Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

I Energy stress tensor of dust

  1. Sep 6, 2016 #1
    If a large cloud of dust of constant ρ is moving with a given ##\vec v ## in some frame, then at any given time and position inside the cloud there should not be no net energy or i-momentum flow on any surface of constant ##x^i ## (i=1,2,3) because the particles coming in cancels those going out off the opposite side of the volume element. So all the spatial ##T^{ij}## components of the energy-stress tensor should be zero. Is my reasoning correct?
     
  2. jcsd
  3. Sep 6, 2016 #2

    haushofer

    User Avatar
    Science Advisor

    From the definition of the stressfensor of a dust it is clear that those components vanish for comoving observers. So i'd say your condition is only satisfied by those observers.
     
  4. Sep 6, 2016 #3

    pervect

    User Avatar
    Staff Emeritus
    Science Advisor

    The stress-energy density of the dust in it's rest frame is
    $$T_{ij} = \begin {bmatrix} \rho & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \end {bmatrix} $$

    To change frames to one where the dust is moving, we perform a Lorentz boost on ##T_{ij}##

    $$T'_{uv} = T_{ij} \, \Lambda^i{}_u \Lambda^j{}_v$$

    Letting ##\beta = ||v||/c## and ##\gamma = 1/\sqrt{1-\beta^2}## for a Lorentz boost in the ##x^1## direction we can write:

    $$\Lambda = \begin {bmatrix} \gamma & -\beta \gamma & 0 & 0 \\ -\beta \gamma & \gamma & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \end {bmatrix} $$

    So we get that the only nonzero components are

    $$T'_{00} = \Lambda^0{}_0 \, \Lambda^0{}_0 \,T_{00} = \gamma^2 \, \rho \quad T'_{01} = T'_{10} = \Lambda^0{}_0 \, \Lambda^0{}_1 \, T_{00} = -\beta \gamma^2 \rho \quad T'_{11} = \Lambda^0{}_1 \, \Lambda^0{}_1 \, T_{00} = \beta^2 \gamma^2 \rho $$

    i.e.

    $$T'_{ij} = \begin {bmatrix} \gamma^2 \rho & -\beta \gamma^2 \rho & 0 & 0 \\ -\beta \gamma^2 \rho & \beta^2 \gamma^2 \rho & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \end {bmatrix} $$

    So no, all the spatial components of ##T'_{ij}## are not zero, in particular ##T'_{11}## which is what I think you mean by "spatial component"(?) is nonzero in the example where v points in the ##x^1## direction.
     
  5. Sep 6, 2016 #4
    Okay. Thank you very much. I think that my interpretation of the definition of the stress energy tensor was wrong. I assumed that the flow of α-momentum across a surface of constant ##x^β## meant the net flow of the α-momentum in an element of volume in the β direction. That was an unwarranted assumption, so the reasoning in my original post is invalid.
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Have something to add?
Draft saved Draft deleted



Similar Discussions: Energy stress tensor of dust
  1. Stress-Energy Tensor (Replies: 5)

Loading...