## Monday, 27 January 2020

### Exercise 1.08 un-conserved dust energy momentum tensor

 Very bad dust by George E. Marsh
Question

If the energy momentum tensor $\partial_\nu T^{\mu\nu}=Q^\mu$ what physically does the spatial vector $Q^i$ represent? Use the dust energy momentum tensor to make your case.

Answer
We had at 1.110 that the dust energy momentum tensor was$$T_{dust}^{\mu\nu}=p^\mu N^\nu=mnU^\mu U^\nu=\rho U^\mu U^\nu$$$p_i$ or $p^i$ is the pressure (not momentum) in the $x^i$ direction, that is force per unit area. I'm not sure if the index should be up or down.

$n$ is the number density as measured in the dust's rest frame, $n=$ particles of dust per unit volume in rest frame.

$N^\nu=nU^\nu$ is the number-flux four-vector,  $N^0$ is the number density of particles in any frame, $N^i$ is the flux of particles in the $x^i$ direction. So if there is no flux, that's the rest frame, $N^\mu=\left(n,0,0,0\right)$

$m$ is the mass of each dust particle (in the dust's rest frame)) which we assume to be the same.

Moreover the dust particles are all moving with the same four-velocity $U^\mu$ - I think.

$\partial_\mu T^{\mu\nu}=0$ was the conservation equation for $T^{\mu\nu}$ so if $\partial_\mu T^{\mu\nu}=Q^\mu$ then clearly $T^{\mu\nu}$ is not being conserved and it is $Q^\mu$ that is disturbing the equilibrium. That answer is correct but rather feeble. I did a bit better with help from Valter Moretti on physics.stackexchange and learnt about the theorem of divergence and that $T^{i0}=T^{0i}$ components of this energy momentum tensor are roughly momentum.
More at Ex 1.08 Dust Energy  Momentum tensor.pdf (2 pages and a bit)