Minowski's Spacetime in Heterogeneous Medium

Let's suppose that both locations $L_{1}(x_{1}, y_{1}, z_{1})$ and $L_{2}(x_{2}, y_{2}, z_{2})$ are under water, somewhere in the Pacific Ocean. Now light in the water has a smaller speed (c$_{\mathrm{w}})$ than in vacuum, i.e. $c_{w}$\textit{ \textless c}. Therefore within the same interval of time $t_{2\thinspace }- t_{1}$, light travels in the water a lesser distance than $L_{1}L_{2}$. Thus $d(E_{1}, E_{2})$ has a different representation now $L_{1}L$. And, if instead of water we consider another liquid, then $d(E_{1}, E_{2})$ would give another new result. Therefore, if we straightforwardly extend Minkowski's spacetime for an aquatic only medium, i.e. all locations $L_{i}(x_{i}, y_{i}, z_{i})$ are under water, but we still refer to the light speed but in the water $(c_{w})$ then the coordinates of underwater events $E_{w}$ would be $E_{w}(x_{i}, y_{i}, z_{i} ,c_{w}, t_{i})$ and Minkowski underwater distance would be: \[ d_{w}^{2}(E_{w1} ,E_{w2} )=c_{w}^{2}(t_{2} -t_{1} )^{2}-[(x_{2} -x_{1} )^{2}+(y_{2} -y_{1} )^{2}+(z_{2} -z_{1} )^{2}] \] But if the underwater medium is completely dark it might be better to consider the speed of sound in order to communicate (similarly as submarines use sonar). Let's denote by s$_{\mathrm{w}}$ the underwater speed of sound. Then the underwater events $E_{ws}(x_{i}, y_{i}, z_{i} ,s_{w}.t_{i})$ with respect to the speed of sound has the Minkowski underwater distance: \[ d_{ws}^{2}(E_{ws1} ,E_{ws2} )=s_{w}^{2}(t_{2} -t_{1} )^{2}-[(x_{2} -x_{1} )^{2}+(y_{2} -y_{1} )^{2}+(z_{2} -z_{1} )^{2}] \] Similarly for any medium $M$ where all locations $L_{i}(x_{i}, y_{i}, z_{i})$ are settled in, and for speed of any waves W that can travel from a location to another location in this medium.