I was asked in a recent homework to "show" that the for the stress-energy tensor $$T^{\mu\nu}=[(\rho +p)]U^{\mu}U^{\nu}+P\eta^{\mu\nu}$$
For a perfect fluid that the conservation law holds, that is $T^{\mu\nu},_{\nu}=0$ (all in Minkowski space-time). I was quite surprised and confused about this question, as it seemed that this, while not strictly postulated, came from the field equations as the Einstein tensor does banish. And while I understand some physical arguments in term of local conservation of energy and momentum can be given, I was told that I could obtain the law by tensor algebra. What some peers did was to consider the four-velocity on MCRF such that $U^{T}=(c^2,0,0,0)$ and that here the derivative must banish while also saying that the density and pressure must be constant for it to be a perfect fluid.
I think this is incorrect as the MCRF is only valid at a particular point, because if not that would mean you could find a global frame in which the velocity field is always 0 for the 3-velocity component, but I can't seem to find any viable construction for such a frame. Also, $\frac{\partial U^{\beta}}{\partial {x^{\alpha}}}$ must require to consider changes in a neighborhood of that point, so while at the that particular point the four velocity is 0, I don't think you can conclude the derivative must banish. Also, I haven't found a definition of a perfect fluid that imposes such strong hypothesis for the pressure and density, and I've seen examples were they aren't constant.
I suspect the question is ill-posed, or either just a trivial consequence of it being a stress energy tensor (we saw in class the Einstein tensor banishes so we can use this fact). So I would like to ask if it makes sense as it is formulated?