Stress energy tensor of perfect fluid in general frame

Question

Good day to everyone. I have a problem with the stress energy tensor of a perfect fluid. In the frame of reference of the fluid the stress energy tensor is

$T^{\mu \nu} = \left( \begin{array}{cccc} \rho & 0 & 0 & 0 \\ 0 & P & 0 & 0 \\ 0 & 0 & P & 0 \\ 0 & 0 & 0 & P \end{array}\right)$

Instead, in a general inertial frame the stress energy tensor is

$T^{\mu \nu} = (\rho+P)u^{\mu}u^{\nu} + \eta^{\mu\nu}$

(I am working in the convention such that $\eta^{00}=-1$ and $\eta^{11}=\eta^{22}=\eta^{33}=1$, and $u^{\mu}=(\gamma, \gamma\vec{\beta})$). This general form of the stress energy tensor should be verified, for example, applying a Lorentz boost along the x direction: $\Lambda^{\mu}_{\nu}=\left( \begin{array}{cccc} \gamma & -\beta\gamma & 0 & 0 \\ -\beta\gamma & \gamma & 0 & 0 \\ 0 & 0 & \gamma & 0 \\ 0 & 0 & 0 & \gamma \end{array}\right)$

So

$T'^{\mu\nu}=\Lambda^{\mu}_{\rho}\Lambda^{\nu}_{\sigma}T^{\rho\sigma} \to T'=\Lambda T \Lambda^T$

$T'^{\mu\nu}=\left( \begin{array}{cccc} \gamma & -\beta\gamma & 0 & 0 \\ -\beta\gamma & \gamma & 0 & 0 \\ 0 & 0 & \gamma & 0 \\ 0 & 0 & 0 & \gamma \end{array}\right) \left( \begin{array}{cccc} \rho & 0 & 0 & 0 \\ 0 & P & 0 & 0 \\ 0 & 0 & P & 0 \\ 0 & 0 & 0 & P \end{array}\right) \left( \begin{array}{cccc} \gamma & -\beta\gamma & 0 & 0 \\ -\beta\gamma & \gamma & 0 & 0 \\ 0 & 0 & \gamma & 0 \\ 0 & 0 & 0 & \gamma \end{array}\right)=$

$=\left( \begin{array}{cccc} \gamma^2\rho+\beta^2\gamma^2P & -\beta\gamma^2\rho-\beta\gamma^2P & 0 & 0 \\ -\beta\gamma^2\rho-\beta\gamma^2P & \beta^2\gamma^2\rho+P\gamma^2& 0 & 0 \\ 0 & 0 & \gamma & 0 \\ 0 & 0 & 0 & \gamma \end{array}\right)$

For the components $T'^{00}$ and $T'^{11}$ I have effectively obtained

$T'^{00} = (\rho+P)\gamma^2 - P$

and

$T'^{11} = (\rho+P)\gamma^2\beta^2 + P$

But my result for the component $T'^{01}$ is

$T'^{01} = -(\rho+P)\gamma\cdot\gamma\beta $

So I have obtained the opposite sign respect to the form $T'^{01}=(\rho+P)u^0u^1.$

Where am I wrong ? If someone makes me understand where I have made any mistakes I will be infinitely grateful.

Answer 1

All of your calculations are correct, however, you simply used the incorrect sign for $\beta \gamma$ in your Lorentz transformation. In other words, your Lorentz transformation should be

$$ \Lambda^\mu_\nu= \left( \begin{array}{cccc} \gamma & \beta\gamma & 0 & 0 \\ \beta\gamma & \gamma & 0 & 0 \\ 0 & 0 & \gamma & 0 \\ 0 & 0 & 0 & \gamma \end{array}\right)$$

Now, the physical interpretation of this is that, originally, we were in the "moving frame" of our perfect fluid which gives us the familiar diagonal stress energy tensor. However, if we want to go into a frame where the four velocity of the perfect fluid is $u^{\mu}=(\gamma, \gamma\vec{\beta})$, then we have to boost to a frame moving at a three velocity of $-\vec{\beta} $ relative to our initial frame. With this change, the calculation works out as expected.