let $\mu$ and $\nu$ be to probability measure on $\mathbb{R}^n$ such that second moment matrix of $\nu$, is $S_\nu:=\int yy^Td\nu(y)=I_n$. So we define the following measure on the unit sphere $\mathbb{S}^{n-1}$ as $$d\nu^*(x)=\int_0^{+\infty}\frac{r^2}{n}d\nu(xr)\qquad \forall x\in\mathbb{S}^{n-1}$$ by this definition of $\nu^*$ i get a probability measure on the unit sphere and its second moment matrix $S_{\nu^*}=\frac{1}{n}I_n$ because $$S_\nu:=\int_{\mathbb{S}^{n-1}} xx^Td\nu^*(x)=\frac{1}{n}\int_{\mathbb{S}^{n-1}}xx^T\int_0^{+\infty}\frac{r^2}{n}d\nu(xr)=\frac{1}{n}\int_{\mathbb{R}^{n}}yy^Td\nu(y)=\frac{1}{n}I_n$$
so can we find any relation between $W_2(\mu,\nu)$ and $W_2(\mu,\nu^*)$? Where $W_2(.,.)$ is the 2-Wasserstein metric means $$W_2^2(\mu,\nu)=\inf_{\pi\in\Gamma(\mu,\nu)}\int\Vert x-y\Vert^2d\pi(x,y)$$ where $\Gamma(\mu,\nu)$ is the set of coupling between $\mu$ and $\nu$.
Can someone please recommend any sources that are related to my question.