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.(adsbygoogle = window.adsbygoogle || []).push({});

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 defintion of the stress energy tensor. When Wiki talks about the stress-energy tensor, they make the following notes:

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? wiki said:In solid state physics and fluid mechanics, thestress tensoris defined to be the spatial components of thestress–energy tensorin the proper frame of reference. In otherwords, thestress energy tensorin engineering differs from thestress–energy tensorhere by a momentumconvective term.

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# A Pressure redux

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