I Why is Number-Flux Four-Vector Frame-Independent?

[GR191511]

Recently I started studying 《A First Course in General Relativity》 and I came across a question in my book:
$\vec N =n\vec U$where n is number density,U is four-velocity,N is number-flux four-vector .The following sentence confused me:
In Galilean physics,number density was a scalar,the same in all frames(no Lorentz contraction),while flux was quite another thing:a three-vector that was frame dependent,since the velocities of particles are a frame-dependent notion.Our relativistic approach has unified these two notions into a single,frame-independent four-vector...
I wonder Why the number-flux four-vector is frame-independent？

 

[malawi_glenn]

All four-vectos are frame independent

You should read page 88 (in the second edition) where it is explained what 'frame independent' means in this case

 

[Ibix]

I wonder Why the number-flux four-vector is frame-independent？

Because it's a four vector. They're frame independent by construction.

I think what you need to do is convince yourself that $nU^a$ actually represents something meaningful, and that it represents something meaningful whether $U^a$ is parallel to your frame's timelike axis or not. The usual way to go about it is to imagine a very large number of particles each with velocity $U^a$. Then convince yourself that the number of particles crossing an infinitesimal plane depends on the number density $n$ and the velocity field $U^a$, then convince yourself that the transformation properties of $U^a$ supply the right factors of $\gamma$ so that the transformed flux density behaves as you expect.

 

[Orodruin]

The components of the 4-velocity are still frame dependent even if the 4-velocity itself is an invariant geometric object. This means that also the number density itself is frame dependent in relativity ($n$ is the number density in the rest frame and is a scalar) just as the flux, which is number density times velocity just as in Galilean spacetime.

 

[vanhees71]

Yes, and that's why one defines (!) such medium-related "intrinsic" quantities in the local rest frame(s) of the fluid and thus becoming a scalar. That holds also for thermodynamic quantities like temperature or chemical potential.