho$ is allowed to depend on time. We note 