# How does Galilean invariance lead to equality of "mass" flux and momentum density?

Let us consider a fluid, with spacetime translation symmetry, and one internal $$U(1)$$ symmetry. Corresponding to the spatial translation symmetry, we can write down a momentum conservation equation for momentum density $$G_i$$ as : $$\partial_t G_i + \nabla_j(P\delta_{ji} + v_j G_i) = 0$$ ; where $$P$$ is the Pressure (assuming no dissipation)

Similarly we can write, for a conservation law corresponding to the $$U(1)$$ symmetry: $$\partial_t n + \nabla_j (n_j) = 0$$ ; where, $$n$$ is the "charge" density, and $$n_j$$ is $$j$$th component of the charge current.

We can interpret $$\frac{n_j}{n} = v_j$$ , where $$v$$ is the "fluid velocity". This is the same $$v$$ as appears in the momentum conservation equation, and is the velocity with which the fluid advects conserved quantities.

If we have an isotropic fluid, we know that $$G_i = \rho v_i$$, (basically, because the only object that transforms under rotations like we expect $$G$$ to, is $$v$$, hence the two must be proportional) where we can take $$\rho$$ to be some arbitrary function (scalar under rotation) of the hydrodynamic variables ; $$\rho = \rho(n,|v|...)$$ .

Now comes the part I do not understand : Apparently, the requirement that the above equations must be Galilean invariant (ie, must retain their form under Galilean Transforms) forces us to choose $$\rho(n,|v|...) = \alpha n$$, where $$\alpha$$ is some constant. That is, not only is the direction of $$G$$ fixed by the direction of $$v$$, but also, the magnitude of $$G$$ now scales linearly with the magnitude of $$v$$.

My question is, what goes wrong if we do not assume this? (I can see how this works when we start off with a "fluid" composed of point particles of some "mass", and then invoke the mechanical definition of momentum of a single particle to work out the momentum density of the fluid. What I am looking for is a proof using only the fact of Galilean invariance).

Addendum : the exchange of comments below made me think where I have seen any derivation of conservation of mass in the non-relativistic limit. As far as I can see, I can think of two places:

$$(1)$$ : $$\S 133$$, Fluid Mechanics, Landau (Relativistic Fluid mechanics). Towards the end of this section, Landau makes the following statement : "The non-relativistic case is that of small velocities...of the internal(microscopic) motion of fluid particles. In passing to this limit it must be borne in mind that relativistic internal energy $$e$$ includes the rest energy $$nmc^2$$ of the fluid particles ($$m$$ being the rest mass of $$1$$ particle) "

$$(2)$$ Weinberg (I), page 62, where he is discussing how to obtain Galilean Algebra as low velocity limit of Poincare Algebra : "..For a system of typical particle mass $$m$$, velocity $$v$$, the momentum operator ... is expected to be $$P \approx mv$$..."

I understand $$(1)$$ a lot better than $$(2)$$, but what is clear is that in both cases, we have an additional input, that is, a particle of mass $$m$$.

So, I reiterate my question:

What does this construction of a single particle of mass $$m$$ have to do with Galilean invariance? (And actually, what IS Galilean invariant about say, Navier Stokes equation ? )

• Footnote at end of section 49, fluid mechanics, Landau Commented Mar 25, 2020 at 11:11
• Are you getting this derivation from any particular text/paper? You reference section 49 of Landau & Lifschitz, but I cannot find this derivation in there. Commented Mar 27, 2020 at 5:31
• @talrefae Sorry, this was just meant to be a note to self ; as a reminder to think about implications of what is being said there . The footnote says "It must be borne in mind that, whatever the definitions used, the mass flux density j must always be the momentum of a unit volume of fluid ". This is exactly what is assumed for Galilean field theories. Landau seems to take it as a definition. So, yes, you are right, there is no proof there. Commented Mar 27, 2020 at 6:04
• Sure. I am still curious to know where you found the derivation you listed in your main question, as I find this topic pretty interesting. Commented Mar 27, 2020 at 6:07
• @talrefae Also, $\S 139$ (The equations of superfluid dynamics) also has a discussion of the implications of Galilean relativity. Don't understand it though. Too hung up on the details of super-fluidity (that I do not understand) Commented Mar 27, 2020 at 6:10