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I think you are missing my point and no doubt the fault is mine due to my informal language. When I said "the pressure used in the stress-energy tensor" I was referring to the directional components of p' rather than the isotropic (proper) pressure represented by the symbol p. By analogy if I said length is not a Lorentz invariant in SR, I am referring to L' rather than the proper length ##L_0## in the length contraction formula ##L' = L_0/\gamma##.What you said regarding the pressure in the stress-energy tensor is incorrect. ... This means that the pressure is measured in a local Lorentz frame in which the fluid 3-velocity field ##\vec{v}## vanishes: ##\vec{v} =0##. In other words, by definition of ##p##, the pressure gauge that measures ##p## is at rest with respect to the fluid element.

Clearly the quantity ##p'_x## you describe here is not a Lorentz invariant and is not isotropic and this is the quantity I meant when I said "the pressure" which was a bit sloppy of me. It is this non isotropic transformed pressure that can be thought of as the total pressure that includes terms related to static pressureThen ##p'_x = T^{x'x'} = \Lambda^{x'}{}{}_{\mu}\Lambda^{x'}{}{}_{\nu}T^{\mu\nu} = \rho\Lambda^{x'}{}{}_{0}\Lambda^{x'}{}{}_{0}+ p\Lambda^{x'}{}{}_{x}\Lambda^{x'}{}{}_{x} = \gamma^2 \beta^2 \rho + \gamma^2 p## ... whereas ##p'_y = p'_z = p##.

*and*dynamic pressure. The isotropic pressure (p) measured by a comoving observer can be thought of as the static pressure. When v is zero, the dynamic pressure term of the covariant total pressure (p') vanishes and you are left with only the static isotropic pressure (p). While static pressure is defined as the measurement made by a comoving observer, it can be calculated or even physically measured by a non comoving observer. The air speed indicator on an aircraft measures the static pressure of the air (even though the instrument is moving relative to the air) and compares this to the total pressure which includes static and dynamic pressure. The difference is the dynamic pressure which is a measure of the air speed.

Here Peter is using the phrase "the pressure" in the same way that I used the same phrase, to mean the covariant transformed pressure, (that includes terms related to dynamic pressure).... but if you transform to a different local inertial frame, the tensor components transform too, sothe pressure won't be the same in the new frame.

When I said "the pressure" in the s-e tensor is not a Lorentz invariant, I was trying to suggest that the components do not transform in an invariant way under a Lorentz boost as you describe here. Your terminology is confusing to me here. I thought the definition of a "scalar" quantity was a quantity that does not have directional components?... For example above I noted that the (isotropic) pressure of a perfect fluid isdefinedin the rest frame of the fluid. Now as noted the scalar pressure comprises three components of the stress-energy tensor in the fluid rest frame and as such transforms under a Lorentz boost in a non-trivial way ...

I am not sure which of my two statements you are saying saying is incorrect. (probably both :). My statement that "static pressure as used in the gas laws is isotropic, scalar and Lorentz invariant" is very similar to your statement that the scalar field ##p## is scalar, isotropic and and invariant by definition. I guess the subtle difference is the difference between the meanings of a proper quantity and Lorentz invariant quantity. Perhaps you could clarify.This is also incorrect I'm afraid, at least when applied to a perfect fluid. See my post above where I show that the scalar field ##p## that appears in ##T_{\mu\nu}## for a perfect fluid is related to the magnitude of the force ##dF## on a space-like infinitesimal area element ##dA## as measured by a comoving observer by ##p = \frac{d\left \| F \right \|}{dA}##. Hence the scalar field ##p## is an isotropic pressure field. It is invariant by definition since it's always measured in the rest frame of the fluid.It would seem that static pressure as used in the gas laws is isotropic, scalar and Lorentz invariant. The total pressure used in the stress energy tensor is none of those things.

Perhaps your objection is to my statement "The total pressure used in the stress energy tensor is none of those things", in which case, hopefully I have explained what I meant by that statement in the above replies. I have no formal training in the language used here, so hopefully you will be generous and skilled enough to interpret my intended meaning from the context.

Finally and most importantly in the context of the OP, given:

##P' = f\frac{nRT_0}{(V_0/\gamma)}## and ##P' =g P_0##

what are the functions f and g?

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