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)

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