# How to prove that Navier-Stokes equations are Lorentz invariant?

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.

• Where did you read that NS is Lorentz invariant? – AccidentalFourierTransform May 3 '18 at 16:40
• I had heard from college friends, then i decide to try to prove it . is it incorrect? – Ricardo Guzman May 3 '18 at 16:53
• No, but there is a relativistic generalization of the NS equation. – Thomas May 3 '18 at 16:56
• @RicardoGuzman I think it's great that you, as an undergrad, are trying to learn physics beyond the syllabus. I want to encourage you in the future to always cite your sources: every claim you make has to be come hand in hand a link to a reputable source where you read it. If you heard it from someone else, ask them for a source. Making claims without a source is inadmissible in science. It takes some time to learn that, but you'll have to sooner or later. Good luck! – AccidentalFourierTransform May 3 '18 at 17:07
• You're right. Sorry for my bad english (i'm spanish speaker). – Ricardo Guzman May 3 '18 at 22:43