Exploring the Stress-Energy Tensor of a Perfect Fluid

[pervect]

I believe this thread is sufficiently different from one that was recently closed to not violate any guidelines, though there are unfortunately some similarities as the closed thread sparked the questions in my mind.

If we look at the stress energy tensor of a perfect fluid in geometric units with the minus time sign convention, we find:

$$T^{ab} = (\rho + P) \, u^a \times u^b + P \, \eta^{ab}$$

where $u^a$ is the four-velocity of the frame, $\rho$ and P can be described as the density and pressure in the rest frame of the fluid. and $\eta^{ab}$ is the (inverse) metric tensor.

$$\eta^{ab} = \begin{bmatrix} -1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{bmatrix}$$In the rest frame of the fluid, $u^a$ has components [1,0,0,0] and we find the stress-energy tensor:

$$T^{ab} = \begin{bmatrix}\rho & 0 & 0 & 0 \\ 0 & P & 0 & 0 \\ 0 & 0 & P & 0 \\ 0 & 0 & 0 & P \end{bmatrix}$$
If we consider a moving fluid, with 4-velocity $u^a$ has components $[\gamma, \beta \gamma, 0, 0]$ where $\beta=v/c$ and $\gamma = 1/\sqrt{1-\beta^2}$, I get:
$$(T')^{ab} = \left[ \begin{array}{rrr} \gamma^2 (\rho+P) - P & \beta \gamma^2 (\rho+P) & 0 & 0 \\ \beta \gamma^2 (\rho+P) & \beta^2 \gamma^2 (\rho+P) + P & 0 & 0 \\ 0 & 0 & P & 0 \\ 0 & 0 & 0 & P \end{array} \right] $$

We can extract various components from this stress energy tensor. I want to focus on the component which represents the pressure in the direction of the motion of the fluid, which I'll call $P'_x$. This is:

$P'_x = \beta^2 \gamma^2 \rho + (\beta^2\gamma^2+1)P$

The first question is, how do we best explain the term $\beta^2 \gamma^2 \rho$ in the pressure? Is calling it the "dynamic pressure" good enough?

The second, related question regards the engineering vs physics definition of the stress energy tensor. When Wiki talks about the stress-energy tensor, they make the following notes:

In solid state physics and fluid mechanics, the stress tensor is defined to be the spatial components of the stress–energy tensor in the proper frame of reference. In other words, the stress energy tensor in engineering differs from the stress–energy tensor here by a momentum convective term.

I used the physics definition of the stress-energy tensor in my analysis, I'm not sure what answer the engineering definition would give. If I'm reading the wiki right, the engineering definition is only defined in a frame co-moving with the fluid, and is simply not define it in any other frame?

 

[Ibix]

I have a copy of Tolman's Relativity, Thermodynamics and Cosmology[1]. In it, Tolman says that pressure does not change under Lorentz transformation, because $p=F/A$ (eqn 69.3, p154) and the transformation of the force happens to cancel with the transformation of the area. He notes that this also follows from his definition of the transformation of stress:$$\begin{array}{rclrclrcl}
t'_{xx}&=&t_{xx}&t'_{xy}&=&\gamma t_{xy}&t'_{xz}&=&\gamma t_{xz}\\
t'_{yx}&=&t_{yx}/\gamma&t'_{yy}&=&t_{yy}&t'_{yz}&=& t_{yz}\\
t'_{zx}&=&t_{zx}/\gamma&t'_{zy}&=&t_{zy}&t'_{yz}&=& t_{zz}
\end{array}$$...of which he says "[this] definition in terms of force per unit area as measured in either system of coordinates is chosen to agree with the usage of Laue, Das Relativitatsprinzip, second edition, Braunschweig, 1913".

He then defines a new quantity $p_{ij}=t_{ij}+g_iu_j$ (i=1,2,3, j=1,2,3) where $g_i$ is a momentum density and $u_j$ is the velocity (of the material, I suppose?) at the point. Then he uses that as the space-space components of the energy momentum tensor (his name - it looks like the stress-energy tensor to me).

So, if I understand Tolman correctly, the thing you've called $P'_x$ is the pressure plus a momentum flux term, and the engineering definition is minus the momentum flux.

It's more than possible that I'm missing one or more points here.

[1]Side note - my copy has the cover upside down. I haven't yet read it in public, but intend to do so with the cover ostentatiously visible at some point.

 

[pervect]

I have a copy of Tolman's Relativity, Thermodynamics and Cosmology[1]. In it, Tolman says that pressure does not change under Lorentz transformation, because $p=F/A$ (eqn 69.3, p154) and the transformation of the force happens to cancel with the transformation of the area. He notes that this also follows from his definition of the transformation of stress:$$\begin{array}{rclrclrcl}
t'_{xx}&=&t_{xx}&t'_{xy}&=&\gamma t_{xy}&t'_{xz}&=&\gamma t_{xz}\\
t'_{yx}&=&t_{yx}/\gamma&t'_{yy}&=&t_{yy}&t'_{yz}&=& t_{yz}\\
t'_{zx}&=&t_{zx}/\gamma&t'_{zy}&=&t_{zy}&t'_{yz}&=& t_{zz}
\end{array}$$...of which he says "[this] definition in terms of force per unit area as measured in either system of coordinates is chosen to agree with the usage of Laue, Das Relativitatsprinzip, second edition, Braunschweig, 1913".

He then defines a new quantity $p_{ij}=t_{ij}+g_iu_j$ (i=1,2,3, j=1,2,3) where $g_i$ is a momentum density and $u_j$ is the velocity (of the material, I suppose?) at the point. Then he uses that as the space-space components of the energy momentum tensor (his name - it looks like the stress-energy tensor to me).

So, if I understand Tolman correctly, the thing you've called $P'_x$ is the pressure plus a momentum flux term, and the engineering definition is minus the momentum flux.

It's more than possible that I'm missing one or more points here.

[1]Side note - my copy has the cover upside down. I haven't yet read it in public, but intend to do so with the cover ostentatiously visible at some point.

Thanks - I have a copy of that book tucked away - somwhere. If I can find it, I'll definitely look up the section in question.

 

[Ibix]

Thanks

You're welcome.

I didn't reference terribly well above. The derivation of the transformation of $p$ is where I said. The $t'_{ij}$ are defined in equation 34.5, p64, and $p_{ij}$ is defined in equation 36.1, p69, leading to $T^{\alpha\beta}$ in equation 37.3, p71.