Let $\mathcal{P}_{2,ac}(\mathbb{R}^d)$ be the space of all probability measures on $\mathbb{R}^d$ with finite 2nd momentum and absoulute continuous w.r.t. the Lesbegue measure. It is known that if we implement $\mathcal{P}_{2,ac}(\mathbb{R}^d)$ with $L^2(\mu)$ metric at $\mu$, the geodesic $\mu,\nu\in\mathcal{P}_{2,ac}(\mathbb{R}^d)$ is given by an "optimal transport" and
$$W_2(\mu,\nu)=\inf_{\mu_0=\mu,\mu_1=\nu}\left\{\int_0^1\|v_t\|_{L^2(\mu_t)}\bigg|\partial_t\mu_t+\text{div}(\mu_tv_t)=0\right\},$$
where $W$ is for Wasserstein distance.
So a natrual question is, what will be the new geodesic if we change the metric? I want to solve this problem but have no clue: $$\inf_{\mu_0=\mu,\mu_1=\nu}\{\int_0^1\|v_t\|_{L^2(\mathbb{R}^d)}|\partial_t\mu_t+\text{div}(\mu_tv_t)=0\}.$$