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?
Best Answer
It makes sense to prove this using incopressibility and the equations of motion for the fluid. Start with:
$\nabla _{\mu } T^{\mu\nu}=(\rho +p)\nabla _{\mu }(U^{\mu}U^{\nu})+p\nabla _{\mu }g^{\mu\nu} $
The second term vanishes for the Levi-Civita connection (metric compatibility). The first term is :
$\nabla _{\mu }(U^{\mu}U^{\nu})=U^{\nu}\nabla_{\mu}U^{\mu} +U^{\mu}\nabla_{\mu}U^{\nu}$
If the fluid is incompressible: $\nabla_{\mu}U^{\mu}=0$
If the fluid flows on geodesics: $U^{\mu}\nabla_{\mu}U^{\nu}=(\nabla _{U}U)^{\nu}=0$
So those 2 conditions imply conservation of energy-momenum.
Edit: As pointed out by OP, this treatment is partial and assumes both incompressibility and constant pressure. In the full treatment, one take the equation $\nabla _{\mu } T^{\mu\nu}=0$ , substitutes T for a perfect fluid as above, and projects the equation on different directions to get the 2 equations:
$U^\mu\partial _\mu \rho =-(\rho +p)\nabla_{\mu}U^{\mu} $
$(\rho +p)U^{\mu}\nabla_{\mu}U_{\nu}=-\nabla_{\nu}p-U_{\nu }U^{\mu }\nabla_{\mu}p $
The first equation is a conservation equation $\rho$, and the second is a conservation equation for the momentum.
Reference: Relativistic Fluid Dynamics notes by Jason Olsthoorn Sections 3.1-3.3 http://mathreview.uwaterloo.ca/archive/voli/2/olsthoorn.pdf