Friday, 9 April 2021

Exercise 1.9 Energy momentum tensor of point particles

Question

Point particles in proton

For a system of discrete point particles the energy-momentum tensor takes the form$$
T_{\mu\nu}=\sum_{a}{\frac{p_\mu^{\left(a\right)}p_\nu^{\left(a\right)}}{p^{0\left(a\right)}}\delta^{\left(3\right)}\left(\mathbf{x}-\mathbf{x}^{\left(a\right)}\right)}
$$where the index ##a## labels the different particles. Show that, for a dense collection of particles with isotropically distributed velocities, we can smooth over the individual particle worldlines to obtain the perfect-fluid energy-momentum tensor 1.114. 

1.114 was$$
\mathbf{T}^{\mu\nu}=\left(\rho+p\right)U^\mu U^\nu+p\eta^{\mu\nu}
$$where ##\rho## is the energy density of the fluid measured in its rest frame (rest-frame energy density) and ##p## is the isotropic rest-frame pressure. ##T^{\mu\nu}## is the energy-momentum tensor of a fluid element with a four velocity ##U^\mu##. 

Answer

Carroll is setting plenty of traps here. 
  • The ##p##'s in second equation are rest-frame pressure and the ##p##'s in the first are 4-momentum components.
  • What does ##\delta^{\left(3\right)}\left(\mathbf{x}-\mathbf{x}^{\left(a\right)}\right)## mean? Carroll gives no hint.
  • How do you smooth over the individual particle worldlines?
  • Having indices down on the first equation is just perverse. Raise them immediately! 
  • There is a conceptual problem. The first equation concerns a point particle and the second concerns a particle of fluid which contains lots of point particles. I try to call the fluid particle a fluid element. It's a bit weird. We are doing summations over the point particles and we will be integrating and differentiating over fluid elements which are bigger but nevertheless still small enough to be treated as points in our calculus.
First I found out what ##\delta^{\left(3\right)}\left(\mathbf{x}-\mathbf{x}^{\left(a\right)}\right)## is and justified the expression for the energy-momentum of the point particle. I then got stuck until I found a solution by Alexey Bobrick on physics.stackexchange. It starts well but ends not so well, I believe. I think I did a little better.

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