Fisher's variance-stabilising $z$-transformation

Suppose $\{(X_i,Y_i)\}$ is a random sample from a bivariate normal distribution with correlation coefficient $\rho$, and

$$\sqrt{n}\,(r_n-\rho) \overset{d}{\longrightarrow} N\bigl(0,(1-\rho^{2})^{2}\bigr).$$

Which transformation $g(\cdot)$ gives $\sqrt{n}\bigl(g(r_n)-g(\rho)\bigr)\overset{d}{\longrightarrow} N(0,1)$?