I've been hearing recently that the Navier-Stokes (NS) equations are invariant under a Lorentz transformation, so I tried to prove it just changing terms of transformed velocity instead of the velocity,
$$u(\vec r,t)_{i}=\frac{u'(\vec r,t)_{i}+v}{1+\frac{v}{c}u'(\vec r,t)_{i}}$$
in
$$\rho[\partial_{t}u(\vec r,t)_{i}+u(\vec r,t)_{j}\partial^{j}u(\vec r,t)_{i}]=\eta\partial^{j}\partial^{j}u(\vec r,t)_{i}-\partial^{i}P+\rho\vec f$$
but suddenly I noted that in the NS equations the velocity depends on space and time $v(\vec r,t)$ and in special relativity does not depends on, I guess.
Now I feel a little bit confused about. Do I have to find a new Lorentz transformation for the velocity $v(\vec r,t)$? Do I need to introduce information about the metric tensor? is there another way to prove it? What about the pressure and the force terms?
I just want a way to proceed, I don't really want all the answers. Thanks for the read. I'm an undergraduate.
