Is the Energy Stress Tensor of Dust Always Zero Inside a Moving Cloud?

[epovo]

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?

 

[haushofer]

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.

 

[pervect]

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.

 

[epovo]

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.