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)

Why I can’t open the file.

ReplyDeleteSorry, the link was broken. Should be OK now

DeleteIn the question, the partial derivative is with respect to "nu", not "mu". I'm very new to this tensor math, but I figure it might be significant to the problem.

ReplyDeleteNo it's not. And if it was, the indices in the equation would be unbalanced.

DeleteThe Q also has the wrong superscript which should be "nu" which balances the indices. You did it correctly in your pdf.

Delete