Four-Vector For Number Flux of Particles

Question

Imagine a four-velocity $U^mu(x)$ defined all over spacetime. Define the number-flux four-vector as
$$N^mu = nU^mu$$
where $n$ is the number density of a collection of particles in their rest frame. So in this rest frame, the number-flux four-vector reduces to $(n, 0, 0, 0)$. My question is how this vector transforms under a general Lorentz boost with velocity $vec{v}$. Inspecting how other four-vectors, such as the four-velocity and four-momentum transform, one would guess that the number-flux four vector transforms as $(gamma n, gamma n v^1, gamma n v^2, gamma n v^3),$
where $v^1 = frac{dx^1}{dt}$ etc. Is this correct? If so, what is the intuition behind how the number density of the particles, $n$, transforms to $gamma n$?
A derivation of the correct Lorentz boosted number-flux four-vector would be very much appreciated, as this is a topic I still find quite confusing.

Frobenius · Answer

I think that the question has its exact analog in Classical Electrodynamics. Consider that in an inertial system $,S,$  we have a system of particles with
begin{align}
nleft(mathbf x,tright) & boldsymbol{=}texttt{particle number volume density at } mathbf x,t   
tag{01a}label{01a}
&text{moving with 3-velocity}
nonumber
mathbf uleft(mathbf x,tright) & boldsymbol{=}texttt{velocity of particles at } mathbf x,t 
tag{01b}label{01b}  
end{align}
Suppose now that we $''$charge$''$ each particle with the unit quantum charge $,texttt{e},$ of the electron. Then we have the following quantities of Classical Electrodynamics (omitting for convenience the space-time dependence)
begin{align}
varrho & boldsymbol{=}texttt{e}, nboldsymbol{=}texttt{electric charge volume density}  
tag{02a}label{02a}
boldsymbol{j} & boldsymbol{=}varrho,mathbf uboldsymbol{=}texttt{e}, n,mathbf uboldsymbol{=}texttt{electric charge current density}
tag{02b}label{02b}
end{align}
If particles  are  neither created nor destroyed then in the electrodynamics analog the electric charge obeys the easily proved conservation law
begin{equation}
dfrac{partial varrho}{partial t}boldsymbol{+}boldsymbol{nablacdot}boldsymbol{j} boldsymbol{=}0
tag{03}label{03} 
end{equation}
In electrodynamics we define the 4-dimensional electric charge current density
$,mathbf J,$
begin{equation}
mathbf J stackrel{texttt{def}}{boldsymbol{=!=}}left(varrho,c,boldsymbol{j}right)boldsymbol{=}varrholeft(c,mathbf uright)
tag{04}label{04}  
end{equation}
so that the conservation law is expressed in terms of the 4-divergence
begin{equation}
boldsymbol{square!!!!square,cdot} mathbf J boldsymbol{=}partial_{mu}mathrm J^{mu}boldsymbol{=}partial^{mu}mathrm J_{mu}boldsymbol{=}0
tag{05}label{05} 
end{equation}
By analogy we define the 4-dimensional particle flux  $,mathbf N,$
begin{equation}
mathbf N boldsymbol{=}dfrac{mathbf J }{texttt{e}}boldsymbol{=}dfrac{left(varrho,c,boldsymbol{j}right)}{texttt{e}}boldsymbol{=}nleft(c,mathbf uright)boldsymbol{=}left(n,c,boldsymbol{f}right)
tag{06}label{06} 
end{equation}
where
begin{equation}
boldsymbol{f}boldsymbol{=}n,mathbf uboldsymbol{=}texttt{the 3-dimensional particle flux}
tag{07}label{07} 
end{equation}
The  particle flux obeys the particle number conservation law corresponding to equation eqref{03}
begin{equation}
dfrac{partial n}{partial t}boldsymbol{+}boldsymbol{nablacdot}boldsymbol{f} boldsymbol{=}0
tag{08}label{08} 
end{equation}
or its 4-dimensional version corresponding to equation eqref{05}
begin{equation}
boldsymbol{square!!!!square,cdot} mathbf N boldsymbol{=}partial_{mu}mathrm N^{mu}boldsymbol{=}partial^{mu}mathrm N_{mu}boldsymbol{=}0
tag{09}label{09} 
end{equation}
Now, in electrodynamics it has been proved that the 4-dimensional electric charge current density $,mathbf J,$ of equation eqref{04} is a Lorentz 4-vector expressed also as the Lorentz 4-velocity $mathbf Uboldsymbol{=}gamma_{rm u}left(c,mathbf uright)$ times the invariant scalar rest charge density $varrho_{0}boldsymbol{=}varrho/gamma_{rm u}$
begin{equation}
mathbf J boldsymbol{=}varrho_{0}mathbf U
tag{10}label{10} 
end{equation}
By analogy the 4-dimensional particle flux  $,mathbf N,$ is a Lorentz 4-vector expressed also as the Lorentz 4-velocity $mathbf Uboldsymbol{=}gamma_{rm u}left(c,mathbf uright)$ times the invariant scalar rest particle number $n_{0}boldsymbol{=}n/gamma_{rm u}$
begin{equation}
mathbf N boldsymbol{=}n_{0}mathbf U
tag{11}label{11} 
end{equation}
So the proof that $,mathbf N,$ is a Lorentz 4-vector is essentially the proof that the 4-dimensional electric charge current density $,mathbf J,$ of equation eqref{04} is a Lorentz 4-vector.

Proofs that $,mathbf J,$  is  transformed as a Lorentz 4-vector based upon the conservation law  eqref{03} are false. That electric charge is constant in an inertial system doesn't provide any information about how it is transformed between inertial systems. It's a confusion between what is a constant (it concerns what happens in a system) and what is an invariant (it concerns what happens between two systems).

For an elegant  proof by L.D.Landau and E.M.Lifshitz  see my $color{blue}{textbf{ANSWER A}}$ here How do we prove that the 4-current jμ transforms like xμ under Lorentz transformation?. The proof is based on one hand upon the fact that the charge on a particle is, from its very definition, an invariant quantity and on the other hand upon the fact that the 4-dimensional infinitesimal $''$volume$''$ $mathrm dV=mathrm dx^0mathrm dx^1mathrm dx^2mathrm dx^3$ is a Lorentz invariant scalar. A proof of the latter is given in footnote (1) of the aforementioned answer.

Four-Vector For Number Flux of Particles

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