Are the Pearson and Spearman rank correlation coefficients related in a specific way for uniform random variables? Specifically, is the relationship $\rho_{spearman} = 2*\sin(\frac{\pi}{6}\rho_{pearson})$? If so, why?

This formula is from Pearson 1907, see e.g. Rank Correlation and ProductMoment Correlation Author(s): P. A. P. Moran Source: Biometrika, Vol. 35, No. 1/2 (May, 1948), pp. 203206 Johan 


This is only valid for normal distribution. 

